Volume 4 Number 14 EJTPe-mail: [email protected] [email protected] Leonardo Chiatti Medical...
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Volume 4 Number 14
EJTPElectronic Journal of Theoretical Physics
ISSN 1729-5254
http://www.ejtp.com March, 2007 Email:[email protected]
Volume 4 Number 14
Electronic Journal of Theoretical Physics
EJTP
http://www.ejtp.com March, 2007 Email:[email protected]
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Copyright © 2003-2007 Electronic Journal of Theoretical Physics (EJTP) All rights reserved
Table of Contents
No Articles Page
1 On the Dynamics of a n-D Piecewise Linear Map Zeraoulia Elhadj
1
2 Flow of Unsteady Dusty Fluid under Varying Pulsatile Pressure Gradient in Anholonomic Co-ordinate System J.Gireesha, C.S.Bagewadi and B.C.Prasanna Kumara
9
3 Exact Solutions for Nonlinear Evolution Equations via Extended Projective Riccati Equation Expansion Methods M A Abdou
17
4 Evolutionary Neural Gas: A Model of Self Organizing Network from Input Categorization Luigi Lella and Ignazio Licata
31
5 Discrete Groups Approach to Non Symmetric Gravitation TheoryN.Mebarki, F.Khelili and J.Mimouni
51
6 Quantization of the Scalar Field Coupled Minimally to the Vector PotentialW. I. Eshraim and N. I. Farahat
61
7 A Generalized Option Pricing Model J. P. Singh
69
8 Derivation of the Radiative Transfer Equation inside a Moving Semi-Transparent Medium of Non Unit Refractive Index Le Dez and H. Sadat
87
9 Quantum Images and the Measurement Process Fariel Shafee
121
EJTP 4, No. 14 (2007) 1–8 Electronic Journal of Theoretical Physics
On the Dynamics of a n-D Piecewise Linear Map
Zeraoulia Elhadj∗
Department of Mathematics, University of Tebessa, (12000), Algeria.
Received 27 September 2006, Accepted 6 January 2007, Published 31 March 2007
Abstract: This paper, derives sufficient conditions for the existence of chaotic attractors in ageneral n-D piecewise linear discrete map, along the exact determination of its dynamics usingthe standard definition of the largest Lyapunov exponent.c© Electronic Journal of Theoretical Physics. All rights reserved.
Keywords: Chaos, Discrete Mapping, Lyapunov ExponentsPACS (2006): 05.45.a, 95.10.Fh, 05.45.Ra
1. Introduction
There are many works that focus on the topic of the rigorous mathematical proof
of chaos in a discrete mapping ( continuous or not). For example it has been studied
rigorously from a control and anti-control schemes or from the use of Lyapunouv ex-
ponents, see for example [1-2-3-4-5-6], to prove the existence of chaos in n-dimensional
dynamical discrete system, since a large number of physical and engineering systems have
been found to exhibit a class of continuous or discontinuous piecewise linear maps [12-
13] where the discrete-time state space is divided into two or more compartments with
different functional forms of the map separated by borderlines [14-15-16-17-18]. The
theory for discontinuous maps is in the preliminary stage of development, with some
progress reported for 1-D and n-D discontinuous maps in [19-20-21-22-23], these results
are restrictive, and cannot be obtained in the general n-dimensional context [23].
This paper, derives sufficient conditions for the existence of chaotic attractors in a
general n-D piecewise linear discrete map, along the exact determination of its dynamics
using the standard definition of the Lyapunov exponents as the usual test for chaos.
In the following, we present the standart definition of the Lyapunov exponents for a
discrete n-D mapping.
2 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 1–8
Theorem 1. (Lyapunouv exponent): Considered the following n-D discrete dynamical
system:
xk+1 = f(xk), xk ∈ Rn, k = 0, 1, 2, ... (1)
where f : Rn −→ R
n, is the vector field associated with system (1), let J (x) be its
Jacobian evaluated at x , let also the matrix:
Tr (x0) = J (xr−1) J (xr−2) ...J (x1) J (x0) . (2)
Moreover, let Ji(x0, l) be the module of the ith eigenvalue of the lthmatrix Tr (x0) ,where
i = 1, 2, ..., n and r = 0, 1, 2, ...
Now, the Lyapunov exponents of a n-D discrete time systems are defined by:
ωi(x0) = ln
(lim
r−→+∞Ji(x0, r)
1r
), i = 1, 2, ..., n. (3)
2. The main result
Let us consider the following n-D map of the form: f : D → D, D ⊂ Rn, defined by:
xk+1 = f (xk) = Aixk + bi, if xk ∈ Di, i = 1, 2, ..., m. (4)
where Ai =(aijl
)1≤j,l≤n
and bi =(bji)1≤j≤n
,are respectively n × n and n × 1 real
matrices, for all i = 1, 2, ..., m, and xk =(xjk
)1≤j≤n
∈ Rn is the state variable, and m is
the number of disjoint domains on which D is partitioned. Due to the shape of the vector
field f of the map (4) the plane can be divided into m regions denoted by (Di)1≤i≤m ,
and in each of these regions the map (4) is linear.
The Jacobian matrix of the map (4) is:
J (xk) =
⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩
A1, if xk ∈ D1,
A2, if xk ∈ D2,
...
Am, if xk ∈ Dm,
(5)
In the following we will compute analytically all the Lyapunov exponents of the map
(1) and we will show that these exponents are the same in each linear regions (Di)1≤i≤m de-
fined above. The essential idea of our proof is the assumption that the matrices (Ai)1≤i≤m
has the same eigenvalues, i.e. they are equivalent, then, if one compute analytically a
Lyapunov exponent of the map (4) in a region Di (which is the logarithm of the absolute
value of an eigenvalue of a matrix Ai) then, one can find that these exponents are identi-
cal in each linear region Di, for all i ∈ {1, 2, .., m}. Thus, one can consider the Jacobian
matrix J (xk) of the map (4) as any matrix Ai, denoted by A = (ajl)1≤j,l≤n.
Electronic Journal of Theoretical Physics 4, No. 14 (2007) 1–8 3
Assume that the eigenvalues of A are listed in order as follow:∣∣∣λ1
((ajl)1≤j,l≤n
)∣∣∣ ≥ ∣∣∣λ2
((ajl)1≤j,l≤n
)∣∣∣ ≥ ... ≥∣∣∣λn
((ajl)1≤j,l≤n
)∣∣∣ , (6)
where the notation λi
((ajl)1≤j,l≤n
)indicate that the eigenvalue λi depend only the co-
efficients (ajl)1≤j,l≤n , then Tr (x0) = Ar , and its eigenvalues are λr1
((ajl)1≤j,l≤n
), ..., λr
n
((ajl)1≤j,l≤n
),
then the Lyapunov exponents of the map (4) are:
ωi(x0) = ln
(lim
r−→+∞
(∣∣∣λi
((ajl)1≤j,l≤n
)∣∣∣r) 1r
)= ln
∣∣∣λi
((ajl)1≤j,l≤n
)∣∣∣ , i = 1, 2, .., n. (7)
Hence, according to (6) all the Lyapunov exponents are listed as follow:
ω1
((ajl)1≤j,l≤n
)≥ ω2
((ajl)1≤j,l≤n
)≥ ... ≥ ωn
((ajl)1≤j,l≤n
), (8)
Define the following subsets of Rn2
in term of the vector (ajl)1≤j,l≤n as follow:
Ω1 ={
(ajl)1≤j,l≤n ∈ Rn2
,∣∣∣λn
((ajl)1≤j,l≤n
)∣∣∣ > 1}
, (9)
Ω2 ={
(ajl)1≤j,l≤n ∈ Rn2
,∣∣∣λ1
((ajl)1≤j,l≤n
)∣∣∣ < 1}
, (10)
Ω3 ={
(ajl)1≤j,l≤n ∈ Rn2
,∣∣∣λ1
((ajl)1≤j,l≤n
)∣∣∣ = 1}
, (11)
Ω4 ={
(ajl)1≤j,l≤n ∈ Rn2
,∣∣∣λi
((ajl)1≤j,l≤n
)∣∣∣ < 1, i = 2, ..., n}
, (12)
Ω5 ={
(ajl)1≤j,l≤n ∈ Rn2
,∣∣∣λ2
((ajl)1≤j,l≤n
)∣∣∣ = 1}
, (13)
Ω6 ={
(ajl)1≤j,l≤n ∈ Rn2
,∣∣∣λi
((ajl)1≤j,l≤n
)∣∣∣ < 1, i = 3, ..., n}
, (14)
Ω7 ={
(ajl)1≤j,l≤n ∈ Rn2
,∣∣∣λi
((ajl)1≤j,l≤n
)∣∣∣ = 1, i = 1, 2, ..., K, where 1 ≤ K ≤ n}
,
(15)
Ω8 ={
(ajl)1≤j,l≤n ∈ Rn2
,∣∣∣λi
((ajl)1≤j,l≤n
)∣∣∣ < 1, i = K + 1, ..., n}
, (16)
Ω9 =
{(ajl)1≤j,l≤n ∈ R
n2
,∣∣∣λ1
((ajl)1≤j,l≤n
)∣∣∣ > 1, andi=n∏i=2
∣∣∣λi
((ajl)1≤j,l≤n
)∣∣∣ < 1
}, (17)
Finally, one obtain the following results:
4 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 1–8
(1) The map (4) is super chaotic when all its Lyapunov exponents are positive,
i.e. ωn
((ajl)1≤j,l≤n
)> 0, according to inequalities (6) and (8). Thus, one may obtain
(ajl)1≤j,l≤n ∈ Ω1.
(2) The map (4) converges to a stable fixed point when all the Lyapunov exponents
are negative, i.e.∣∣∣λ1
((ajl)1≤j,l≤n
)∣∣∣ < 1, according to inequalities (6) and (8). Thus, one
may obtain (ajl)1≤j,l≤n ∈ Ω2.
(3) The map (4) converges to a circle attractor when ω1 = 0, and 0 > ω2 ≥ ... ≥ ωn,
i.e.∣∣∣λ1
((ajl)1≤j,l≤n
)∣∣∣ = 1, and∣∣∣λi
((ajl)1≤j,l≤n
)∣∣∣ < 1, for i = 2, ..., n. Thus, one may
obtain (ajl)1≤j,l≤n ∈ Ω3 ∩ Ω4.
(4) The map (4) converges to a torus attractor when ω1 = ω2 = 0, and 0 > ω3 ≥... ≥ ωn, i.e.
∣∣∣λ1
((ajl)1≤j,l≤n
)∣∣∣ =∣∣∣λ2
((ajl)1≤j,l≤n
)∣∣∣ = 1, and∣∣∣λi
((ajl)1≤j,l≤n
)∣∣∣ < 1, for
i = 3, ..., n . Thus, one may obtain (ajl)1≤j,l≤n ∈ Ω3 ∩ Ω5 ∩ Ω6.
(5) The map (4) converges to a K-torus attractor when ω1 = ω2 = ... = ωK =
0, and 0 > ωK+1 ≥ ... ≥ ωn, i.e.∣∣∣λ1
((ajl)1≤j,l≤n
)∣∣∣ =∣∣∣λ2
((ajl)1≤j,l≤n
)∣∣∣ = ... =∣∣∣λK
((ajl)1≤j,l≤n
)∣∣∣ = 1, and∣∣∣λi
((ajl)1≤j,l≤n
)∣∣∣ < 1, i = K + 1, ..., n . Thus, one may
obtain (ajl)1≤j,l≤n ∈ Ω7 ∩ Ω8.
(6) The map (4) converges to a chaotic attractor when ω1 > 0, andn∑
i=2
ωi < 0, i.e.∣∣∣λ1
((ajl)1≤j,l≤n
)∣∣∣ > 1, andi=n∏i=2
∣∣∣λi
((ajl)1≤j,l≤n
)∣∣∣ < 1. Thus, one may obtain (ajl)1≤j,l≤n ∈Ω9.
Generally, for a continuous map positive Lyapunov exponent indicate chaos, nega-
tive exponent indicate fixed points, and if the Lyapunov exponent is equal to 0, then
the dynamics is periodic, while for a discontinuous map a zero Lyapunouv exponent
is not indicate periodic behavior, in this case the map generates a symbolic sequence
s = {s0; s1; ...; sj; ...} composed of symbols sj = i if xj = f j(x0) ∈ Di, i = 1, .., m. Each
of those symbolic sequences is called “admissible” and its symbols describe the order in
which trajectories, starting from any initial condition x0, visit the various sub regions Di,
i = 1, ...m. In [7] a general approach for finding periodic trajectories in piecewise-linear
maps, this procedure is based on the decomposition of the initial state via the eigenvec-
tors of their jacobian and it is applied to digital filters with two’s complement overflow
and ΣΔ modulators [8-9-10-11]. Finally, one conclude that there is some cases ( depend
mainly on position of the initial conditions) where the behavior of a map is not periodic
in spite of its Lyapunouv exponent is zero.
Hence, the following theorem is proved.
Theorem 2. Considered a general n-D piecewise linear map of the form:
f (xk) = xk+1 = Aixk + bi, if xk ∈ Di ⊂ Rn, i = 1, 2, ..., m, (18)
and assume the following:
(a) The map (18) is piecewise linear. i.e the integer m verify m ≥ 2, and there exist
Electronic Journal of Theoretical Physics 4, No. 14 (2007) 1–8 5
i, j ∈ {1, 2, ..., m} such that bi = 0 and bi = bj.
(b) The map (18) has a set of fixed point. i.e. There is a set of integers i in {1, 2, ..., m}such that the equations Aix + bi = x, has at least a zero x in the subregion Di.
(c) All the matrices Ai and Aj are equivalent. i.e. there exist invertible matrices Pij
such that: Ai = PijAjP−1ij , for all i, j ∈ {1, 2, ..,m} .
Then, the dynamics of the map (18) is known in term of the vector (ajl)1≤j,l≤n ∈ Rn2
in the following cases:
(1) if (ajl)1≤j,l≤n ∈ Ω1, then the map (18) is super chaotic.
(2) if (ajl)1≤j,l≤n ∈ Ω2, then the map (18) converges to a stable fixed point.
(3) if (ajl)1≤j,l≤n ∈ Ω3 ∩ Ω4, then the map (18) converges to a circle attractor.
(4) if (ajl)1≤j,l≤n ∈ Ω3 ∩ Ω5 ∩ Ω6, then the map (18) converges to a torus attractor.
(5) if (ajl)1≤j,l≤n ∈ Ω7 ∩ Ω8, then the map (18) converges to a K-torus attractor.
(6) if (ajl)1≤j,l≤n ∈ Ω9, then the map (18) is chaotic.
3. Conclusion
We have reported a rigorous proof of chaos in a general n-D piecewise linear map,
along the exact determination of its dynamics using the standard definition of the largest
Lyapunov exponent.
6 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 1–8
References
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[10] M. J. Ogorzalek (1992), Complex behavior in digital filters, Int. J. Bifur. Chaos, 2(1),11–29.
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[13] Tse T K, (2003), Complex Behavior of Switching Power Converters (CRC Press,Boca Raton, USA) .
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[19] Feely O, Chua L O, (1992), Nonlinear dynamics of a class of analog-to-digitalconverters, Inter. J. Bifur.Chaos 22 325–40.
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8 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 1–8
EJTP 4, No. 14 (2007) 9–16 Electronic Journal of Theoretical Physics
Flow of Unsteady Dusty Fluid Under VaryingPulsatile Pressure Gradient in Anholonomic
Co-ordinate System
B.J.Gireesha, C.S.Bagewadi∗ and B.C.Prasanna Kumara†
Department of Mathematics, Kuvempu University, Shankaraghatta-577451,Shimoga, Karnataka, India.
†Department of Mathematics, SBM Jain College of Engineering,Jakkasandra, Bangalore.
Received 22 September 2006, Accepted 6 January 2007, Published 31 March 2007
Abstract: An analytical study of unsteady viscous dusty fluid flow with uniform distributionof dust particles between two infinite parallel plates has been studied by taking into the accountof the influence of pulsatile pressure gradient. The flow analysis is carried out using differentialgeometry techniques and analytical solutions of the problem is obtained with the help of LaplaceTransform technique and which are discussed with the help of graphs.c© Electronic Journal of Theoretical Physics. All rights reserved.
Keywords: Frenet frame field system; laminar flow, dusty gas; velocity of dust gas and fluidphase, unsteady dusty fluid, pulsatile pressure gradient, relaxation zone and density.PACS (2006): 47.15.x, 47.15.Cb,AMS Subject Classification(2000): 76T10, 76T15
1. Introduction
A dusty fluid is a mixture of fluid and fine dust particles. Its study is important
in areas like environmental pollution, smoke emission from vehicles, emission of effluents
from industries, cooling effects of air conditioners, flying ash produced from thermal
reactors and formation of raindrops, etc. Also it is useful in the study of lunar ash flow
which explains many features of lunar soil.
P.G.Saffman [15] has discussed the stability of the laminar flow of a dusty gas in
which the dust particles are uniformly distributed. Liu [11] has studied the Flow induced
by an oscillating infinite plat plate in a dusty gas. Michael and Miller[12] investigated
∗ prof [email protected]
10 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 9–16
the motion of dusty gas with uniform distribution of the dust particles occupied in the
semi-infinite space above a rigid plane boundary. Samba Siva Rao [16] have obtained the
analytical solutions for the dusty fluid flow through a circular tube under the influence of
constant pressure gradient, using appropriate boundary conditions. Later T.M.Nabil [13]
studied the Effect of couple stresses on pulsatile hydromagnetic poiseuille flow, N.Datta
[5] obtained the solutions for Pulsatile flow of heat transfer of a dusty fluid through
an infinitely long annular pipe. A.Eric [6] have studied the Quantitative Assessment of
Steady and Pulsatile Flow Fields in a Parallel Plate Flow Chamber.
Some researchers like Kanwal [10], Trusdell [17], Indrasena [9], Purushotham [14],
Bagewadi, Shantharajappa and Gireesha [1, 2, 3] have applied differential geometry tech-
niques to investigate the kinematical properties of fluid flows in the field of fluid mechan-
ics. Further, the authors [2, 3] have studied two-dimensional dusty fluid flow in Frenet
frame field system. Recently the authors [7, 8] have studied the flow of unsteady dusty
fluid under varying different pressure gradients like constant, periodic and exponential.
The present work is on the laminar flow of a dusty fluid between two infinite station-
ary parallel plates with a pulsatile pressure gradient in anholonomic co-ordinate system.
Further by considering the fluid and dust particles are at rest initially, the analytical
expressions are obtained for velocities of fluid and dust particles. The changes in the
velocity profiles at different times are shown graphically.
2. Equations of Motion
The equations of motion of unsteady viscous incompressible fluid with uniform dis-
tribution of dust particles are given by [15]:
For fluid phase
∇.−→u = 0 (Continuity) (1)
∂−→u∂t
+ (−→u .∇)−→u = −ρ−1∇p + υ∇2−→u +kN
ρ(−→v −−→u ) (2)
(Linear Momentum)
For dust phase
∇.−→v = 0 (Continuity) (3)
∂−→v∂t
+ (−→v .∇)−→v =k
m(−→u −−→v ) (Linear Momentum) (4)
We have following nomenclature:−→u −velocity of the fluid phase, −→v −velocity of dust phase, ρ−density of the gas,
p−pressure of the fluid, N−number of density of dust particles, υ−kinematic viscos-
ity, k = 6πaμ−Stoke’s resistance (drag coefficient), a−spherical radius of dust particle,
m−mass of the dust particle, μ−the co-efficient of viscosity of fluid particles, t−time.
Let −→s ,−→n ,−→b be triply orthogonal unit vectors tangent, principal normal, binormal
respectively to the spatial curves of congruences formed by fluid phase velocity and dusty
Electronic Journal of Theoretical Physics 4, No. 14 (2007) 9–16 11
phase velocity lines respectively, Geometrical relations are given by Frenet formulae [4]
i)∂−→s∂s
= ks−→n ,
∂−→n∂s
= τs−→b − ks
−→s ,∂−→b
∂s= −τs
−→n
ii)∂−→n∂n
= k′n−→s ,
∂−→b
∂n= −σ′n
−→s ,∂−→s∂n
= σ′n−→b − k′n
−→n(5)
iii)∂−→b
∂b= k′′b
−→s ,∂−→n∂b
= −σ′′b−→s ,
∂−→s∂b
= σ′′b−→n − k′′b
−→b
iv) ∇.−→s = θns + θbs; ∇.−→n = θbn − ks; ∇.−→b = θnb
where ∂/∂s, ∂/∂n and ∂/∂b are the intrinsic differential operators along fluid phase
velocity (or dust phase velocity ) lines, principal normal and binormal. The functions
(ks, k′n, k
′′b ) and (τs, σ
′n, σ
′′b ) are the curvatures and torsion of the above curves and θns
and θbs are normal deformations of these spatial curves along their principal normal and
binormal respectively.
3. Formulation and Solution of the Problem
In the present problem we consider unsteady laminar flow of an incompressible viscus
fluid with uniform distribution of dust particles between two infinite stationary parallel
plates separated by a distance h in the absence of body force. The flow is due to the
influence of pulsatile pressure gradient with respect to time. Both the fluid and the dust
particle clouds are supposed to be static at the beginning. The dust particles are assumed
to be spherical in shape and uniform in size. The number density of the dust particles
is taken as a constant throughout the flow. Under these assumptions the flow will be a
parallel flow in which the streamlines are along the tangential direction and the velocities
are varies along binormal direction and with time t, since we extended the fluid to infinity
in the principal normal direction.
Fig. 1 Geometry of the flow
12 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 9–16
By virtue of system of equations (5) the intrinsic decomposition of equations (2) and
(4) give the following forms;
∂us
∂t= −1
ρ
∂p
∂s+ ν
[∂2us
∂b2− Crus
]+
kN
ρ(vs − us) (6)
2u2sks = −1
ρ
∂p
∂n+ ν
[2σ′′b
∂us
∂b− usk
2s
](7)
0 = −1
ρ
∂p
∂b+ ν
[usksτs − 2k′′b
∂us
∂b
](8)
∂vs
∂t=
k
m(us − vs) (9)
2v2sks = 0 (10)
where Cr = (σ′2n + k′2n + k′′2b + σ′′2b ) is called curvature number [3].
From equation (10) we see that v2sks = 0 which implies either vs = 0 or ks = 0.
The choice vs = 0 is impossible, since if it happens then us = 0, which shows that the
flow doesn’t exist. Hence ks = 0, it suggests that the curvature of the streamline along
tangential direction is zero. Thus no radial flow exists.
Equation (6) and (9) are to be solved subject to the initial and boundary conditions;⎧⎪⎨⎪⎩ Initial condition; at t = 0; us = 0, vs = 0
Boundary condition; for t > 0; us = 0, at b = 0 and b = h
⎫⎪⎬⎪⎭ (11)
Since we have assumed that a pulsatile pressure gradient is impressed on the system
for t > 0, we can write
−1
ρ
∂p
∂s= c + αcos(βt)
where c and α are constants and β is the frequency of oscillation.
We define Laplace transformations of us and vs as
U =
∞∫0
e−stusdt and V =
∞∫0
e−stvsdt (12)
Applying the Laplace transform to equations (6), (9) and to boundary conditions,
then by using initial conditions one obtains
sU =c
s+
αs
(s2 + β2)+ ν
[∂2U
∂b2− CrU
]+
L
τ(V − U) (13)
sV =1
τ(U − V ) (14)
U = 0, at b = 0 and b = h (15)
Electronic Journal of Theoretical Physics 4, No. 14 (2007) 9–16 13
where L = mNρ
and τ = mk. Equation (14) implies
V =U
1 + sτ(16)
Eliminating V from (13) and (16) we obtain the following equation
d2U
db2− Q2U = −
[c
νs+
αs
ν(s2 + β2)
](17)
where Q2 =(Cr + s
ν+ sL
ν(1+sτ)
).
The velocities of fluid and dust particle are obtained by solving the equation (17)
subjected to the boundary conditions ((15)) as follows
U =1
νQ2
[c
s+
αs
s2 + β2
]{sinhQ(b − h) − sinh(Qb)
sinh(Qh)+ 1
}
Using U in (16) we obtain V as
V =1
(νQ2)(1 + sτ)
[c
s+
αs
s2 + β2
]{sinhQ(b − h) − sinh(Qb)
sinh(Qh)+ 1
}
By taking inverse Laplace transform to U and V, one can obtain
us =c
νλ2
(sinh(λ(b − h)) − sinh(λb)
sinh(λh)+ 1
)+
4c
π
∞∑n=0
1
2n + 1sin
(2n + 1
hπb
)(ex1t(1 + x1τ)2
((1 + x1τ)2 + L)+
ex2t(1 + x2τ)2
((1 + x2τ)2 + L)
)+
α
ν
((AE + BF )M1 − (BE − AF )M2
[(y1y2 − β2)2 + (βy1 + βy2)2] (E2 + F 2)
)+
4α
π
∞∑n=0
1
2n + 1sin
(2n + 1
hπb
)×[
x1ex1t(1 + x1τ)2
(x21 + β2)((1 + x1τ)2 + L)
+x2e
x2t(1 + x2τ)2
(x22 + β2)((1 + x2τ)2 + L)
]
14 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 9–16
vs =c
νλ2
(sinh(λ(b − h)) − sinh(λb)
sinh(λh)+ 1
)+
4c
π
∞∑n=0
1
2n + 1sin
(2n + 1
hπb
)[ex1t(1 + x1τ)
((1 + x1τ)2 + L)+
ex2t(1 + x2τ)
((1 + x2τ)2 + L)
]+
α
ν
((M1A − M2B)(E − Fβτ) + (M2A + M1B)(Eβτ + F )
[(y1y2 − β2)2 + (βy1 + βy2)2] (E2 + F 2)(1 + β2τ 2)
)+
4α
π
∞∑n=0
1
2n + 1sin
(2n + 1
hπb
)×[
x1ex1t(1 + x1τ)
(x21 + β2)((1 + x1τ)2 + L)
+x2e
x2t(1 + x2τ)
(x22 + β2)((1 + x2τ)2 + L)
]where
x1 = − 1
2τ
(1 + L + νCrτ + ντ
n2π2
h2
)+
1
2τ
√(1 + L + νCrτ + ντ
n2π2
h2
)2
− 4τν
(Cr +
n2π2
h2
)x2 = − 1
2τ
(1 + L + νCrτ + ντ
n2π2
h2
)− 1
2τ
√(1 + L + νCrτ + ντ
n2π2
h2
)2
− 4ντ
(Cr +
n2π2
h2
)y1 = − 1
2τ(1 + L + νCrτ) +
1
2τ
√(1 + L + νCrτ)2 − 4Crντ
y2 = − 1
2τ(1 + L + νCrτ) − 1
2τ
√(1 + L + νCrτ)2 − 4Crντ
A = sinh(α1(b − h))cos(β1(b − h)) − sinh(α1b)cos(β1b) + sinh(α1h)cos(β1h)
B = cosh(α1(b − h))sin(beta1(b − h)) − cosh(α1b)sin(β1b) + cosh(α1h)sin(β1h)
M1 = (cosβt − βτsinβt)(y1y2 − β2) + (sinβt + βτcosβt)(βy1 + βy2)
M2 = (sinβt − βτcosβt)(y1y2 − β2) + (cosβt + βτsinβt)(βy1 + βy2)
E = sinh(α1h)cos(β1h), F = cosh(α1h)sin(β1h)
δ1 =y1y2 − β2 − β2τ(y1 + y2)
ν(1 + β2τ 2), δ2 =
β2τ − β(y1 + y2) − y1y2βτ
ν(1 + β2τ 2)
α1 =
√δ1 +
√δ21 + δ2
2
2and β1 =
√−δ1 +
√δ21 + δ2
2
2, λ =
√x1x2
ν
Electronic Journal of Theoretical Physics 4, No. 14 (2007) 9–16 15
4. Conclusion
The figures 2 and 3 represents the velocity profiles for the fluid and dust particles
respectively, which are parabolic in nature. It is observed that velocity of fluid particles
is parallel to velocity of dust particles and velocity decreases with increase in time t.
Further one can observe that if the dust is very fine i.e., mass of the dust particles is
negligibly small then the relaxation time of dust particle decreases and ultimately as
τ → 0 the velocities of fluid and dust particles will be the same. Also we see that the
fluid particles will reach the steady state earlier than the dust particles. This difference
is due to the fact that pulsatile pressure gradient is directly exerted on the fluid.
Fig. 2 Variation of fluid velocity with b
Fig. 3 Variation of dust phase velocity with b
16 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 9–16
References
[1] C.S.Bagewadi and A.N.Shantharajappa, A study of unsteady dusty gas flow in FrenetFrame Field, Indian Journal Pure Appl. Math., 31 (2000) 1405-1420.
[2] C.S.Bagewadi and B.J.Gireesha, A study of two dimensional steady dusty fluid flowunder varying temperature, Int. Journal of Appl. Mech. & Eng., 09(2004) 647-653.
[3] C.S.Bagewadi and B.J.Gireesha, A study of two dimensional unsteady dusty fluid flowunder varying pressure gradient, Tensor, N.S., 64 (2003) 232-240.
[4] Barret O’ Nell, Elementary Differential Geometry, Academic Press, New York &London, 1966.
[5] N.Datta & D.C.Dalal, Pulsatile flow of heat transfer of a dusty fluid through aninfinitly long annlur pipe, Int. J. Multiphase flow, 21(3) (1995) 515-528.
[6] A.Eric, Nauman, J.Kurtis, Risic, M.Tony, Keaveny, & L.Robert Satcher, QuantitativeAssessment of Steady and Pulsatile Flow Fields in a Parallel Plate Flow Chamber, Annals of Biomedical Engineering, 27 (1999) 194-199.
[7] B.J.Gireesha , C. S. Bagewadi & B.C.Prasannakumara, Flow of unsteady dusty fluidunder varying periodic pressure gradient, ’Journal of Analysis and Computation’,2(2), (2006) 183-189.
[8] B.J.Gireesha , C. S. Bagewadi & B.C.Prasannakumara, Flow of unsteady dusty fluidbetween two parallel plates under constant pressure gradient, Tensor.N.S. 68 (2007)
[9] Indrasena, Steady rotating hydrodynamic-flows, Tensor, N.S., (1978) 350-354.
[10] R.P.Kanwal, Variation of flow quantities along streamlines, principal normals andbi-normals in three-dimensional gas flow, J.Math., 6 (1957) 621-628.
[11] J.T.C.Liu, Flow induced by an oscillating infinite plat plate in a dusty gas, Phys.Fluids, 9 (1966) 1716-1720.
[12] D.H.Michael and D.A.Miller, Plane parallel flow of a dusty gas, Mathematika, 13(1966) 97-109.
[13] T.M.Nabil, EL-Dabe, M.G.Salwa and EL-Mohandis, Effect of couple stresses onpulsatile hydromagnetic poiseuille flow, Fluid Dynamic Research, 15 (1995) 313-324.
[14] G.Purushotham and Indrasena, On intrinsic properties of steady gas flows ,Appl.Sci. Res., A 15(1965) 196-202.
[15] P.G.Saffman, On the stability of laminar flow of a dusty gas, Journal of FluidMechanics, 13(1962) 120-128.
[16] P.Samba Siva Rao, Unsteady flow of a dusty viscous liquid through circular cylinder,Def. Sci. J., 19(1969) 135-138.
[17] C.Truesdell, Intrinsic equations of spatial gas flows, Z.Angew.Math.Mech, 40 (1960)9-14.
EJTP 4, No. 14 (2007) 17–30 Electronic Journal of Theoretical Physics
Exact Solutions for Nonlinear Evolution EquationsVia Extended Projective Riccati Equation
Expansion Method
M A Abdou∗
Theoretical Research Group, Physics Department,Faculty of Science, Mansoura University, 35516 Mansoura, Egypt
Received 21 June 2006, Accepted 6 January 2007, Published 31 March 2007
Abstract: By means of a simple transformation, we have shown that the generalized-Zakharovequations, the coupled nonlinear Klein-Gordon-Zakarov equations, the GDS, DS and GZequations and generalized Hirota-Satsuma coupled KdV system can be reduced to the elliptic-like equations. Then, the extended projective Riccati equation expansion method is used toobtain a series of solutions including new solitary wave solutions,periodic and rational solutions.The method is straightforward and concise, and its applications is promising.c© Electronic Journal of Theoretical Physics. All rights reserved.
Keywords: Extended projective Riccati equation, Nonlinear evolution equations, New solitarywave solutions, Periodic and rational solutions.PACS (2006): 02.30.Hq, 02.30.Jr, 47.35.Fg, 94.05.Fg, 02.90.+p
1. Introduction
The investigation of the exact travelling wave solutions of nonlinear evolution equa-
tions plays an important role in the study of nonlinear physical phenomena. For exam-
ple,the wave phenomena observed in fluid dynamics, plasma,elastic media,optical fibers,
etc. In the past several decades, both mathematicians and physicists have made signifi-
cant progression in this direction.
Many effective methods [1 − 13] have been presented such as variational iteration
method [6], homotopy perturbation method [3], Exp-function method [8, 12], and others.
A complete review on the field is available on [4].
The rest of this paper is organized as follows: In Section 2, first we briefly give the
steps of the method and apply the method to solve the elliptic-like equation. In Section
18 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 17–30
3, by using the results obtained in Section 3, the corresponding solutions of some class of
nonlinear evolution equations in mathematical physics can be obtained. The last section
is devoted to the conclusion.
2. Method and its Applications
To illustrate the basic idea of the extended projective Riccati equation expansion
method, we consider the nonlinear evolution equation with independent variables,say in
two variables x, t,
Q(u, ux, uxx, ....) = 0, (1)
we consider its travelling wave solutions
u(x, t) = u(ξ), ξ = x − λt + ξ0, (2)
then Eq.(1) is reduced to an ordinary differential equation (ODE)
Q(u, u,, u,,, ....) = 0, (3)
where a prime denotes ddξ
.
Step (1). We assume that Eq.(1) has the following formal solution :
u(ξ) = a0 +M∑i=1
f i−1(ξ)[aif(ξ) + big(ξ)], (4)
where a0,ai and bi are constants to be determined later. The parameter M can be
determined by balancing the highest order derivative term with nonlinear term in Eq.(3),
f′(ξ) = pf(ξ)g(ξ), (5)
g′(ξ) = q + pg2(ξ) − rf(ξ), (6)
g2 = −1
p[q − 2rf +
r2 + δ
qf 2], (7)
where p = 0 is a real constant, q, r, δ are real constants.
Step (2). Substituting Eq.(4) into (3) and making use of Eqs.(5-7) yields a set of
algebraic polynomials for f i(ξ)gj(ξ)(i = 0, 1, ...; j = 0, 1, ...). Eliminating all the coeffi-
cients of the power of f i(ξ)gj(ξ), yields a series of alegbraic equations, from which the
parameters ai,bi and λ are explicitly determined.
Step (3). It is easy to see that Eqs.(5) and (6) admits the following solutions:
Case(1):δ = h2 − s2,q = 0, and pq < 0,
f1 =q
r + scosh(√−pqξ) + hsinh(
√−pqξ), (8)
Electronic Journal of Theoretical Physics 4, No. 14 (2007) 17–30 19
g1 =
√−pq
p
ssinh(√−pqξ) + hcosh(
√−pqξ)
r + scosh(√−pqξ) + hsinh(
√−pqξ), (9)
g21 = −1
p[q − 2rf1 +
r2 + h2 − s2
qf2
1 ], (10)
where h, p, s, q and r are constants.
Case(2): δ = −h2 − s2,q = 0 , and pq > 0,
f2 =q
r + scos(√
pqξ) + hsin(√
pqξ), (11)
g2 =
√pq
p
ssin(√
pqξ) − hcos(√
pqξ)
r + scos(√
pqξ) + hsin(√
pqξ), (12)
g22 = −1
p[q − 2rf2 +
r2 − h2 − s2
qf 2
2 ], (13)
where h, p, s, q and r are constants.
Case(3): q = 0,
f3 =1
(pr/2)ξ2 + mξ + n, (14)
g3 =−1
p
prξ + m
(pr/2)ξ2 + mξ + n, (15)
g23 =
2r
pf3 + [
m2
p2− 2rn
p]f2
3 , (16)
where m, n, p, r are arbitrary constants.
Case(4): p = ±1, δ = −r2,
f4(ξ) =q
6r+
2
prψ(ξ), (17)
g4(ξ) =12ψ
′(ξ)
q + 12ψ′(ξ), (18)
where ψ(ξ) satisfies
ψ′2(ξ) = 4ψ3(ξ) − γ2ψ(ξ) − γ3,
where γ2 = q2
12, γ3 = pq3
216,
g24 =
2r
pf4 − p
q(19)
Case(5):p = ±1,δ = − r2
25,
f5(ξ) =5q
6r+
5pq2
72ψ(ξ), (20)
20 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 17–30
g5(ξ) = − qψ′(ξ)
ψ(ξ)(pq + 12ψ(ξ)), (21)
g25 = −1
p[q − 2rf5 +
24r2
25qf2
5 ] (22)
3. The Exact Solutions of Elliptic-like Equations
Let us consider the elliptic-like equation in [7]
Aφ′′(ξ) + Bφ(ξ) + Dφ3(ξ) = 0, (23)
where A, B, D are arbitrary constants. In this section,the exact solutions of Eq.(23)
are derived using the coupled projective Riccati Eqs.(5) and (6).Considering the homo-
geneous balance between φ′′(ξ) and φ3(ξ) in Eq.(23), the solution of Eq.(23) is given
by
φ(ξ) = a0 + a1f(ξ) + b1g(ξ), (24)
where a0,a1 and b1 are constants to be determined later, and f(ξ) and g(ξ) satisfy
Eqs.(5-7).Substituting Eq.(24) into (23) and making use of Eqs.(5-7), becomes a polyno-
mials for f i1(i = 0, 1, 2, 3) and f j
1g1(j = 0, 1, 2), setting the coefficients of the polynomials
to zero yields a set of algebraic equations. Solving the system of algebraic equations with
the aid of Maple, we have
a0 = 0, a21 =
Ap(r2 + h2 − s2)
2qD, b2
1 = −Ap2
2D(25)
Case(1):pq < 0,q = 0,g21 = −1
p[q − 2rf1 + r2+h2−s2
qf2
1 ].Substituting Eq.(25) into
Eq.(24) and using Eqs.(5-7),the exact solution of Eq.(23) are derived as
φ1(ξ) =a1q
r + scosh(√−pqξ) + hsinh(
√−pqξ)−b1
√−pq
p
ssinh(√−pqξ) + hcosh(
√−pqξ)
r + scosh(√−pqξ) + hsinh(
√−pqξ),
(26)
a21 = Ap(r2+h2−s2)
2qD,AD < 0 and b2
1 = −Ap2
2D.
Case(1.1): a0 = a1 = 0, r = 0,the exact solution of Eq.(23) are derived as
φ2(ξ) = −b1
√−pq
p
ssinh(√−pqξ) + hcosh(
√−pqξ)
r + scosh(√−pqξ) + hsinh(
√−pqξ), (27)
b21 = −Ap2
2Dand A
D< 0.
Case(1.2): a0 = b1 = 0,r = 0, the exact solution of Eq.(23) yields
φ3(ξ) =a1q
scosh(√−pqξ) + hsinh(
√−pqξ), (28)
Electronic Journal of Theoretical Physics 4, No. 14 (2007) 17–30 21
a21 = 2Ap(h2−s2)
qD.
Case(2):pq > 0,q = 0,g22 = −1
p[q − 2rf2 + r2−h2−s2
qf 2
2 ].
Case(2.1): a0 = 0, pq > 0,
φ4(ξ) =a1q
r + scos(√
pqξ) + hsin(√
pqξ)+
b1√
pq
p
ssin(√
pqξ) − hcos(√
pqξ)
r + scos(√
pqξ) + hsin(√
pqξ), (29)
a21 = Ap(r2−h2−s2)
2qDand b2
1 = −Ap2
2D.
Case(2.2): a0 = a1 = 0, r = 0, pq > 0,
φ5(ξ) =b1√
pq
p
ssin(√
pqξ) − hcos(√
pqξ)
scos(√
pqξ) + hsin(√
pqξ), (30)
b21 = −2Ap2
D.
Case(2.3):a0 = b1 = 0,r=0,pq > 0,
φ6(ξ) =a1q
scos(√
pqξ) + hsin(√
pqξ), (31)
a21 = 2Ap(−h2−s2)
qDand pq > 0.
Case(3): p = ±1, δ = −r2, g24 = 2r
pf4 − q
p. The exact solution of Eq.(23) admits
φ7(ξ) =12b1ψ
′(ξ)
q + 12pψ′(ξ), (32)
b21 = −Ap2
2Dand A
D< 0
Case(4): p = ±1,g25 = −1
p[q−2rf5+ r2−h2−s2
qf 2
5 ].The exact solution of Eq.(23) admits
φ8(ξ) = a1[5q
6r+
5pq2
72rψ(ξ)] − b1qψ
′(ξ)
ψ(ξ)(pq + 12ψ(ξ)), (33)
a21 = 12r2AP
25Dq, b2
1 = −AP 2
2D, p
q< 0,p = ±1 and A
D< 0.
4. Exact Solutions of Some Class of Nonlinear Evolution Equa-
tions
In this section,by using the results obtained in section (3), we will constract the
corresponding solutions of the generalized-Zakharov equations, the coupled nonlinear
Klein-Gordon-Zakarov equations, the GDS,DS and GZ equations and generalized Hirota-
Satsuma coupled KdV system.
4.1 The generalized-Zakharov equations
The generalized Zakharov equations for the complex envelope ψ(x, t) of the high-frequency
wave and the real low-frequency field v(x, t) reads [13]
22 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 17–30
iψt + ψxx − 2λ|ψ|2ψ + 2ψv = 0, (34)
vtt − vxx + (|ψ|2)xx = 0, (35)
where the cubic term in Eq.(34) describes the nonlinear-self interaction in the high
frequency subsystem,such a term corresponds to a self-focusing effect in plasma physics.
The coefficient λ is a real constant that can be a postive or negative number. Let us
assume the travelling wave solution of Eqs.(34) and (35) in the form
ψ(x, t) = eiηφ(ξ), v = v(ξ),
η = αx + βt, ξ = k(x − 2αt), (36)
where φ(ξ) and v(ξ) are real functions, the constants α, β and k are to be determined.
Substituting (36) into Eqs.(34) and (35), we have
k2φ′′(ξ) + 2φ(ξ)v(ξ) − (α2 + β)φ(ξ) − 2λφ3(ξ) = 0, (37)
k2(4α2 − 1)v′′(ξ) + k2(φ2)
′′(ξ) = 0 (38)
In order to simplify ODEs (37) and (38), integrating Eq.(38) once and taking inte-
gration constant to zero, and integrating yields
v(ξ) =φ2(ξ)
(1 − 4α2)+ C, ifα2 = 1
4, (39)
where C-integration constant. Inserting Eq.(39) into (37), we have
Aφ′′(ξ) + Bφ(ξ) + Dφ3(ξ) = 0 (40)
Eq.(40) coincides with Eq.(23), where A, B and D are defined by
A = k2,
B = [2C − α2 − β],
D = 2[1
1 − 4α2− λ] (41)
Then the solution of Eqs.(34) and (35) are
ψ(x, t) = eiηφ(ξ),
v(x, t) =φ2(ξ)
(1 − 4α2)+ C, (42)
Electronic Journal of Theoretical Physics 4, No. 14 (2007) 17–30 23
where φ(ξ) is given by Eqs.(26-33), η = αx + βt, ξ = k(x− 2αt) and A, B and D are
defined by Eq.(41).
4.2 The coupled nonlinear Klein-Gordon-Zakarov equations
The coupled nonlinear Klein-Gordon-Zakarov equations [14] read
utt − c20∇2u + f2
0 u + δuv = 0,
vtt − c20∇2v − β∇2|u|2 = 0, (43)
where c0, f0, β and δ are constants. We seek its following wave packet solution
u(x, y, z, t) = φ(ξ)ei(kx+ly+nz−Ωt), v = v(ξ), ξ = px + qy + rz − wt, (44)
where φ(ξ) and v(ξ) are real functions.Substituting Eq.(44) into Eqs.(43) yields
[w2 − c20P
2)φ′′(ξ) + 2i[wΩ − c2
0K.P )φ′(ξ) − (w2 − K2c2
0 − f 20 )φ(ξ) + δv(ξ)φ(ξ) = 0,
[w2 − c20P
2)v′′(ξ) − βP 2(φ2(ξ))
′′= 0, (45)
K = (k, l, n), P = (p, q, r), K.P = kp + lq + nr
If we take w.Ω = c20K.P , then Eqs.(43) leads to
[w2 − c20P
2]φ′′(ξ) − (w2 − K2c2
0 − f 20 )φ(ξ) + δv(ξ)φ(ξ) = 0, (46)
[w2 − c20P
2)v′′(ξ) − βP 2(φ2(ξ))
′′= 0 (47)
Integrating (47) twice with respect to ξ, we get
v(ξ) =c
w2 − c20P
2+
βP 2
w2 − c20P
2φ2(ξ), (48)
where c is an integration constant. Substituting (48) into (46) the obtained equation
can be expressed as Eq.(23), while the parameters A, B and D are defined by
A = [w2 − c20P
2]2,
B = [(w2 − c20P
2)(−w2 + c20K
2c20 + f2
0 ) + δc],
D = δβP 2 (49)
24 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 17–30
Then the solution of Eqs.(43) are defined as follows
u(x, y, z, t) = φ(ξ)ei(kx+ly+nz−Ωt),
v(x, y, z, t) =c
w2 − c20P
2+
βP 2
w2 − c20P
2φ2(ξ),
Ω =c20K.P
w, (50)
where φ(ξ) appearing in these solutions is given by Eqs.(26-33) and A, B and D are
defined by (49) and ξ = px + qy + rz − wt.
4.3 The GDS,DS and GZ equations
We consider a class of NLPDEs with constant coefficients [15]
iut + ν(uxx + D1uyy) + E1|u|2u + C1uv = 0,
D2vtt + (vxx − E2uyy) + C2(|u|2)xx = 0, (51)
where ν, Di, Ei, Ci are real constants and ν = 0, D1 = 0, C1 = 0, C2 = 0. Eqs.(51)
are a class of physically important equations.In fact, if one takes
ν =1
2k2, D1 = 2ν, E1 = α, C1 = −1, D2 = 0, E2 = D1, C2 = −2α, k2 = ±1, (52)
then Eqs.(51) represent the DS equations [16]
iut +1
2k2(uxx + k2uyy) + α|u|2u − uv = 0,
vxx − k2uyy − 2α(|u|2)xx = 0 (53)
If one takes
ν = v(x, t), i.e., vy = 0, ν = 1, D1 = 0, E1 = −2σ,E2 = −1, C2 = −1, C1 = 2, (54)
then Eqs.(51) represent the GZ equations [17]
iut + uxx − 2σ|u|2u + 2uv = 0,
vtt − vxx + (|u|2)xx = 0 (55)
Since u is a complex function,we assume that
Electronic Journal of Theoretical Physics 4, No. 14 (2007) 17–30 25
u(x, y, t) = φ(ξ)ei(kx+ly−Ωt), v(x, y, t) = v(ξ), ξ = px + qy − wt, (56)
where both φ(ξ) and v(ξ) are real functions,and k, l, p, q, Ω and w are constants to
be determined later.Substituting Eq.(56) into (51),we have the following ODE for φ(ξ)
and v(ξ)
ν(p2+D1q2)φ
′′(ξ)+[Ω−ν(k2+D1l
2)]φ(ξ)+E1φ3(ξ)+i[−w+2ν(kp+D1lq)]φ
′(ξ)+C1φ(ξ)v(ξ) = 0,
(57)
[D2w2 + p2 − E2q
2]v′′(ξ) + C2p
2(φ2(ξ))′′
= 0 (58)
if we set
w = 2ν(kp + D1lq), (59)
then Eq.(57) reduces to
ν(p2 + D1q2)φ
′′(ξ) + [Ω − ν(k2 + D1l
2)]φ(ξ) + E1φ3(ξ) + C1φ(ξ)v(ξ) = 0 (60)
Integrating Eq.(58) twice, we get
v(ξ) =c
D2w2 + p2 − E2q2− C2p
2
D2w2 + p2 − E2q2φ2(ξ), (61)
where c is an integration constant. Substituting Eq.(61) into (60) yields
ν(p2+D1q2)(D2w2+p2−E2q2)φ′′(ξ)+C1c−(D2w2+p2−E2q2)[ω−ν(k2+D1l2)]φ(ξ)+E1(D2w2+p2−E2q2)−C1C2p2φ3(ξ)=0,
(62)
Eq.(62) can be rewritten as Eq.(23), while A, B and D are given by the following
equation,
A = ν(p2 + D1q2)(D2w
2 + p2 − E2q2),
B = C1c − (D2w2 + p2 − E2q
2)[Ω − ν(k2 + D1l2)],
D = E1(D2w2 + p2 − E2q
2) − C1C2p2 (63)
Then the solution of Eqs.(51) are
u(x, y, t) = φ(ξ)ei(kx+ly−Ωt), (64)
v(x, y, t) =c
D2w2 + p2 − E2q2− C2p
2
D2w2 + p2 − E2q2φ2(ξ), (65)
26 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 17–30
w = 2ν(kp + D1lq) (66)
The expression φ(ξ) appearing in these solutions is given by Eqs.(26-33) and ξ =
px + qy − wt. We may obtain from Eq.(53) that
u(x, y, t) = φ(ξ)ei(kx+ly−Ωt), (67)
v(x, y, t) =c
p2 − k2q2+
2αp2
p2 − k2q2φ2(ξ), (68)
w = k2(kp + k2lq), (69)
where φ(ξ) satisfy the elliptic-like Eq.(23) with A, B and D defined as follows
A = k2(p2 + k2q2)(k2q2 − p2),
B = 2c + (p2 − k2q2)[2Ω − k2(k2 + k2l2)],
D = 2α(p2 + k2q2) (70)
The expression φ(ξ) are defined by Eqs.(26-33) and ξ = px + qy − wt.Then From
Eq.(55) we have that
u(x, y, t) = φ(ξ)ei(kx−Ωt), (71)
v(x, y, t) =c
p2 − w2+
p2
p2 − w2φ2(ξ), (72)
w = 2kp, (73)
where φ(ξ) satisfies Eq.(23),while A, B and D are given by
A = p2(p2 − w2),
B = 2c − (p2 − w2)[Ω − k2],
D = 2[p2 − σ(p2 − w2)] (74)
The expression φ(ξ) appearing in these solutions is given by Eqs.(26-33) and ξ =
px − wt.
Electronic Journal of Theoretical Physics 4, No. 14 (2007) 17–30 27
4.4 Generalized Hirota-Satsuma coupled KdV equation
Consider the Hirota-Satsuma coupled KdV system in [18]
ut =1
4uxxx + 3uux + 3(w − v2)x,
vt = −1
2vxxx − 3uvx,
wt = −1
2wxxx − 3uwx (75)
When w = 0,Eqs.(75) reduces to be the well-known Hirota-Satsuma coupled KdV
system [19]. We seek travelling wave solutions for Eqs.(75) in the form
u(x, t) = u(ξ), v(x, t) = v(ξ), w(x, t) = w(ξ), ξ = k(x − ct) (76)
Substituting Eq.(76) into (75), we get
−cku′=
1
4k3u
′′′+ 3kuu
′+ 3k(w − v2)
′, (77)
−ckv′= −1
2k3v
′′′ − 3kuv′, (78)
−ckw′= −1
2k3w
′′′ − 3kuw′
(79)
Let
u = αv2 + βv + γ,
w = A0v + B0, (80)
where α, γ, β, A0 and B0 are constants. Inserting Eq.(80) into (78) and (79) inte-
grating once we know that (78) and (79) give rise to the same equation
k2v′′
= −2αv3 − 3βv2 + 2(c − 3γ)v + c1, (81)
where c1 is an integration constant.Integrating (81) we have
k2v′2 = −αv4 − 2βv3 + 2(c − 3γ)v2 + 2c1v + c2, (82)
where c2 is an integration constant. By means of Eqs.(80-82) we get
k2u′′
= 2αk2v′2 + k2(2αv + β)v
′′= 2α[−αv4 − 2βv3
+2(c − 3γ)v2 + 2c1v + c2] + (2αv + β)[−2αv3 − 3βv2 + 2(c − 3γ)v + c1] (83)
28 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 17–30
Integrating (77) once we have
1
4k2u
′′+
3
2u2 + cu + 3(w − v2) + c3 = 0, (84)
where c3 is an integration constant.Inserting (80) and (83) into (84) gives
3αc − 3αγ +3
4β2 − 3 = 0,
1
2[αc1 + βc + γβ) + A0 = 0,
1
4(2αc2 + βc1) +
3
2γ2 + cγ + 3B0 + c3 = 0 (85)
Let
c1 =1
2α2[β3 + 2cαβ − 6αβγ),
v(ξ) = aφ(ξ) − β
2α(86)
Therefore from Eq.(81), we have
k2φ′′(ξ) − a(
3β2
2α+ 2c − 6γ)φ(ξ) + 2αa3φ3(ξ) = 0, (87)
then Eq.(87) can be written as
Aφ′′(ξ) + Bφ(ξ) + Dφ3(ξ) = 0 (88)
Eq.(88) is the same with Eq.(23) where A, B and D are defined by
A = k2, B = −a((3β2/2α) + 2c − 6γ), D = 2αa3 (89)
Then the solutions of Eqs.(75) are given by
u(x, t) = α[aφ(ξ) − β
2α]2 + γ, (90)
v(x, t) = [aφ(ξ) − β
2α], (91)
w(x, t) = A0[aφ(ξ) − β
2α] + B0, (92)
the expression φ(ξ) appearing in these solutions are defined by Eqs.(26-33).
Electronic Journal of Theoretical Physics 4, No. 14 (2007) 17–30 29
5. Conclusion
In this paper, with the aid of a simple transformation technique, we have shown
that the generalized-Zakharov equations, the coupled nonlinear Klein-Gordon-Zakarov
equations, the GDS,DS and GZ equations and generalized Hirota-Satsuma coupled KdV
system can be reduced to the elliptic-like equation.
The validity of the proposed method has been tested by applying it successfully to the
generalized-Zakharov equations, the coupled nonlinear Klein-Gordon-Zakarov equations,
the GDS,DS and GZ equations and generalized Hirota-Satsuma coupled KdV system.
As a result, many exact wave solutions are obtained which include new solitary wave
solutions, periodic and rational solutions.
Finally, it is worthwhile to mention that the proposed method is straightforward and
concise, more applications to other nonlinear physical systems should be concerned and
deserve further investigation. This is our task in the future work.
Acknowledgement
The author is thankful to Prof. Dr. S. A. El-Wakil for his suggestions, reviews and
continuous encouragement.
30 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 17–30
References
[1] El-Wakil S A, Abdou M A, Chaos, Solitons and Fractals 31(2007)840-852
[2] El-Wakil S A, Abdou M A. The Adomian decomposition method for solving nonlinearphysical models, Chaos, Solitons and Fractals (2007) in Press
[3] Ji-Huan He, Chaos, Solitons and Fractals 26(2005)695
[4] Ji-Huan He,Int.J.Modern Phys.B 20(2006)1141
[5] El-Wakil S A, Abdou M A, Phys. Lett. A 358(2006)275-282
[6] Abdou M A, Soliman A A, Physica D 2005;211:1
[7] Abdou M A, Chaos, Solitons and Fractals 31(2007)95-104.
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[10] El-Wakil S A, Abdou MA, Elhanbaly A, Phys. Lett. A 353(2006)40
[11] Abdou M A, Elhanbaly A.Construction of periodic and solitary wave solutions bythe extended Jacobi elliptic function expansion method, Comm. Non. Sci. and Numer.Sim. (2007) in Press
[12] Ji-Huan He,Abdou M A. New periodic solutions for nonlinear evolution equationsusing Exp-function method, Chaos, Solitons and Fractals (2007) in Press
[13] Wang M, X Li, Phys. Lett. A 343(2005)48
[14] Ablowitz M,Clarkson P A 1991. Solitons, Nonlinear evolution equations and inversescattering transform, New York, Cambridge, University Press.
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EJTP 4, No. 14 (2007) 31–50 Electronic Journal of Theoretical Physics
Evolutionary Neural Gas (ENG) : A Model of SelfOrganizing Network from Input Categorization
Ignazio Licata1∗, Luigi Lella2
1Ixtucyber for Complex Systems, Marsala, TP andInstitute for Scientific Methodology, Palermo, Italy
2A.R.C.H.I. - Advanced Research Center for Health Informatics, Ancona, Italy
Received 16 December 2006, Accepted 6 January 2007, Published 31 March 2007
Abstract: Despite their claimed biological plausibility, most self organizing networks have stricttopological constraints and consequently they cannot take into account a wide range of externalstimuli. Furthermore their evolution is conditioned by deterministic laws which often are notcorrelated with the structural parameters and the global status of the network, as it shouldhappen in a real biological system. In nature the environmental inputs are noise affected and“fuzzy”. Which thing sets the problem to investigate the possibility of emergent behaviour in anot strictly constrained net and subjected to different inputs. It is here presented a new model ofEvolutionary Neural Gas (ENG) with any topological constraints, trained by probabilistic lawsdepending on the local distortion errors and the network dimension. The network is consideredas a population of nodes that coexist in an ecosystem sharing local and global resources. Thoseparticular features allow the network to quickly adapt to the environment, according to itsdimensions. The ENG model analysis shows that the net evolves as a scale-free graph, andjustifies in a deeply physical sense- the term “gas” here used.c© Electronic Journal of Theoretical Physics. All rights reserved.
Keywords: Self-Organizing Networks; Neural Gas; Scale-Free Graph; Information in NetworkFunctional SpecializationPACS (2006): 89.75.k, 89.75.Fb, 82.39.Rt,07.05.Mh, 84.35.+i, 87.23.n, 91.62.Np
1. Introduction
Self organizing networks are systems widely used in categorization tasks. A network
can be seen as a set A={c1, c2,. . . ,cn} of units with associated reference vectors wc ∈Rn
where Rn is the same space where inputs are defined. Each unit (or node) can establish
connections with the other ones, the units belonging to the same clusters are subjected
∗ Corresponding author: [email protected]
32 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 31–50
to similar modification affecting their reference vectors.
Self organizing networks can automatically adapt to input distributions without super-
vision by means of training algorithms that are simple sequences of deterministic rules.
Competitive hebbian learning and neural gas are the most important strategies used for
their training.
Neural gas algorithm (Martinetz T.M. and Schulten K.J., 1991) sorts the network units
according to the distance of their reference vector to each input. Then the reference vec-
tors are adapted so that the ones related to the first nodes in the rank order are moved
more close than the others to the considered input.
Competitive hebbian learning (Martinetz and Schulten, 1991; Martinetz, 1993) consists
in augmenting the weight of the link connecting the two units whose reference vectors
are closest to the considered input (the two most activated units).
Both strategies are examples of deterministic rules. As we know there are other rules that
constrain the topology of the network which has a fixed dimensionality. That’s the case
of Self Organizing Maps (Kohonen, 1982) and Growing Cell Structures (Fritzke, 1994).
In other cases the network structures haven’t topological constraints, they take a well
ordered distribution by exactly adapting to the manifold inputs. For example TRN
(Martinetz and Schulten, 1994) and GNG are networks whose final structure is similar to
a Delaunay Triangulation (Delaunay, 1934).We have tried to define a new self organizing
network that is trained by probabilistic rules avoiding any topological constraints.
According to Jefferson (1995) life and evolution are structured at least into four funda-
mental levels: molecular, cellular, organism and population. We propose a population
level based on evolutionary algorithm where the network is seen as a population of units
whose interactions are conditioned by the availability of resources in their ecosystem. The
evolution of the population is driven by a selective process that favours the fittest units.
This approach has a biological plausibility. As stated by recent theories (Edelman, 1987)
human brain evolution is subjected to similar selective pressures.
Obviously we are not interested in recreating the same structure as the human brain.
Our work aims at finding innovative and effective solutions to the categorization problem
adopting natural system strategies. So our system falls within the Artificial Life field
(Langton, 1989).
Our model is a complex system that shows emergent features. In particular its structure
evolves as a scale free graph. In the training phase there arise clusters of units with a
limited number of nodes that establish a great number of links with the others.
Scale free graphs are a particular structure that is really common in natural systems.
Human knowledge, for instance, seems to be structured as a scale free graph (Steyvers,
Tenenbaum 2001). If we represent words and concepts as nodes, we’ll find that some of
these are more connected than the others.
Scale free graphs have three main features.The small world structure. It means there is a
relatively short path between any couple of nodes (Watts, Strogatz, 1998).The inherent
tendency to cluster that is quantified by a coefficient introduced by Watts and Strogatz.
Given a node i of ki degree i.e. having ki edges which connect it to ki other nodes, if those
Electronic Journal of Theoretical Physics 4, No. 14 (2007) 31–50 33
make a cluster, they can establish ki(ki-1)/2 edges at best. The ratio between the actual
number of edges and the maximum number gives the clustering coefficient of node i. The
clustering coefficient of the whole network is the average of all the individual clustering
coefficients.
Scale free graphs are also characterized by a particular degree distribution that has a
power-law tail P(k)∼k−n. That’s why such networks are called “scale free” (Albert,
Barabasi, 2000).
The three previous features are quantified by three parameters: the average path length
between any couple of nodes, the clustering coefficient and the exponent of the power
law tail. We’ll show that the values of these parameters in our model seem to confirm its
scale free nature.
2. An Outline on Self-Organization and Evolutionary Systems
Natural selection mechanism has been successfully used for a lot of industrial appli-
cations spanning from projecting to real-time control and neural networks training.
It was in the 60s that Genetic Algorithms based on the Evolution Theory’s three main
mechanisms - reproduction, mutation and fitness – were first used in dealing with op-
timization problems. Although the solution is reached by a population of individuals,
systems based on this approach are not considered self organizing because their dynam-
ics depend on the external constraint of the fitness function.
In the 80s a new approach to the study of living systems which mixed together self or-
ganization and evolutionary systems came out (Rocha, 1997). Its success was due to the
studies on the way how biological systems work (metabolism, adaptability, autonomy,
self repairing, growth, evolution etc.). The hybrid systems make us possible to get a bet-
ter simulation both of the evolutionary optimization processes and the internal structure
modification to reach a greater biological plausibility in the fitness.
Neuroevolutionary systems are an example of this approach. In classic neuroevolutionary
models the network parameters are genetically set, whereas the connection weights are
modified according to a training strategy. This solution follows the classic vision of cere-
bral development where genes control the formation of synaptic connections while their
reinforcement depends on neural activity.
More recent neuroevolutionary systems are characterized by different forms of self or-
ganizing processes which are cooperative coevolution (Paredis, 1995; Smith, Forrest and
Perelson, 1993) and synaptic Darwinism (Edelman, 1987).
Cooperative co evolutionary systems offer a promising alternative to classic evolutionary
algorithms when we face complex dynamical problems. The main difference with respect
to classic EA is the fact that each individual represents only a partial solution of the
problem. Complete solutions are obtained by grouping several individuals. The goal of
each individual is to optimize only a part of the solution, cooperating with other individ-
uals that optimize other parts of the solution. It is so avoided the premature convergence
towards a single group of individuals. An example of such approach is given by the Sym-
34 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 31–50
biotic Adaptive Neuroevolution System (Moriarty and Miikkulainen, 1998) that operates
on populations of neural networks.
While in most neuroevolutionary systems each individual represents a complete neural
network, in SANE each individual represents a hidden unit of a two-layered network.
Units are continuously combined and the resulting networks are evaluated on the basis
of the performances shown in a given task. The global effect is equal to schemas promot-
ing in standard EAs. In fact during the evolution of the population the neural schemas
having the highest fitness values are favoured and the possible mutations in the copies of
these schemas don’t affect the other copies in the population.
Other recent strategies focus on the evolution of connection schemas in the network. In
the human brain the number of synapses established by a single neuron is always much
lower than the overall number of neurons. That gives the network a sparsely connected
aspect. In the last years several models have been proposed to emulate the mechanism
involved in the selection of links without referring to the physical and chemical properties
of neurons.
The Chialvo and Bak model (Chialvo and Bak, 1999) is based on two simple and bio-
logical inspired principles. First, the neural activity is kept low selecting the activated
units by a winner takes all strategy. Second, the external environment gives a negative
feedback that inhibits active synapses if the network behaviour is not satisfying. With
these simple rules the model operates in a highly adaptive state and in critical conditions
(extreme dynamics). The fundamental difference of this strategy based on the synaptic
inhibition with respect to the classic one based on synaptic reinforcement is that the
reinforcement-based learning is a continuative process by definition, while the inhibition-
based learning stops when the training goal is achieved. The synaptic inhibition is also
biologically plausible. According to Young (Young, 1964; Young, 1966) learning is the
result of the elimination of synaptic connections (closing of unneeded channels). Dawkins
(Dawkins R., 1971) stressed that pattern learning is achieved by synaptic inhibition. As
stated by the neural groups’ selection theory developed by Edelman (Edelman, 1978;
Edelman, 1987), brain development is characterized by generating a structural and dy-
namical variability within and between populations of neurons, by the interaction of the
neural circuit with the environment and by the differential attenuation or amplification
of synaptic connections. Research in neurobiology seems to confirm the validity of the
negative feedback model and the fact that neural development follows the process of Dar-
winian evolution.
The Chialvo and Bak model is a simple two-layered network. After the training each
input pattern is associated with a single output unit leading to the formation of an as-
sociative map. When an input pattern is presented the most activated input unit i is
selected. Then the neuron j from the hidden layer that establishes the most robust con-
nection with i is selected. Finally the output neuron k that is the most strongly connected
with j is selected. If k is not the desired output the two links connecting i with j and
j with k are inhibited by a coefficient d that is the only parameter of the model. The
iterative application of these rules leads to a rapid convergence towards any input-output
Electronic Journal of Theoretical Physics 4, No. 14 (2007) 31–50 35
mapping. This selective process followed by an inhibitory one is the essence of the natural
selection in the evolutionary context. The fittest individual is selected on the basis of a
strategy that doesn’t reward the best but punishes the worst. That’s the reason why this
model has been considered a particular kind of synaptic Darwinism.
Our neuroevolutionary model is also based on a selection strategy. The structural infor-
mation of our network is not codified by genes. We directly consider the entire network
as a population of nodes that can establish connections, generate other units or die. The
probability of these events depends on the presence of local and global resources. If there
are few resources the population falls, if there is a lot of resources the population grows.
Like in the Chialvo and Bak model we don’t select the fittest nodes reinforcing their
links, but we simply remove the worst nodes when the ecosystem resources are low. This
generates a selective process that indirectly rewards the units which can better model the
input patterns. Our evolutionary strategy can be seen as a selective retention process
(Heylighen, 1992) that removes those units which cannot reach a stable state, remaining
associated with several input patterns. Even if the stability of a unit is quantified by
the minimum distortion error related to it, this information mustn’t be considered to be
environmental information. The minimum distortion error simply quantifies the difficulty
encountered by the unit during the modelling of input patterns.
3. The Evolutionary Algorithm
Research has confirmed (Roughgarden, 1979; Song and Yu, 1988) that in natural
environments the population size along with competition and reproduction rates con-
tinuously changes according to some natural resources and the available space in the
ecosystem.
These mechanisms have been reproduced in some evolutionary algorithms, for example
to optimize the evolution of a population of chromosomes in a genetic algorithm (Annun-
ziato and Pizzuti, 2000). We have tried to use a similar strategy for the evolution of a
population of units in a self organizing network without using the string representation
of genetic programming.
In our model each node is defined by a vector of neighbouring units connected to it,
a reference vector and a variable D that is the smallest distance between its reference
vector and the closest modelled input. The value of this variable quantifies the debility
degree of the unit. The lower is D the higher are the chances for the unit to survive.
At each presentation of the training input set, D is set to the maximum value. After
the presentation of a given input x, if the reference vector w of the unit is modified, the
resulting distance between the two vectors ||x-w|| is calculated. If this value is lower than
D it becomes its new value.
The training algorithm here used can be subdivided in three phases:
(1) Winners are selected. For each input the unit having the closest reference vector is
selected.
(2) The reference vectors of the winners and their neighbours are updated according to
36 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 31–50
the following formula :
w (t + 1) = w (t) + α (x − w (t)) (1)
So the reference vectors w of the selected units are moved towards the relative inputs
x of a certain fraction of the distances that separate them. For winners this fraction
is two or three orders of magnitude higher than the one used for their neighbours.
So winners have the reference vectors moving more quickly towards the inputs.
(3) The population of units evolves producing descendants, establishing new connections
and eliminating the less performing units. All these events can occur with a well
defined probability that depends on the availability of resources.
These rules are iterated until a given goal is achieved. For example the minimization of
the expected quantization error that is the mean of the distances between the winners
and the K inputs they model:
D = 1/K
K∑i=1
‖xi − wj‖ (2)
If this value falls below a certain threshold Dmin, training is stopped.
The first two phases can be considered a kind of winner takes all strategy, where only
the most activated units are selected and enabled to modify their reference vectors. The
third phase is the evolutionary phase (fig. 3.1). Each unit i, i=[1. . . N(t)] where N(t) is
the actual population size can meet the closest winner j with probability Pm:
Fig. 3.1 – The evolutionary phase of the algorithm.
If meeting occurs, the two units establish a link and they can interact by reproducing
Electronic Journal of Theoretical Physics 4, No. 14 (2007) 31–50 37
with probability Pr. In this case two new units are created. One is closer to the first
parent, the other to the second parent:
w1 =wp1+
wp1+wp22
2
w2 =wp2+
wp1+wp22
2
(3)
If reproduction doesn’t take place due to the lack of resources the weaker unit of the
population, i.e. the one with the highest debility degree, is removed.
If unit i doesn’t meet any winner it can interact with the closest node k with probability
Pr establishing a connection and producing a new unit whose reference vector is set
between the parents reference vectors:
w2 =wp1 + wp2
2(4)
When we fix a maximum population size, the ratio between the actual size and the
threshold N(t)/Nmax can be seen as a global resource of the ecosystem affecting the
probabilities of the events. For example if the population size is low the reproduction
rate should be high. So we can reasonably choose Pr = 1-N(t)/Nmax. If the population
size is high, the chance for the units to meet each other will be higher, so we can set Pm
= N(t)/Nmax.
We can also consider a local resource that is the ratio between the threshold Dmin and
the debility degree Di of the unit i. Each unit i should meet a winner with a probability
Pm=(N(t)/Nmax)(1-Dmin/Di) and Pr = 1 – Pm. In this way winners are not encouraged
to migrate to other groups of nodes and weaker units don’t participate in reproduction
activities.
We can estimate the population grow rate in the following way:
N (t + 1) = N (t) + 2PmPrN (t) − Pm (1 − Pr) N (t) + (1 − Pm) N (t) − (1 − Pm) PdN (t) =
= 2N (t) − 2P 2mN (t) =
= 2N (t)(1 − N(t)2
M2P
)⇒ X (t + 1) = 2X (t)
(1 − X (t)2) (first model)
= 2N (t)(1 − N(t)2
M2P
(1 − Dmin
D
)2)⇒ X (t + 1) = 2X (t)(1 − X (t)2 (1 − Dmin
D
)2)(second model)
(5)
where X(t) is the normalized size N(t)/Nmax. This is the quadratic-logistic map of An-
nunziato and Pizzuti(Annunziato and Pizzuti, 2000):
X (t + 1) = aX (t)(1 − X (t)2) (6)
They proved that by varying the parameter different chaotic regimes arise. For a<1.7 the
behaviour is not chaotic, for 1.7<a<2.1 we have chaotic regimes with simple attractors
localized in a fixed part of the plane of the phases. Theoretically for the first model we
expect to obtain a chaotic regime that is described by a simple attractor. In the second
38 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 31–50
model the factor (1 – Dmin/D) might reduce the influence of the negative feedback in the
final part of network training.
It is possible to demonstrate that during the evolution the population size converges to
N(t) = 0.72 Nmax. In this phase the probability that a unit establishes n connections
with the other ones for the first model, considering only clusters of n units, is given by:
P (n) =
(0.72Nmax
Nmax
)n
−0.72Nmax−1−n∑
i=1
(0.72Nmax
Nmax
)i+n
= αn−β (7)
It has to be pointed out we have subtracted the probability that such n links developed
within a cluster of more than a n unit.
The coefficients α and β of the power law are considered constant at the end of the
training. To compute their values, we can take into consideration the cases n=1 and
n=0.72Nmax-1 which correspond to the minimum and maximum number of connection
at the end of the training.
P (1) = 0.72 −0.72Nmax−2∑
i=1
0.72i+1 = α1−β = α (8)
P (0.72Nmax − 1) = 0.720.72Nmax−1 =
(0.72 −
0.72Nmax−2∑i=1
0.72i+1
)(0.72Nmax − 1)−β
⇒ β = log0.72Nmax−1
⎛⎜⎜⎝0.72 −0.72Nmax−2∑
i=1
0.72i+1
0.720.72Nmax−1
⎞⎟⎟⎠The distribution tail of the degrees tends to stretch when the maximum size of the
population increases, it means that in wider networks there are more hubs with a higher
degree.
For the second model we can consider that at the end of the training (1-Dmin/D) ∼ ε
So the probability that a unit establishes n links becomes:
P (n) =
(0.72Nmax
Nmax
ε
)n
−0.72Nmax−1−n∑
i=1
(0.72Nmax
Nmax
ε
)i+n
= αn−β (9)
⇒ β = log0.72Nmax−1
⎛⎜⎜⎝0.72ε −0.72Nmax−2∑
i=1
(0.72ε)i+1
(0.72ε)0.72Nmax−1
⎞⎟⎟⎠ ,
and the considerations made for the first model can be therefore extended to the second
model.
Electronic Journal of Theoretical Physics 4, No. 14 (2007) 31–50 39
4. Training the Net: Simulations
We have compared the performances of our networks with those of a Growing Neural
Gas in categorizing bidimensional inputs.
GNG is a self organizing network which thanks to both the competitive hebbian learning
strategy and the neural gas algorithm can categorize inputs without altering their exact
dimensionality.For the GNG, the parameters of the model aregggggα = 0.5, β = 0.0005
and at each λ = 300 steps a new unit is inserted. The maximum age of the links is set
to 88.
For the two different ENG models, the parameters are α = 0.05, β = 0.0006 and the
maximum size is set to Nmax = 120.
As stopping criterion for both the algorithms we have chosen the minimization of the
expected quantization error that is the average distance between the winners and the
corresponding inputs.
We have considered two different input domains. In the first case inputs are localized
within four square regions, in the second one inputs are uniformly distributed in a ring
region.
As shown in fig.4.1 after the training, GNG reference vectors are all positioned in the
input domain. In the Evolutionary Self Organizing Networks (fig.4.2a and fig.4.2b) some
units fall outside the input domain, but in this way the network remains fully connected.
The nodes’ distribution statistical analysis confirms what appears to be intuitively patent:
the emerging network structure is a typical scale-free one, i.e. a structure where few hubs
manage the links.
Fig. 4.1 – Growing Neural Gas simulations.
We trained 30 networks of each type obtaining the average degree distributions reported
40 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 31–50
Fig. 4.2a – Evolutionary Self Organizing NETwork simulations (first model).
Fig. 4.2b – Evolutionary Self Organizing NETwork simulations (second model).
in fig.4.3-4.5. In tab. 1 – 2 are reported the average values of the structural parameters
of the two networks.
While GNG have a high value for the average path length and a low clustering coefficient,
ENG have a short average path length and a high clustering coefficient which along with
the power law tail of the degree distribution confirm its scale free graph features.
Fig. 4.6 – 4.7 shows the population dynamics of the two ENG models. The structure
shared by the two different ENG models is due to the fact that the winner units tend to
establish the greatest number of connections. These are the favoured units with which
Electronic Journal of Theoretical Physics 4, No. 14 (2007) 31–50 41
Fig. 4.3 – Average degree distribution in GNG (two different input manifolds)
Fig. 4.4 – Average degree distribution in ENG (first model, two different input manifolds)
Fig. 4.5 – Average degree distribution in ENG (second model, two different input manifolds)
each node try to establish a connection. If the probability depends also on the local
distortion error as it happens in the second model, we obtain a final structure that is
more similar to the GNG, which is to say more similar to a gas. In point of fact, the
conditions to create a new link become more restrictive, reducing the interaction among
each cluster and the whole network. The structure of connections seems to extend more
uniformly in the regions where inputs are present as it can be seen in picture 4.2b (more
evident in the circular distribution).
Picture 4.7 shows the dynamics of the populations in the two different models of ENG.
In the first model the population size seems to converge to the final value of 0.72Nmax,
42 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 31–50
Average path length Clustering coefficient Power law exponent
GNG - 0.49 2.04
ESON (1st) 3.82 0.64 1.15
ESON(2nd) 3.92 0.63 1.14
Table 1 Comparison of structural parameters (average values, first input domain)
Average path length Clustering coefficient Power law exponent
GNG 6.4 0.42 2.98
ESON (1st) 3.61 0.58 1.11
ESON(2nd) 3.67 0.59 1.14
Table 2 Comparison of structural parameters (average values, second input domain)
confirming the experimental results of Annunziato and Pizzuti. As it can be noticed in
fig. 4.6, since the d value gradually diminishes during the training, the influence of the
factor (1−Dmin/d) grows reducing the effects of the negative feedback which character-
izes the quadratic logistic map. This justifies the sudden growth of the population at the
end of the training in the second model.
Fig. 4.6 – Network size evolution of the two ENG models (first input manifold)
At the end of training new units connect with the winner units which have a lower d, while
the subgroups of units become more isolated. Considering the function (X(t),X(t+1))
the attractor becomes more marked in the second model. This means that the system
tends to converge more toward a precise final state with a lower interaction among the
groups of units.
Electronic Journal of Theoretical Physics 4, No. 14 (2007) 31–50 43
Fig. 4.7 – population dynamics (X(t),X(t+1)) of the two ESONET models (first inputmanifold).
5. The Role of Information in Functional Specialization and
Integration
We can classify a system as complex when it is made up of different parts hetero-
geneously interacting. In addition, its behaviour and its structure have to be neither
completely casual (as it happens in a gas) nor too regular (as it happens in a crystal). In
Nature we generally observe the co-existence of functionally highly specialized integrated
areas.
That’s what happens in the brain, where different areas and groups of neurons interact
to give rise to an integrated and unitary cognitive scenario (G. M. Edelman, G. Tononi,
2000).
Edelman has introduced the integration, reciprocal information and complexity concepts
in order to mathematically define the functional organization of the cerebral structures.
Within a complex system, a subset of elements can be defined an integrated process if
– on a given temporal scale – the elements interact more strongly with each other than
with the system. In a neural net or in a self-organizing one it means that the units of an
integrated group will tend to simultaneously activate themselves.
When the units in a subset are independent, the system’s entropy reaches its maximum
value which is the sum of the entropies of the single elements (local entropies). On the
contrary, when any kind of interaction occurs, the global entropy decreases so becoming
lower than the sum of the local entropies. The integration measure is, therefore, a natural
indicator of the system informational “capacity”.
So the integration of a subset of network units can be calculated by deducting the sum of
the entropies of each single component (xi) from the entropy of the system considered as
a whole. If each unit can only take two states (activated/not-activated), the amount of
the possible activation patterns of a subset with N units is 2N . So the system maximum
entropy is:
Hmax (X) =n∑
i=1
H (xi) =n∑
i=1
pi log2
(1
pi
)=
2N
2Nlog2
(1
/1
2N
)= log2 2N = N (10)
44 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 31–50
and the integration will be:
I (X) =n∑
i=1
H (xi) − H (X) (11)
for the self-organized net here considered, the integration of a sub-group of units takes
the following expression:
I (X) = N −N−1∑i=1
⎛⎜⎝N
i + 1
⎞⎟⎠Pi log2
(1
Pi
)(12)
where Pi is the probability for a node to establish i connections. The overall number of
the system’ states is equal to the total number of possible groups of i+1 units. Groups
of units having the same dimension (groups of i+1 units) give the same contribution to
the entropy of the system.
If we choose the WTA strategy as activation modality, for each presented input only a
single unit (the winner) and the 1< i < N -1 i units will activate themselves. All the
other ones remain not-activated.
The probability for a node to create connections is ruled by the power lawPi = αk−β,
with αand βdepending on 1) the network dimension, 2) the local distortion errors (for the
second model) and 3) the particular evolution of the network structure, i.e. the dynamic
behaviour of α (t) and β (t).
So the integration of the two self-organizing network here presented is:
I (X) = N −N−1∑i=1
⎛⎜⎝N
i + 1
⎞⎟⎠(αi−β)i
log2
(1
αi−β
)i
(13)
The integration can be seen as a measure of the statistic dependency within a subset of
units. The stronger their interactions are, the higher their integration.
In order to measure the statistic dependency between a subset and the whole system,
Edelman introduced the concept of mutual information. Given an n subset made up of k
elements(Xk
n
)and its complement in the system
(X − Xk
n
), the mutual information is:
IR(Xk
n; X − Xkn
)= H
(Xk
n
)+ H
(X − Xk
n
)− H (X) (14)
The mutual information is essential to evaluate the differentiation degree of a system, i.e.
it is a significant index of the system’ “resolution” degree, calculated on the subdividable
and distinct states.
In order to measure the information of an integrated activation pattern, we calculate how
the states of a given subset can differentiate them from the whole system ones. Which
thing, following Edelman, is equivalent to considering the whole system as the observer
of itself. In fact, if entropy measures the variability of a system according to an external
observer evaluation, the mutual information measures the system variability according to
Electronic Journal of Theoretical Physics 4, No. 14 (2007) 31–50 45
an observer ideally placed within the system itself.
The overall measure of the differentiation degree of a complex system is given by the
mutual information average between each subset and the whole system:
C (X) =
N/2∑k=1
⟨IR(Xk
n; X − Xkn
)⟩(15)
Edelman defined such measure as complexity and its value is high if each subset can aver-
agely take many different states which are statistically depending on the whole system’s
ones, so it shows how the system is differentiated. High complexity values correspond
to an optimal synthesis of functional specialization and functional integration. Systems
whose elements are not integrated (such as a gas) or not specialized ( such as an homo-
geneous crystal) have a minimum complexity.
In the evolutionary neural gas case, the WTA strategy limits the integration among the
activation patterns. So the mutual information between any activation pattern and the
other possible patterns is equal to zero. It justifies the use of the term “gas”, since the
patterns behave like isles of information weakly interacting each other.
If there were selected more winner units for the same input signal in the early training
phase, we could get a given system status characterized by i+ 1 activated units not only
by the activation of just a single winner and its related i units, but also by the activation
of more winners. therefore we should also take into consideration all the possible sub-
groups with j+1 elements.
The mutual information formula between a subgroup with k activated units and the
system is given by:
H(Xkn)=
k−1∑i=1
⎛⎜⎜⎜⎜⎝k
i + 1
⎞⎟⎟⎟⎟⎠⎡⎢⎢⎢⎢⎣(αi−β)
i+
i∑j=1
⎛⎜⎜⎜⎜⎝i + 1
j + 1
⎞⎟⎟⎟⎟⎠(αj−β)j
⎤⎥⎥⎥⎥⎦ log2
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝1
(αi−β)i+
i∑j=1
⎛⎜⎜⎜⎜⎜⎜⎜⎝i + 1
j + 1
⎞⎟⎟⎟⎟⎟⎟⎟⎠(αj−β)
j
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠(16)
H(X−Xkn)=
k−2∑i=1
⎛⎜⎜⎜⎜⎝N
i + 1
⎞⎟⎟⎟⎟⎠⎡⎢⎢⎢⎢⎣(αi−β)
i+
i∑j=1
⎛⎜⎜⎜⎜⎝i + 1
j + 1
⎞⎟⎟⎟⎟⎠(αj−β)j
⎤⎥⎥⎥⎥⎦ log2
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝1
(αi−β)i+
i∑j=1
⎛⎜⎜⎜⎜⎜⎜⎜⎝i + 1
j + 1
⎞⎟⎟⎟⎟⎟⎟⎟⎠(αj−β)
j
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠+
+
⎛⎜⎜⎜⎜⎝⎛⎜⎜⎜⎜⎝
N
k
⎞⎟⎟⎟⎟⎠−1
⎞⎟⎟⎟⎟⎠⎡⎢⎢⎢⎢⎣(α(k−1)−β)
k−1+
k−1∑j=1
⎛⎜⎜⎜⎜⎝k
j + 1
⎞⎟⎟⎟⎟⎠(αj−β)j
⎤⎥⎥⎥⎥⎦ log2
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝1
(α(k−1)−β)k−1
+k−1∑j=1
⎛⎜⎜⎜⎜⎜⎜⎜⎝k
j + 1
⎞⎟⎟⎟⎟⎟⎟⎟⎠(αj−β)
j
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠+
46 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 31–50
+N−1∑i=k
⎛⎜⎜⎜⎜⎝N
i + 1
⎞⎟⎟⎟⎟⎠
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣(αi−β)
i+
i∑j=1
j �=k−1
⎛⎜⎜⎜⎜⎝i + 1
j + 1
⎞⎟⎟⎟⎟⎠(αj−β)j+
⎛⎜⎜⎜⎜⎝⎛⎜⎜⎜⎜⎝
i + 1
k
⎞⎟⎟⎟⎟⎠−1
⎞⎟⎟⎟⎟⎠(αj−β)j
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦·
log2
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
1
(αi−β)i+
i∑j=1
j �=k−1
⎛⎜⎜⎜⎜⎜⎜⎜⎝i + 1
j + 1
⎞⎟⎟⎟⎟⎟⎟⎟⎠(αj−β)
j+
⎛⎜⎜⎜⎜⎜⎜⎜⎝
⎛⎜⎜⎜⎜⎜⎜⎜⎝i + 1
k
⎞⎟⎟⎟⎟⎟⎟⎟⎠−1
⎞⎟⎟⎟⎟⎟⎟⎟⎠(αj−β)
j
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
H(X)=N−1∑i=1
⎛⎜⎜⎜⎜⎝N
i + 1
⎞⎟⎟⎟⎟⎠⎡⎢⎢⎢⎢⎣(αi−β)
i+
i∑j=1
⎛⎜⎜⎜⎜⎝i + 1
j + 1
⎞⎟⎟⎟⎟⎠(αj−β)j
⎤⎥⎥⎥⎥⎦ log2
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝1
(αi−β)i+
i∑j=1
⎛⎜⎜⎜⎜⎜⎜⎜⎝i + 1
j + 1
⎞⎟⎟⎟⎟⎟⎟⎟⎠(αj−β)
j
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠To provide the system with a greater level of complexity, in order to favouring the inte-
gration among the network unit subgroups, it is, therefore, necessary adopting a strategy
different from the WTA in the early training phases so as to select more winner units.
6. Conclusions and Future Works
The here presented self-organizing network can be considered as an example of au-
topoietic system which evolves by means of a closed network of interactions and based
upon the production of components (the categorization units). In the course of the re-
productive dynamics, those ones produce other components, also belonging to the system
(i.e. other categorization units) which maintain the system identity over time with re-
spect to the experimental task.
In particular, it has to be noticed that they are not just the environmental information to
lead the evolution of the network of connections, but rather the network internal status,
which is individuated globally by the size that the population has reached and locally by
the values of the parameters of the units. The latter show the difficulties that the units
encounter in modelling the presented input, such difficulty is directly proportional to the
amount of variations their reference vectors are subjected to.
Learning and the capability to model the system external inputs, therefore, emerges more
Electronic Journal of Theoretical Physics 4, No. 14 (2007) 31–50 47
by means of the population internal dynamics than by means of a learning algorithm.
The appearing of a scale-free structure emerging from the choice of the population dynam-
ics is peculiarly significant for the model’s biological plausibility. Which thing describes
a quite phase-transition-like status where cluster “float” as informational “isles” in a
“gaseous” configuration. It is worthy noticing that the WTA strategy and the environ-
mental noise (probabilistic laws) suffice to create a kind of basic informational skeleton
around which more interconnected functional structures can then aggregate. In the ner-
vous system, it plausibly happens according to an essentially genetic design. Such kind of
neural dynamics guarantees flexibility and redundancy to the informational nuclei which
are ready to synchronize and connect through signals. Actually, what we tried here to
describe is a proto-neural scenario with low integration of clusters which are specialized
in easy categorization tasks.
Developing the ENG model requires to investigate different synchronization scenarios
among clusters and their ensuing functional integration to execute more complex tasks.
In particular, it is necessary to modify the evolutive dynamics so as to mane the connec-
tions among units active. In this way, it should be possible to create a dynamic neural
topology susceptible of hierarchical organization.
Everything seems to confirm not only the deep reasons for the scale-free structures re-
curring in nature (Z. Toroczkai, K. E. Bassler, 2004), but also the fundamental lesson
associating complexity with a thin border zone between integration and differentiation
among the functional modules of a system.
Acknowledgements
The authors thank Eliano Pessa and Graziano Terenzi for their precious suggestions.
48 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 31–50
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50 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 31–50
EJTP 4, No. 14 (2007) 51–60 Electronic Journal of Theoretical Physics
Discrete Groups Approach to Non SymmetricGravitation Theory
N.Mebarki, F.Khelili and J.Mimouni∗
Laboratoire de Physique Mathematique et Subatomique,Mentouri University, Constantine, Algeria
Received 22 August 2006, Accepted 6 January 2007, Published 31 March 2007
Abstract: A generalized discrete group formalism is obtained and used to describe the NonSymmetric Gravity theory (NGT) coupled to a scalar field. We are able to derive explicitly thevarious terms of the NGT action including the interaction term without any ad-hoc assumptions.c© Electronic Journal of Theoretical Physics. All rights reserved.
Keywords: General Relativity, Non Commutative Geometry, Non Symmetric GravityPACS (2006): 04.20.Cv, 04.90.+e, 95.30.Sf, 02.40.Gh, 11.10.Nx
1. Introduction
During the past few years, a renewed interest in the non commutative geometry ap-
proach [1], [2], [3], [4] of the standard model and some of the grand unified theories, has
appeared among the physicists and mathematicians. The motivation is to find probable
answers to the remaining outstanding problems. One of the promising approach is the
one using the discrete groups [5], [6], [7] where it is shown that it has an intimate relation
to non commutative geometry in which the scalar particles are treated in an equal foot-
ing with the usual gauge boson. Recently, this formalism has been applied to the case
of General Relativity [8] where it was shown that the gravitational field is completely
decoupled from the scalar one.
The purpose of this paper is to generalize this approach based essentially on the
work presented in references [9],[10] , and derive explicitly the various terms of the Non
Symmetric Gravitation theory (NGT) action [11],[12],[13],[14]. In section 2 we present
the mathematical formalism, in section 3 we derive the NGT action together with the
scalar field interaction terms. Finally, in section 4 we draw our conclusions.
52 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 51–60
2. Formalism
An alternative to A.Cones’s Non Commutative Geometry [1], [2], [3], [4] is the dis-
crete groups approach [5], [6], [7] based on the algebra of 2× 2 matrices having as entries
the p-differential forms. In this formulation, a generalized product denoted by � is used
to define the structure of a Z2 graded associative algebra. Thus, the product of two
elements of this algebra is given by [8]:⎛⎜⎝ A C
D B
⎞⎟⎠�
⎛⎜⎝A′ C ′
D′ B′
⎞⎟⎠ =
⎛⎜⎝ A ∧ A′ + (−)∂C C ∧ D′ C ∧ B′ + (−)∂A A ∧ C ′
D ∧ A′ + (−)∂B B ∧ D′ B ∧ B′ + (−)∂D D ∧ C ′
⎞⎟⎠ (1)
where A, B, C D, A′, B′, C′, D′ are p-forms, ∂ stands for degree of these p-forms, and ∧denotes the exterior product.
One can also define a nilpotent differential operator d satisfying a generalized Leibnitz
rule as follows [8]:
dX = d
⎛⎜⎝A C
D B
⎞⎟⎠ =
⎛⎜⎝ dA + C + D −dC − (A − B)
−dD + (A − B) dB + C + D
⎞⎟⎠d (X � X ′) = dX � X ′ + (−)∂X X � dX ′
(2)
This formulation was applied to describe the Einstein-Hilbert action with a minimal
coupling of the gravitation with scalar fields [8].
Concerning NGT, one can define the following generalized spin connection Ωab:
Ωab =
⎛⎜⎝ωab φab
φab
ωab
⎞⎟⎠ , a = { i = 1, 2, , n
.a = n + 1, , N
(3)
where ωab and ωab(resp.φab and φ
ab)are the generalized hyperbolic complex 1-forms
(resp.0-forms)where their components in the holonomic basis{ei, i = 1, n
}are given by:
ωab = ωabμ dXμ , ωab = ωab
μ dXμ , dXμ = Eμi ei (4)
and the generalized vierbein is defined as:
Eμi =
⎛⎜⎝ 0 eμi
eμi 0
⎞⎟⎠Here eμ
i is the hyperbolic complex and eμi its hyperbolic complex conjugate
eμi = αμ
i + εβμi , ε = −ε, ε2 = 1
eμi = αμ
i − εβμi , αμ
i , βμi ∈ C∞
R (X)(5)
Electronic Journal of Theoretical Physics 4, No. 14 (2007) 51–60 53
A generalized orthonormal basis can be defined such that:
ξa =
⎛⎜⎝ ρa sa
−sa ρa
⎞⎟⎠ , a = { i = 1, 2, , n
.a = n + 1, , N
(6)
where (resp.sa and sa) ρa is a 1-form (resp.0-forms) given by:
ρi = eiμdXμ , i = 1, 2, , n
ρ.a = 0 ,
.a = n + 1, , N
(7)
si = 0, si = 0, i = 12, , n
s.a = Mλ
.a, s
.a = Mλ
.a,
.a = n + 1, , N
(8)
with eμj is the inverse of the vierbein verifying:
eiμe
μj = δi
j , eiμe
νi = δν
μ (9)
and M , M are the following 2 × 2 matrices:
M =
⎛⎜⎝ 0 1
0 0
⎞⎟⎠ ,M =
⎛⎜⎝ 0 0
1 0
⎞⎟⎠ (10)
here λ.a and its hyperbolic complex conjugate λ
.a are arbitrary functions.
The exterior product and differential operator for the generalized spin connection
components are defined by:
ωab ∧ ωcd = ωabμ ωcd
ν dXμ ∧ dXν = E[μi E
ν]j ωab
μ ωcdν .ei.ej
ωab ∧ ϕcd = Eμi Mωab
μ ϕcdei
ϕab ∧ ωcd = MEμi ϕabωcd
μ
ϕab ∧ ϕcd = MMϕabϕcd
(11)
and
dωab = d(ωab
μ dXμ)
=(dωab
μ
)dXμ = ∂μω
abν dXμ ∧ dXν
dφab = ∂μφabdXμ = MEμ
i ∂μϕab ei
dφab
= ∂μφab
dXμ = Eμi M∂μϕ
ab ei
(12)
with:
φab = Mϕab , φab
= Mϕab (13)
Now imposing the unitarity condition:(Ωab)∗
= Ωba (14)
54 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 51–60
where * is an involution such that:(ei)∗
= −ei , (dXμ)∗ = −dXμ (15)
we obtain the following constraints:
ωabμ = −ωba
μ , ωab
μ = −ωbaμ
ϕab = ϕba , ϕab
= ϕba(16)
As for the 2-form curvature Rab, it is given by [8]:
Rab = dΩab + Ωac � Ωcb
Straightforward calculations lead to:
Rab11 = dωab + ωac ∧ ωcb + φacφ
cb+ φab + φ
ab= Rab + τϕacϕcb + Mϕab + Mϕab
Rab12 = −dφab + φacωcb − ωacφcb − (ωab − ωab
)= −∇φab − (ωab − ωab
)Rab
21 = −dφab
+ φac
ωcb − ωacφcb
+(ωab − ωab
)= −∇φ
ab − (ωab − ωab)
Rab22 = dωab + ωac ∧ ωcb + φ
acφcb + φab + φ
ab= R
ab+ τϕacϕcb + Mϕab + Mϕab
with
τ =
⎛⎜⎝ 1 0
0 0
⎞⎟⎠ τ =
⎛⎜⎝ 0 0
0 1
⎞⎟⎠and
∇φab = ejeμj∇μϕ
abτ − ejeμj ω
acμ ϕcbτ3
∇μϕab = ∂μϕ
ab − ϕacωcbμ + ωac
μ ϕcb
∇φab
= ej eμj∇μϕ
abτ + ej eμj ω
acμ ϕcbτ3
∇μϕab = ∂μϕ
ab + ωacμ ϕcb − ϕacωcb
μ
It is worth mentioning that Rab and Rab
have the following expressions:
Rab =(∂μω
abν + ωac
μ ωcbν
)dXμ ∧ dXν = 1
2Rab
μνdXμ ∧ dXν
Rab
=(∂μω
abμ + ωac
μ ωcbν
)dXμ ∧ dXν = 1
2R
ab
μνdXμ ∧ dXν
Rabμν = ∂μω
abν + ωac
μ ωcbν − (μ ↔ ν) = −Rab
νμ
Rab
μν = ∂μωabν + ωac
μ ωcbν − (μ ↔ ν) = −R
ab
νμ
The torsion is defined by[8]:
T a = dξa + Ωab � ξb (17)
Using the fact that:
dXμ ∧ dXν = E[μi E
ν]j ei.ej =
[ημνij e[i.ej] + εgμν
ij e(i.ej)τ3
]where ημν
ij and gμνij are the real and imaginary parts of the product eμ
i eνj that is:
Gμνij = eμ
i eνj = ημν
ij − εgμνij (18)
Electronic Journal of Theoretical Physics 4, No. 14 (2007) 51–60 55
with ε is a pur imaginary hyperbolic complex number (ε2 = 1) and τ3 is the usual Pauli
matrix:
τ3 =
⎛⎜⎝ 1 0
0 −1
⎞⎟⎠the notations () and []mean symmetric and antisymmetric parts respectively. Direct
simplifications lead to:
(T a)11 = dρa + ωab ∧ ρb − φabsb + sa − sa
=(∂μρ
aν + ωab
μ ρbν
)dXμ ∧ dXν − τϕabλb + sa − sa
(T a)22 = dρa + ωab ∧ ρb + φab
sb + sa − sa
=(∂μρ
aν + ωab
μ ρbν
)dXμ ∧ dXν + τϕabλb + sa − sa
(T a)12 = −dsa + φabρb − ωabsb = −ejeμj
(∂μλ
a − ϕabρbμ
)τ − ejeμ
j ωabμ λbτ
(T a)21 = dsa + φab
ρb + ωabsb = ej eμj
(∂μλ
a − ϕabρbμ
)τ + ej eμ
j ωabμ λbτ
The components of T i are given by:
(T i)11 =(ημνkl e[k.el] + εgμν
kl e(k.el)τ3
) (∂μe
iν + ωij
μ ejν
)− τϕi.aλ
.a
(T i)22 =(ημνkl e[k.el] + εgμν
kl e(k.el)τ3
) (∂μe
iν + ωij
μ ejν
)+ τϕi
.aλ
.a
(T i)12 = .ejϕijτ − ejeμj ω
i.aμ λ
.aτ
(T i)21 = ejϕijτ + ej eμj ω
i.aμ λ
.aτ
while those of T.a are:(
T.a)
11=(ημνkl e[k.el] + εgμν
kl e(k.el)τ3
)ω
.akμ ek
ν − τϕ.a
.bλ
.b + Mλ
.a − Mλ
.a(
T.a)
22=(ημνkl e[k.el] + εgμν
kl e(k.el)τ3
)ω
.akμ ek
ν + τϕ.a
.bλ
.b + Mλ
.a − Mλ
.a(
T.a)
12= −ejeμ
j
(∂μλ
.a − ϕ
.akek
μ
)τ − ejeμ
j ω.a
.b
μ λ.bτ(
T.a)
12= −ej eμ
j
(∂μλ
.a+ ϕ
.akek
μ
)τ + ej eμ
j ω.a
.b
μ λ.bτ
3. The NGT Action
If one defines the scalar product (·, ·) as:
(X, Y ) =
∫∗tr (X � Y ) =
∫ √eed4xtrXi1...iP ,Yj1...jq ∗
(ei1 . . . eip
) (ej1 . . . ejq
)(19)
where X = Xi1...iP ei1 . . . eip and Y = Yj1...jqej1 . . . ejq ,and ∗ is the Hodge star operator
verifying the following equations:
∗ (ei.ej) = −δij = ∗e(i.ej)
∗ (ei.ej.ek.el)
= δijδkl − δikδjl + δilδjk
∗ (ei.ej.e(k.el))
= δijδkl + δikδjl + δilδjk
∗ (ei) = 0 = ∗ (ej1 . . . .ej2k+1) = 0
∗ (1) = 0 , ∗e[i.ej] = 0
then the NGT action takes the form:
I =1
2
∫ √eed4x ∗ Tr
[Ea � Eb∗ − Eb∗ � Ea
]� Rba (20)
56 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 51–60
where Ea are given by:
Ea =
⎛⎜⎝ τρa Mλa
−Mλa τρa
⎞⎟⎠ , a = { i = 1, 2, , n
.a = n + 1, , N
(21)
that is:
Ei =
⎛⎜⎝Mei 0
0 Mei
⎞⎟⎠ , E.a =
⎛⎜⎝ 0 Mλ.a
−Mλ.a
0
⎞⎟⎠After a direct calculation we obtain:
I = I(1) + I(2) (22)
with
I(1) =
∫ √eed4x
(−Gμν
(Rμν + Rμν
)+
1
2
(ϕiaϕai − ϕiaϕai
))(23)
I(2) = −1
2
∫ √eed4x{λ
.a
eμi
(∂μϕ
.ai + ω
.abμ ϕbi − ϕ
.abωbi
μ
)(24)
+λ.aeμ
i
(∂μϕ
i.a + ωib
μ ϕb.a − ϕibωb
.a
μ
)}
Now, in order to get dynamical fields, we impose the following weak torsionless con-
ditions: ⎛⎜⎝ 0 M
−M 0
⎞⎟⎠� T i = 0 (25)
and
Tr (τ3 ⊗ 1) � T.a = 0
Here Tr denotes the trace over the 2 × 2 matrices algebra.
After some straightforward simplifications, the action becomes (see Appendix A):
I =
∫ √eed4xL
where
L = L(1)+L(2)+L
with
L(1) = Gμν(Rμν + Rμν
)= 2GμνRμν
L(2) = −12
(ϕiaϕai − ϕiaϕai
)= 0
and
L(3) = 12λ
.aeμi
(∂μϕ
.ai + ω
.abμ ϕbi − ϕ
.abωbi
μ
)+ 1
2λ
.aeμ
i
(∂μϕ
i.a + ωib
μ ϕb.a − ϕibωb
.a
μ
)Note that Gμν = eμ
i eνi is the NGT metric.
Setting λ = exp (εΦ), we get:
Electronic Journal of Theoretical Physics 4, No. 14 (2007) 51–60 57
L = GμνRμν + 12G(μν)WμWν − 1
2G[μν]∂νWμ + 1
2G(μν)∂μΦ∂νΦ − G(μν)εWμ∂νΦ
Notice that one can also add the following cosmological term J (see Appendix B):
J =1
2
∫∗Tr[Ea � Eb∗ − Eb∗ � Ea
]� (ξb � ξa∗) (26)
which may be also written as:
J = −∫ √
eed4x(−2G[μν]G[μν] − 4λλ − 8 − Gνμ
ji Gjiμν
)(27)
4. Conclusions
We have shown that we can consistently generalize the discrete groups formalism
to the case of Non Symmetric Gravitation theory, and have obtained in the process a
Lagrangian density containing the pure NGT action an interaction term, as well as the
kinetic term for the scalar field Φ. Thus, the various terms that Moffat has introduced
by hand for mere physical consistency, are here seen to be the result of the generalized
discrete group approach. Moreover, a dynamical scalar field was found to be also neces-
sary in this formalism, but contrary to General Relativity, it couples to the gravitational
field (term proportional to G(μν)εWμ∂νΦ).
Appendix A
In order to get dynamical fields, we impose a weak torsionless condition:⎛⎜⎝ 0 M
−M 0
⎞⎟⎠� T i = 0 , T rτ
((τ3 ⊗ 1) � T
.a)
= 0
where Trτ denotes the trace over M2 (K) ( M, M ,τ ,τ , τ3).
We thus get the following constraints:
ωi.aμ = ωi
.aμ = 0(
∂μeiν + ωij
μ ejν
)= 0(
∂μeiν + ωij
μ ejν
)= 0
∂μλ.a − ϕ
.akek
μ − ω.a
.b
μ λ.b = 0
∂μλ.a + ϕ
.akek
μ − ω.a
.b
μ λ.b = 0
Consequently we obtain:
Rijμν = R
ij
μν
Rμν = Rμν
Now by imposing also that Tr (T i) = 0, we get:
λ.aϕi
.a = λ
.aϕi
.a
which implies:
ϕi.aϕ
.ai − ϕi
.aϕ
.ai = 0
Using the fact that:
58 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 51–60
ϕijϕji − ϕijϕji = 0
we obtain:
ϕiaϕai − ϕiaϕai = 0
and thus
L(2) = −12
(ϕiaϕai − ϕiaϕai
)= 0
By taking into account the above constraints, L(3) takes the form:
L(3) = 12λeμ
i
(∂μϕ
5i − ω55μ ϕ5i − ϕ5jωji
μ
)+ 1
2λeμ
i
(∂μϕ
i5 + ωijμ ϕj5 − ϕi5ω55
μ
)Putting Wμ = ω55
μ , Wμ = ω55μ = −Wμ, and using the compatibility condition:
∇μeσi = ∂μe
σi − ωji
μ eσj + W σ
αμeαi = 0
we end up with:
2L(3) = λGσμ∂μ (∂σ (λ − Wσ) λ) − Wσλ + GνμW σνμλ (∂μ − Wσ) λ
−λGσμWμ (∂σλ − Wσλ) + h.c.c
where here h.c.c. means hyperbolic complex conjugate.
Using the parametrization λ = exp (εΦ) , L(3)becomes:
L(3) = G(μν)WμWν − G[μν]∂νWμ + G(μν)∂μΦ∂νΦ − 2G(μν)εWμ∂νΦ
Finally we get for the action I :
I =
∫ √eed4xL
with
L = 2GμνRμν + G(μν)WμWν − G[μν]∂νWμ + G(μν)∂μΦ∂νΦ − 2G(μν)εWμ∂νΦ
Appendix B
The cosmological term can be obtained from the following expression:
J = 12
∫ ∗Tr[Ea � Eb∗ − Eb∗ � Ea
]� (ξb � ξa∗) = 12
(J(1)−J(2)
)where:
J(1)=∫ ∗Tr{(Ea � Eb∗ − Eb∗ � Ea
)11∧ (ξb � ξa∗)
11
+(Ea � Eb∗ − Eb∗ � Ea
)22∧ (ξb � ξa∗)
22}
and
J(2) =∫ ∗Tr{(Ea � Eb∗ − Eb∗ � Ea
)12∧ (ξb � ξa∗)
21
+(Ea � Eb∗ − Eb∗ � Ea
)21∧ (ξb � ξa∗)
12}
Straightforward calculations give:
J(1)=2∫ ∗Tr{(Ei � Ej∗ − Ej∗ � Ei)11 ∧ (ξj � ξi∗)11
= −2∫ √
eed4x(GνμGμν − Gνμ
ij Gjiμν − 12
)and:
J(2) =∫ ∗Tr{
((E
.a � Ei∗
)−(Ei∗ � E
.a))
12∧ (ξi � ξ
.a∗)21
+((
E.a � Ei∗
)−(Ei∗ � E
.a))
21∧ (ξb � ξa∗)12
+((Ei � E.b∗) − (E
.b∗ � Ei))21 ∧ (ξ
.b � ξi∗)12
+((Ei � E.b∗) − (E
.b∗ � Ei))12 ∧ (ξ
.b � ξi∗)21}
Electronic Journal of Theoretical Physics 4, No. 14 (2007) 51–60 59
J(2) = −8∫ √
eed4xλ.aλ
.a
Finally we obtain:
J = − ∫ √eed4x(GνμGμν − Gνμ
ij Gjiμν − 12 − 4λ
.aλ
.a)
= − ∫ √eed4x(−2G[μν]G[μν] − 4λλ − 8 − Gνμ
ji Gjiμν
)
60 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 51–60
References
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[2] A. Connes,in ” Essay on Physics and Non-commutative Geometry”, The Interfaceof Mathematics and Particle Physics, Clarendon Press, Oxford (1990).
[3] D.Kastler, T.Schucker, J.Geom.Phys. 24 (1997)61.
[4] A.H.Chamseddine, G.Felder, J.Frohlich, Nucl. Phys. B395 (1993) 672
[5] A.Sitarz, J. Geom. Phys. 15 (1995) 123.
[6] R.Coquereaux, G.Esposito-Farese, G.Vaillant, Nucl.Phys. B 353, (1991) 689.
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[10] N.Mebarki, F.Khelili, J.Mimouni, in ” Extended Non Symmetric Gravitation Theorywith a Scalar Field in Non Commutative Geometry”, Mentouri Univ. Preprints August2006.
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[12] J.Legare, J.W.Moffat,in ”Field Equations and Conservation Laws in theNonsymmetric. Gravitational Theory” arXiv:gr-qc/9412009
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EJTP 4, No. 14 (2007) 61–68 Electronic Journal of Theoretical Physics
Quantization of the Scalar Field Coupled Minimallyto the Vector Potential
W. I. Eshraim1∗ and N. I. Farahat2†
Department of PhysicsIslamic University of Gaza
P.O.Box 108, Gaza, Palestine
Received 6 July 2006, Accepted 16 August 2006, Published 31 March 2007
Abstract: A system of the scalar field coupled minimally to the vector potential is quantized byusing canonical path integral formulation based on Hamilton-Jacobi treatment. The equation ofmotions are obtained as total differential equation and the integrability conditions are examined.c© Electronic Journal of Theoretical Physics. All rights reserved.
Keywords: Hamilton-Jacobi Formalism, Path Integral Quantization, Constrained SystemsPACS (2006): 11.10.Ef, 03.65.-w, 11.10.z, 31.15.Kb
1. Introduaction
Dirac approach [1,2] is widely used for quantizing the constrained Hamilton systems.
The path integral is another approach used for the quantization of constrained systems
of classical singular theories which is initiated by Faddeeve [3]. Faddeeve has applied this
approach when only first-class constraints in the canonical gauge are present. Senjanovic
[4] generalized Faddeev’s method to second-class constraints. Fradkin and Vilkovisky
[5,6] rederived both results in a broader context, where they improved procedure to the
Grassman variables. Gitman and Tyutin [7] discussed the canonical quantization of sin-
gular theories as well as the Hamiltonian formalism of gauge theories in an arbitrary
gauge.
The Hamilton-Jacobi approach [8-10] is most powerful approach for treating con-
strained systems. The equations of motion for singular system are obtained as total
differential equations in many variables. The integrability conditions for the system lead
us to obtain the canonical reduced phase-space coordinates without using any fixing con-
∗ wibrahim−[email protected]† [email protected]
62 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 61–68
ditions . Muslih and Guler’s have constructed the desired path integral in the context of
canonical formalism [11-14], which is based on the Hamilton-Jacobi approach.
In this paper, we shall treat the scalar field coupled minimally to the vector potential
as constrained system. The path integral quantization is obtained using both Hamilton-
Jacobi approach and Faddeeve approach and the results are compared.
2. Path Integral Formulation
In this section, we briefly review the Faddeeve method and the Hamilton-Jacobi
method for studying the path integral for constrained systems.
2.1 Fadeeve Pop Method
Consider a mechanical system with n degrees of freedom and having α first-class con-
straints φα, but no second-class constraints, Fadeeve has formulated the transition am-
plitude as [3]
〈Out | S | In〉 =
∫exp
[i
∫ ∞
−∞(piqi − H0) dt
]∏t
dμ(qi(t), pi(t)), (1)
where H0 is the Hamiltonian of the system. The measure of integration is defined by
dμ(q, p) =
(α∏
a=1
δ(χa)δ(φa)
)det||{χa, φa}||
n∏i=1
dpi dqi. (2)
and χa(pi, qi) are the gauge-fixing condition with
1. {χa, χa′} = 0,
2. det||{χa, φa}|| = 0.
2.2 Hamilton-Jacobi Path Integral Quantization
One starts from singular Lagrangian L ≡ L(qi, qi, τ), i = 1, 2, . . . , n, with the Hess matrix
Aij =∂2L(qi, qi, τ)
∂qi ∂qji, j = 1, 2, . . . , n, (3)
of rank (n − r), r < n. Then r momenta are dependent. The generalized momenta pi
corresponding to the generalized coordinates qi are defined as
pa =∂L
∂qa, a = 1, 2, . . . , n − r, (4)
pμ =∂L
∂qμ, μ = n − r + 1, . . . , n. (5)
The singular value of the system enables us to solve Eq.(4) for qa as
qa = qa(qi, qμ, pb; τ) ≡ wa. (6)
Electronic Journal of Theoretical Physics 4, No. 14 (2007) 61–68 63
Substituting Eq. (6), into Eq. (5), we get
pμ =∂L
∂qμ
∣∣∣∣qa≡ωa
≡ −Hμ(qi, qμ, pa; τ). (7)
Relations (7) indicate the fact that the generalized momenta Pμ are independent of Pa
which is a natural result of the singular nature of the Lagrangian.
The canonical Hamiltonian H0 is defined as
H0 = −L(qi, qμ, qa ≡ wa; τ) + paqa + Pμqμ
∣∣∣∣pμ=−Hμ
. (8)
The set of Hamilton-Jacobi Partial Differential Equations (HJPDE) is expressed as
H ′α
(τ, qμ, qa, pi =
∂S
∂qi, p0 =
∂S
∂τ
)= 0, α = 0, n − p + 1, . . . , n, (9)
where
H ′α = pα + Hα , (10)
The equations of motion are obtained as total differential equations in many variables
as follows:
dqr =∂H ′
α
∂pr
dtα, r = 0, 1, . . . , n, (11)
dpa = −∂H ′α
∂qadtα, a = 0, . . . , n − p, (12)
dpμ = −∂H ′α
∂qμdtα, α = 0, n − p + 1, . . . , n, (13)
dZ =
(− Hα + pa
∂H ′α
∂pa
)dtα, (14)
where Z = S(tα, qa) being the action. The set of Eqs. (11-14) are integrable if
dH ′α = 0, α = 0, n − p + 1, . . . , n. (15)
If conditions (15) are not satisfied identically, one may consider them as new constraints
and a gain test the integrability conditions, then repeating this procedure, a set of con-
ditions may be obtained.
In this case the path integral representation may be written as [11-14].
〈Out | S | In〉 =
∫ n−r∏a=1
dqadpa exp
[i
∫ t′α
tα
(−Hα + pa
∂H ′α
∂pa
)dtα
], (16)
One should notice that the integrate (16) is an integration over the canonical phase-space
coordinates qa, pa.
64 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 61–68
3. The Scalar Field Coupled Minimally to the Vector Potential
Consider the action integral for the scalar field coupled minimally to the vector
potential as
S =
∫d4x L, (17)
where the Lagrangian L is given by
L = −1
4Fμν(x)F μν(x) + (Dμϕ)∗(x)Dμϕ(x) − m2ϕ∗(x)ϕ(x), (18)
where
F μν = ∂μAν − ∂νAμ, (19)
and
Dμϕ(x) = ∂μϕ(x) − ieAμ(x)ϕ(x). (20)
Let us first discuss the system using Hamilton-Jacobi approach. In this approach the
canonical momenta (4) and (15) take the forms
πi =∂L
∂Ai
= −F 0i, (21)
π0 =∂L
∂A0
= 0, (22)
pϕ =∂L
∂ϕ= (D0ϕ)∗ = ϕ∗ + ieA0ϕ
∗, (23)
pϕ∗ =∂L
∂ϕ∗= (D0ϕ) = ϕ − i e A0 ϕ, (24)
From Eqs. (21), (23) and (24), the velocities Ai, ϕ∗ and ϕ can be expressed in terms
of momenta πi, pϕ and pϕ∗ respectively as
Ai = −πi − ∂iA0, (25)
ϕ∗ = pϕ − ieA0ϕ∗, (26)
ϕ = pϕ∗ + ieA0ϕ. (27)
The canonical Hamiltonian H0 is obtained as
H0 =1
4F ijFij − 1
2πiπ
i + πi ∂iA0 + pϕ∗pϕ + ieA0ϕpϕ
− ieA0ϕ∗pϕ∗ − (Diϕ)∗(Diϕ) + m2ϕ∗ϕ. (28)
Making use of (9) and (10), we find for the set of HJPDE
H ′0 = π4 + H0, (29)
Electronic Journal of Theoretical Physics 4, No. 14 (2007) 61–68 65
H ′ = π0 + H = π0 = 0, (30)
Therefore, the total differential equations for the characteristic (11-13) are obtained as
dAi =∂H ′
0
∂πi
dt +∂H ′
∂πi
dA0,
= −(πi + ∂iA0) dt, (31)
dA0 =∂H ′
0
∂π0
dt +∂H ′
∂π0
dA0 = dA0, (32)
dϕ =∂H ′
0
∂pϕ
dt +∂H ′
∂pϕ
dA0,
= (pϕ∗ + ieA0ϕ) dt, (33)
dϕ∗ =∂H ′
0
∂pϕ∗dt +
∂H ′
∂pϕ∗dA0,
= (pϕ − ieA0ϕ∗) dt, (34)
dπi = −∂H ′0
∂Ai
dt − ∂H ′
∂Ai
dA0,
= [∂lFli + ie(ϕ∗∂iϕ + ϕ ∂iϕ
∗) + 2e2Aiϕϕ∗] dt, (35)
dπ0 = −∂H ′0
∂A0
dt − ∂H ′
∂A0
dA0,
= [∂iπi + ieϕ∗pϕ∗ − ieϕ pϕ] dt, (36)
dpϕ = −∂H ′0
∂ϕdt − ∂H ′
∂ϕdA0,
= [(−→D · −→Dϕ)∗ − m2ϕ∗ − ieA0pϕ] dt, (37)
and
dpϕ∗ = −∂H ′0
∂ϕ∗dt − ∂H ′
∂ϕ∗dA0,
= [(−→D · −→Dϕ) − m2ϕ + ieA0pϕ∗ ] dt.
(38)
The integrability condition (dH ′α = 0) implies that the variation of the constraint H ′
should be identically zero, that is
dH ′ = dπ0 = 0, (39)
which lead to a new constraint
H ′′ = ∂iπi + ieϕ∗pϕ∗ − ieϕ pϕ = 0. (40)
66 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 61–68
Taking the total differential of H ′′, we have
dH ′′ = ∂idπi + iepϕ∗dϕ∗ + ieϕ∗dpϕ∗ − ieϕ dpϕ − iepϕ dϕ = 0. (41)
Then the set of equations (31-38) is integrable. Therefore, the canonical phase space
coordinates (ϕ, pϕ) and (ϕ∗, pϕ∗) are obtained in terms of parameters (t, A0).
Making use of Eqs.(14) and (28-30), one gets the canonical action integral as
Z =
∫d4x(−1
4〉F ijFij − 1
2〉πiπ
i + pϕpϕ∗ +−→Dϕ∗ · −→
Dϕ + m2|ϕ|2), (42)
where −→D =
−→� + ie−→A. (43)
Now the path integral representation (16) is given by
〈out|S|In〉 =
∫ ∏i
dAi dπi dϕ dpϕ dϕ∗ dpϕ∗〉exp
[i
{∫d4x
(−1
2πiπ
i − 1
4F ijFij + pϕpϕ∗ + (Diϕ)∗(Diϕ) − m2ϕ∗ϕ)
}]. (44)
To apply the Faddeeve method to the pervious system, we start with the total Hamil-
tonian
HT =1
4F ijFij − 1
2πiπ
i + πi ∂iA0 + pϕ∗pϕ + ieA0ϕpϕ
− ieA0ϕ∗pϕ∗ − (Diϕ)∗(Diϕ) + m2ϕ∗ϕ + λπ0. (45)
According to Dirac’s method, the time derivative of the primary constraints should be
zero, that is
H ′ = {H ′, HT} = ∂iπi + ieϕ∗pϕ∗ − ieϕ pϕ ≈ 0, (46)
which leads to the secondary constraints
H ′′ = ∂iπi + ieϕ∗pϕ∗ − ieϕ pϕ ≈ 0. (47)
There are no tertiary constraints, since
H ′′ = {H ′′, HT} = 0. (48)
By taking suitable linear combinations of constraints, one has to find the first-class one,
that is
Φ = H ′ = π0. (49)
The equations of motion read as
Ai = {Ai0, HT} = −(πi + ∂iA0), (50)
A0 = {A0, HT} = λ, (51)
Electronic Journal of Theoretical Physics 4, No. 14 (2007) 61–68 67
ϕ = {ϕ,HT} = (pϕ∗ + ieA0ϕ), (52)
ϕ∗ = {ϕ∗, HT} = (pϕ − ieA0ϕ∗), (53)
πi = {πi, HT} = ∂lFli + ie(ϕ∗∂iϕ + ϕ∂iϕ
∗) + 2e2Aiϕϕ∗, (54)
π0 = {π0, HT} = ∂iπi + ieϕ∗pϕ∗ − ieϕ pϕ, (55)
pϕ = {pϕ, HT} = (−→D · −→Dϕ)∗ − m2ϕ∗ − ieA0pϕ, (56)
pϕ∗ = {pϕ∗ , HT} = (−→D · −→Dϕ) − m2ϕ + ieA0pϕ∗ . (57)
We will contact ourselves with a partial gauge fixing by introducing gauge constraints
for the first-class primary constraints only, just to fix the multiplier λ in Eq.(45). Since
there are weakly vanishing, a gauge choice near at hand would be:
φ′ = A0 = 0. (58)
But for this forbids dynamic at all, since the requirement A0 = 0 implies λ = 0.
Making use of Eq.(1), we obtain the path integral quantization
〈out|S|In〉 =
∫exp
[i
∫ +∞
−∞(−1
2〉πiπ
i − 1
4〉F ijFij + pϕpϕ∗
+−→Dϕ∗ · −→Dϕ − m2|ϕ|2
]d4x dAi dπi dϕ dpϕ dϕ∗ dpϕ∗ . (59)
We showed that Eq.(44) and Eq.(59) are identical.
4. Conclusion
Path integral quantization of the scalar field coupled minimally to the vector potential is
obtained by using the canonical path integral formulation [11-14]. The integrability con-
ditions dH ′0 and dH ′ are satisfied, the system is integrable, hence the path integral is ob-
tained directly as an integration over the canonical phase space coordinatesAi, πi, ϕ, Pϕ, ϕ∗
and pϕ∗ without using any gauge fixing conditions.
The Hamilton-Jacobi quantization is simpler and more economical. Also there is no
need to distinguish between first and second-class constraints, and there is no need to
introduce Lagrange multipliers; all that is needed is the set of Hamilton-Jacobi partial
differential equations and the equations of motion. If the system is integrable then one
can construct the canonical phase space.
68 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 61–68
References
[1] P.A.M. Dirac, lectures of Quantum Mechanics, Yeshiva University Press, New york(1964).
[2] P.A.M. Dirac, Can J. Math. 2, 129 (1950).
[3] L.D. FADDEEV, Teoret. Mat. Fiz. 1, 3 (1969)[Theor. Math. Phys. 1, 1 (1970)].
[4] P. Senjanovic, Ann. Phys (NY) 100, 227 (1976).
[5] E. S. Fradkin and G. A Vilkovisky, Phys. Rev. D8,4241 (1973).
[6] E. S. Fradkin and G. A Vilkovisky, Phys. Lett. B55,241 (1975).
[7] D. M. Gitman and I.V. Tyutin, Quantization of Fields with constraints, Springs verlag,Berlin (1990).
[8] Y. Guler, Nuovo Cimento B107, 1389 (1992).
[9] Y. Guler, Nuovo Cimento B107, 1143 (1992).
[10] N. Farahat and Y. Guler, Nuovo Cimento B111, 513 (1996).
[11] S. I. Muslih and Y. Guler, Nuovo Cimento B113, 277 (1998).
[12] S. I. Muslih and Y. Guler, Nuovo Cimento B112, 97 (1997).
[13] S. I. Muslih, Nuovo Cimento B115, 1 (2000).
[14] S. I. Muslih, Nuovo Cimento B115, 7 (2000).
[15] S. I. Muslih, Nuovo Cimento B117, 4 (2002).
[16] S. I. Muslih, Mod. Phys. Lett. A A19, 151 (2004).
EJTP 4, No. 14 (2007) 69–86 Electronic Journal of Theoretical Physics
A Generalized Option Pricing Model
J. P. Singh∗
Department of Management StudiesIndian Institute of Technology Roorkee
Roorkee 247667, India
Received 6 December 2006, Accepted 6 January 2007, Published 31 March 2007
Abstract: The Black Scholes model of option pricing constitutes the cornerstone ofcontemporary valuation theory. However, the model presupposes the existence of severalunrealistic assumptions including the lognormal distribution of stock market price processes.There, now, subsists abundant empirical evidence that this is not the case. Consequently,several generalisations of the basic model have been attempted with relaxation of some ofthe underlying assumptions. In this paper, we postulate a generalization that contemplatesa statistical feedback process for the stochastic term in the Black Scholes partial differentialequation. Several interesting implications of this modification emanate from the analysis andare explored.c© Electronic Journal of Theoretical Physics. All rights reserved.
Keywords: Econophysics, Stochastic Processes, Financial Markets, Black Scholes Model, OptionPricing ModelPACS (2006): 89.65.s, 89.65.Gh, 02.50.Ey, 05.40.a
1. Introduction
With the rapid advancements in the evolution and study of disordered systems and
the associated phenomena of nonlinearity, chaos, self organized criticality etc., the impor-
tance of generalizations of the extant mathematical apparatus to enhance its domain of
applicability to such disordered systems is cardinal to the further development of science.
A possible mechanism for achieving this objective is through deformation of standard
mathematics.
A considerable amount of work has already been done and success achieved in the
broad areas of q-deformed harmonic oscillators [1], representations of q-deformed rotation
and Lorentz groups [2-3]. q-deformed quantum stochastic processes have also been studied
∗ [email protected] and Jatinder [email protected]
70 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 69–86
with realization of q-white noise on bialgebras [4], deformations of the Fokker Planck’s
equation [5], Langevin equation [6] and Levy processes [7-8] have also been analysed and
results reported.
Though at a nascent stage, the winds of convergence of physics and finance are unmis-
takably perceptible with several concepts of fundamental physics like quantum mechanics,
field theory and related tools of non-commutative probability, gauge theory, path integral
etc. being applied for pricing of contemporary financial products and for explaining var-
ious phenomena of financial markets like stock price patterns, critical crashes etc [8-19].
The celebrated Black Scholes formula [20,21] constitutes the cornerstone of contem-
porary valuation theory. However, the model, although very robust and of immense
practical utility is based on several unrealistic and rigid assumptions. Several general-
izations have been attempted through relaxation of one or other assumption, thereby
enhancing its spectrum of applicability.
In this paper, we attempt one such generalization based on the deformation of the
standard Brownian motion. Section 2, which forms the essence of this paper, attempts a
deformation of the standard Black Scholes pricing formula. In Section 3 we illustrate the
theory developed in the previous section with a concrete example.Section 4 looks at the
interpretation of the deformation index. Section 5 addresses issues relating to empirical
relevance of the model. Section 6 the conclusions.
2. The Generalized Black Scholes Model
The standard analysis of the Black Scholes formula for option pricing presupposes
that the stock price follows the lognormal distribution. However, significant empirical
evidence now subsists of the stock returns deviating from the lognormal distribution with
“fat tails” and a “sharp peak” which better fit the truncated Levy flights or other power
law distributions [9, 22, 23]. To broadbase the Black Scholes model, generalizations by
way of “Levy noise” and “jump diffusions” [24] have already been studied. In this paper,
we propose a model that incorporates a “weighted Brownian motion” as the stochastic
(noise) term, where the weights themselves are a function of the “Brownian motion /
noise” i.e.
dW Pt → dUP
t = f(UP
t , t)dW P
t (1)
W Pt is a regular Brownian motion representing Gaussian white noise with zero mean and
δ correlation in time i.e. EP (dWtdWt′) = dtdt′δ (t − t′) and on some filtered probability
space (Ω, (Ft) , P ). We, further, mandate that the function f(UP
t , t)
satisfies the Novikov
condition and that the process UPt =
∫ t
0f(UP
t , s)dW P
s is a local P -martingale wih a non
normal distribution. This requirement is not as restrictive as it may seem at first sight
in context of the applications envisaged. We shall address this issue again in the sequel.
This generalization contemplates a statistical feedback process. In this context, sev-
eral studies on stock market data have shown the existence of nonlinear characteristics
and chaotic behavior that lend credence to the existence of a statistical feedback mech-
anism of market players. Explanations for the existence of “fat tails” in stock market
Electronic Journal of Theoretical Physics 4, No. 14 (2007) 69–86 71
data have been offered through this statistical feedback process e.g. “extremal events”
cause “disproportionate reactions” among market players. This deformed noise may also
capture the “herd behavior” of stock market investors. The model also encompasses time
dependent return processes since f is a function of UPt and t so that the drift term varies
with time.
We define the European call option as a financial contingent claim that entails a right
(but not an obligation) to the holder of the option to buy one unit of the underlying
asset at a future date (called the exercise date or maturity date) at a price (called the
exercise price). The option contract, therefore, has a terminal payoff of max (ST − E, 0) =
(ST − E)+ where ST is the stock price on the exercise date and E is the exercise price.
We consider a non-dividend paying stock, the price process of which follows the ge-
ometric Brownian motion with drift St = e(μt+σUPt ) under the probability measure P
with drift μ and volatility σ. The logarithm of the stock price Yt = In St follows the
stochastic differential equation
dYt = μdt + σdUPt = μdt +
[σf(UP
t , t)]
dW Pt (2)
Application of Ito’s formula yields the following SDE for the stock price process
dSt =
(μ +
1
2
[σf(UP
t , t)]2)
Stdt +[σf(UP
t , t)]
StdW Pt (3)
Let C (St, t) denote the instantaneous price of a call option with exercise price E at any
time t before maturity when the price per unit of the underlying is St. We assume that
C (St, t) does not depend on the past price history of the underlying. Applying the Ito
formula to C (St, t)yields
dCt=[(
μ+ 12 [σf(UP
t ,t)]2)St
∂C∂S
+ ∂C∂t
+ 12 [σf(UP
t ,t)]2S2
t∂2C∂S2
]dt+ ∂C
∂S [σf(UPt ,t)]StdWP
t , (4)
Applying Girsanov’s theorem to the price process (3), we perform a change of measure and
define a probability measure Q such that the discounted stock price process Zt = Ste−rt
or equivalently
dZt =
(μ − r +
1
2
[σf(UP
t , t)]2)
Ztdt +[σf(UP
t , t)]
ZtdW Pt (5)
behaves as a martingale with respect to Q. This is performed by eliminating the drift
term through the transformation(μ − r + 1
2
[σf(UP
t , t)]2)
σf (UPt , t)
→ γt (6)
whence WQt = W P
t + γtt is a Brownian motion without drift with respect to the measure
Q and dZt =[σf(UQ
t , t)]
ZtdWQt which is driftless under the measure Q and hence, Zt
is a Qmartingale.
72 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 69–86
The equivalence of[σf(UP
t , t)]
ZtdW Pt and
[σf(UQ
t , t)]
ZtdWQt follows from the
fact that both WQt ,W P
t are zero mean Weiner processes and that f(UQ
t , t)
can be ex-
pressed in terms of f(UP
t , t)
through dZt =[σf(UQ
t , t)]
ZtdWQt alongwith eq. (5). The
noise terms in dZt =[σf(UQ
t , t)]
ZtdWQt and eq. (5), will, therefore, be equivalent
stochastically.
The two measures P& Q are related through the Radon Nikodym derivative which in
the deformed case takes the form
ξ (t) =dQ
dP= exp
⎛⎝−t∫
0
γtdW Pt − 1
2
t∫0
γ2t dt
⎞⎠ (7)
and the expectation operators under the two measures are related as
EQ (Xt |Fs ) = ξ−1 (s) EP (ξ (t) Xt |Fs ) (8)
Our next step in martingale based pricing is to constitute a Q martingale process that
hits the discounted value of the contingent claim i.e. call option. This is formed by taking
the conditional expectation of the discounted terminal payoff from the claim under the
Q‘measure i.e.
Et = EQ[e−rT (ST − E)+ |Ft
]. (9)
We now constitute a self-financing strategy that exactly replicates the claim and whose
value is known with certainty. For this purpose, we introduce a ‘bond’ in our model that
evolves according to the following price process
dBt
Bt
= rdt, B0 = 1, (10)
where ris the relevant risk free interest rate.
Making use of φt units of the underlying asset and ψt units of the bond, where φt =∂C(St,t)
∂S, Btψt = C (St, t) − φtSt, we can now construct a trading strategy that has the
following properties
(1) it exactly replicates the price process of the call option i.e.
φtSt + ψtBt = C (St, t) ,∀t ∈ [0, T ] . (11)
(2) it is self financing i.e.
φtdSt + ψtdBt = dVt,∀t ∈ [0, T ] . (12)
Using eqs. (1), (3), (11) & (12) we have
dC =
(φtμSt +
1
2φt
[σf(UP
t , t)]2
St + ψtrBt
)dt + φt
[σf(UP
t , t)]
StdW Pt . (13)
Matching the diffusion terms of (3) & (13) and using (11), we get the aforesaid expressions
for φt and ψt respectively. The value of this portfolio at any time tcan be shown to be
Electronic Journal of Theoretical Physics 4, No. 14 (2007) 69–86 73
equal to Vt = ertEt with Et being given by eq.(9). It follows that the value of the
replicating portfolio and hence of the call option at time t is given by
Vt = ertEt = e−r(T−t)EQ[(ST − E)+|Ft] = e−r(T−t)EQ[(ST − E)1(ST≥E)|Ft]
= e−r(T−t)
∫{UQ
T :S(UQT ,T )≥E}
(S(UQT , T ) − E)f(UQ
T , T |UQT , t)dUQ
T (14)
The expectation value of the contingent claim max (ST − E, 0) = (ST − E)+ under the
measure Q depends only on the marginal distribution of the stock price process St under
the measure Q which is obtained by writing it in terms of Q Brownian motion WQt . We
have, from eq.(2), for the deformed stock price process under the measure Qas
d (In St) = μdt +[σf(UP
t , t)]
dW Pt =
(r − 1
2
[σf(UQ
t , t)]2)
dt +[σf(UQ
t , t)]
dWQt
(15)
which on integration yields
St = S0 exp
⎡⎣ t∫0
[σf(UQ
t , t)]
dWQt +
t∫0
(r − 1
2
[σf(UQ
t , t)]2)
ds
⎤⎦ . (16)
The value of the call option can now be computed by using eq. (14). The existence or
otherwise of a closed form solution would depend on the explicit representation of the
function f (U, t).
The following observations are cardinal to the above analysis.
(a) We have, implicitly, made the standard assumption of the market satisfying the
“No Arbitrage” condition. It is well known that long-term market equilibrium cannot
subsist in the presence of arbitrage opportunities. This “No Arbitrage” condition guar-
antees the existence and measurability of γt defined by eq. (6) as is proved below:
For this purpose, we assume that there exist values of UPt for which f
(UP
t , t)
= 0 and
hence, γt does not exist. Let Xt ={UP
t : f(UP
t , t)
= 0}. We construct a portfolio (φ, ψ)
of the normalized stock process(St
)and the bond process
(Bt)
where
φ =
⎧⎪⎨⎪⎩ θ for UPt ∈ Xt
0 for UPt /∈ Xt
⎫⎪⎬⎪⎭and
ψt = ψ0 + φ0S0 +t∫
0
e−rsφsdSs −t∫
0
re−rsφsds − e−rtφtSt,B0 = 1 and the normalized stock
process i.e. the stock process adapted to a market with zero interest rates being given by
St = Ste−rt and dSt = e−rtdSt − re−rtStdt.
The portfolio is self financing since Vt = ψt+φtSt and hence, dVt = ψt+φtdSt. Further,
Vt−V0 =t∫
0
φsdSs =t∫
0
e−rs(μ + 1
2
[of(UP
s , s)]2 − r
)φsSsds+
t∫0
e−rs[of(UP
s , s)]
φsSsdW Ps
=t∫0
ℵXse−rs(μ+ 1
2 [of(UPs ,s)]
2−r)θsSsds+
t∫0
ℵXse−rs[of(UP
s ,s)]θsSsdWPs =
t∫0
ℵXse−rs(μ−r)θsSsds≥0
74 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 69–86
where ℵXt is the characteristic function of the set Xt∀U, t. But under the “No Arbitrage”
condition Vt − V0 ≤ 0. It, therefore, follows that ℵXt = 0 ∀U, t and hence, Xt = φ.
(b) In the standard Black Scholes theory, the Novikov condition is automatically satisfied
due to the constancy of γt ≡ γ. However, in the deformed version, this condition needs
to be explicitly imposed to ensure the applicability of the Girsanov’s theorem and hence,
the existence of the equivalent martingale measure Q. Hence, we require that the func-
tion f (U, t) to be such that EP
{exp
[12
T∫0
(γs)2 ds
]}< ∞. As mentioned above, this
condition is not very restrictive insofar as the applications of this model are concerned,
since f (U, t) would normally take the form of probability distributions and hence, be
non zero bounded functions, thereby, automatically satisfying the square integrability
requirements.
(c) Except for the Novikov condition, which needs to be explicitly imposed in the
deformed model as mentioned in (b) above, our analysis is equivalent to the standard
Black Scholes model since f (U, t) can be expressed as a function of Y , the logarithm of
the stock price Sthrough eq. (2);
(d) The “No Arbitrage” condition together with the Novikov Condition guarantee
the completeness of the market and hence, the availability of replicating portfolios for
the valuation of any contingent claim. This is established by showing that there exists a
self financing portfolio (φ, ψ)defined as in (a) above that exactly replicates the terminal
payoff of any lower bounded contingent claim, say C (St, t). Mathematically, this implies
that there exists a real number ε such that C (ST , T ) = V εT = ε +
T∫0
(φtdSt + ψtdBt) or
equivalently
C (ST , T ) = V εT = ε +
T∫0
(φtdSt + ψtdBt) = erT
(ε +
T∫0
φtdSt
)=erT
[ε+
T∫0
e−rt(μ+ 1
2 [of(UPt ,t)]
2−r)φtStdt+
T∫0
e−rt[of(UPt ,t)]φtStdWP
t
]=erT
[ε+
T∫0
e−rt[of(UQt ,t)]φtStdW
Qt
]
By the Martingale Representation Theorem, there exists a function ηt such that
C (ST , T ) = erT
{EQ[e−rTC (ST , T )
]+
T∫0
ηtStdWQt
}. Hence, we can identify ε =
EQ[e−rTC (ST , T )
]and φt = ert
[of(UQ
t , t)]−1
ηt. By selecting the bond component of
the portfolio (ψ) according to ψt = ψ0 +t∫
0
e−rsdλs where λs =s∫0
φvdSv − φsSs, we can
make our portfolio (φ, ψ) self financing. This is shown below. We have,
dVt=d(ψtert+φtSt)=rertψtdt+ertd(ψt)+d(φtSt)=rertψtdt+ertd(λt)+d(φtSt)=rertψtdt+φtdSt
as required. Furthermore,
V εt =ert
(ε+
t∫0
φvdSv
)=ert
(ε+
t∫0
ηvSvdWQv
)=ertEQ( e−rT V ε
T |Ft)=ertEQ( e−rT C(ST ,T )|Ft)
showing that V εt is lower bounded and hence, establishing the completeness of the market.
Electronic Journal of Theoretical Physics 4, No. 14 (2007) 69–86 75
3. An Illustration of the Deformed Model
We now present a concrete example as an application of the aforesaid analysis. For
the purpose, we consider a Brownian motion of the form
dW Pt → dUP
t = f(UP
t , t)q
dW Pt (17)
where f(UP
t , t)is a probability density function.
The incorporation of probability dependent term in the stochastic force enables us
to describe nonlinear return processes where the randomness is not uniform across the
entire return spectrum. In the standard theory, we envisage a random process that
is independent of the level of returns and hence, if sufficient number of observations
are accumulated, the entire spectrum of possible returns will be traversed. However,
through this deformed noise function we can model return processes that change with
the respective probability of such returns i.e. the degree of randomness changes across
the return spectrum – highly frequented regions of the spectrum may have higher/lower
returns depending on the nature of the deformation function. Hence, a biased yet random
return process can be accommodated. Although, in theory, the entire return spectrum
may still be traversed if sufficient number of observations are made, yet the dependence
on probabilities enable the modeling of systems that require a cleavage of the return
spectrum to create an effectively nonergodic space for the system. The model would also
be versatile enough to encompass a return spectrum having the character of a multifractal
which goes well with contemporary research findings in this area. Furthermore, unlike the
standard case where W Pt =
t∫0
dW Pt is normally distributed, UP
t =∫ t
0f(UP
t , s)dW P
s is no
longer normally distributed but follows a skewed distribution depending on the explicit
representation of the function f(UP
t , t)
and parameter q.
Eq. (17) is equivalent to the Langevin equation [25]
dUPt
dt= f(UP
t , t)q dW P
t
dt= f(UP
t , t)q
η (t) (18)
η (t) is a noise function that satisfies
〈η (t)〉 = 0 (19)
〈η (t′) dt′η (t′′) dt′′〉 = δ (t′ − t′′) dt′ (20)
The time evolution of the probability density f(UP
t , t)
is given by the following equation
[26] (The super(sub)scripts are suppressed for the sake of brevity)
f (U, t + Δt) =
∫f (U, t + Δt |U ′, t) .f (U ′, t) dU ′ (21)
f is the transition probability between states. We now set U ′ = U −ΔU and expand the
integrand as a Taylor’s series around f (U + ΔU, t + Δt |U, t) f (U, t) to obtain
f (U, t + Δt |U ′, t) f (U ′, t) = −ΔU ddU
f (U + ΔU, t + Δt |U, t) f (U, t) +
−ΔU2
2d2
dU2 f (U + ΔU, t + Δt |U, t) f (U, t) + .........(22)
76 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 69–86
Eq. (22) on integration gives
f (U, t + Δt) = − ddU
[∫ΔU f (U + ΔU, t + Δt |U, t)dΔU
]f (U, t) +
−12
d2
dU2
[∫ΔU2 f (U + ΔU, t + Δt |U, t)dΔU
]f (U, t) + .........
(23)
We can further simply the above expression, noting that U is a martingale, as follows:-
∫ΔUf(U + ΔU, t + Δt|U, t)dΔU = Et[ΔU ] = Et[
t+Δt∫t
f(Us, s)qdWs] = 0 (24)
and∫ΔU2f(U + ΔU, t + Δt|U, t)dΔU = Et[ΔU2] = Et[
t+Δt∫t
f(Us, s)2qds] = f(Us, t)
2qΔt + o(Δt)
(25)
where the last step follows from Ito isometry. We have ignored terms of second and higher
orders in Δt. Using the results in eqs. (24) & (25) in eq. (23) and taking the limit as
Δt → 0 we obtain the Fokker Planck equation [26] for the time evolution of the deformed
probability density (17) asdf
dt=
1
2
d2f2q+1
dU2(26)
To obtain an explicit solution of eq. (26) for the probability density f (U, t), we postulate
a normalized scaled solution, which enables the separation of the U and t dependencies
through the ansatz
f (U, t) = g (t) H (Ug (t)) = g (t) H (z) (27)
Substitution from eq. (27) into eq. (26) and simplification yields
.
g (t)
g (t)2q+3
∂
∂z(zH (z)) =
1
2
∂2
∂z2H (z)2q+1 (28)
Writing.
2g(t)
g(t)2q+3 = −k, we have
g (t) = [(q + 1) k (t − t0)]− 1
2(q+1) (29)
which gives the solution of eq. (26) as
f (U, t) = A (t − t0)− 1
2(q+1) exp(1−2q)
{B[(U − U0) (t − t0)
− 12(q+1)
]2}(30)
whereA = [(q + 1) k]−1
2(q+1) B = − kA2
4(2q+1)and expq (x) = [1 + (1 − q) x]
11−q is the qexponential
function. kcan be determined from the normalization condition∞∫−∞
f (U, t) dU = 1,
f (U, t)being a probability density function.
Electronic Journal of Theoretical Physics 4, No. 14 (2007) 69–86 77
The transition probability density f (U, t |U0, t0 ), that is the key element in option
pricing, is the probability density f (U, t)with a special initial condition f (U, t0) =
δ (U − U0) i.e. f (U, t |U0, t0 ) also obeys the Fokker Planck equation (26). Furthermore,
it is seen that the solution for f (U, t) given by eq. (30) meets the δ function initial con-
dition in the limit t → t0, and is, therefore, also a solution for the transition probability
density f (U, t |U0, t0 ).
As an illustration, the conditional probability density of the logarithm of the stock
prices would be
f (Yt+Δt |Yt ) = A (Δt)−1
2(q+1) exp(1−2q)
{B
[(ln
St+ΔtSt
−μΔt)
σ(Δt)−
12(q+1)
]2}under the probability measure P and
f (Yt+Δt |Yt ) = A (Δt)−1
2(q+1) exp(1−2q)
{B[
1σ
(ln St+Δt
St
)(Δt)−
12(q+1)
]2}under Q.
Using the expression (30) for f (U, t) with U0 = 0, t0 = 0(which does not result in
any loss of generality) in eq. (16), we derive the expression for the stock price process
under the martingale measure Q and, thereby, of the contingent claim using eq. (14). To
approximatet∫
0
f (U, s)2q ds we note that for any arbitrary value of time s, the distribution
of the random variable Us can be mapped onto the distribution of a random variable ω
at a fixed time T through the transformation Us =(Ts
)− 12(1+q) UT . Hence,
t∫0
f (U, s)2q ds =
t∫0
f((
Ts
)− 12(1+q) UT , s
)2q
ds
= A2q
t∫0
s−q
(q+1) exp2q(1−2q)
[B(UTT−
12(q+1)
)2]ds = Ct
1(q+1) exp2q
(1−2q)
[B(UTT−
12(q+1)
)2](31)
where C = (q + 1) A2q.
Furthermore,t∫
0
f (U, t)q dW = U (t), in view of eq. (17). Substituting this result and
that of eq. (31) in eq. (16), we get the following expression for the stock price process in
the martingale measure Q
St = S0 exp
{σUt + rt − 1
2σ2Ct
1(q+1) exp2q
(1−2q)
[B(UTT−
12(q+1)
)2]}
(32)
from which the value of the call option can be recovered using (14). It may, however,
be noted that in the standard case the exponential is linear in W and the stock price,
therefore, is a monotonically increasing function of W . Hence, the condition St − E > 0
is satisfied for all values of W that exceed a threshold value. However, in this illustration,
consequent to the noise induced drift, the exponential in the stock price process is now
a quadratic function of the deformed Brownian motion U . We, therefore, have two roots
of Uthat meet the condition St−E = 0. Accordingly, there will exist an interval (U1, U2)
78 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 69–86
within which the inequality St − E > 0 will hold. Furthermore, as q → 0, U2 → ∞thereby recovering the standard case. Hence, we have
Vt = e−r(T−t)
U2∫U1
⎛⎝S0e
{σUT +rT− 1
2σ2CT
1(q+1) exp2q
(1−2q)
[B
(UT T
− 12(q+1)
)2]}
− E
⎞⎠ f (UT , T ) dU
(33)
As in the standard case, in the martingale measure based risk neutral world, the stock
price distribution under Q is dependent on the risk free interest rate r and not on the
average return μ. We easily recover the standard results from the generalized model in
the limit q → 0.
4. Interpretation of the q Index
Towards examining the interpretation of the q index in the context of the ap-
plication being envisaged, we study the impact of the deformation of the standard
exponential distribution g (U, ζ) = CeBU2ζ . For this purpose, we note that f (U, t),
withU0 = 0, t0 = 0, can be expanded in the form of a gamma distribution as f (U, x) =
Aζ1/20
1
Γ[(−2q)−1]
∞∫0
x−(1+ 12q )e−x(1+2qζ0BU2)dxwhere ζ = t−(1+q)−1
. We assume that there ex-
ists a function h (ζ) that modifies the exponential distribution g (U, ζ) to f (U, ζ) i.e.
that f (U, ζ) = A∞∫0
h (ζ) eBU2ζdζ. Identifying −2qζ0x with ζ and comparing the two
expressions for f we obtain h (ζ) = ζ1/20
1
Γ[(−2q)−1]e(2qζ0)−1ζ (−2qζ0)
1/2q ζ−(1+ 12q ). Using
this expression for h (ζ) we obtain the expected values of ζ and ζ2 as 〈ζ〉 = ζ3/20 and
〈ζ2〉 = (1 − 2q) ζ5/20 which gives the coefficient of variation as (1 − 2q) ζ
−1/20 − 1. Hence,
it follows that if f (U, t) is a probability distribution function that satisfies the nonlinear
Fokker Planck eq. (26), then its explicit representation is given as in eq. (30) where
the parameter q is linearly related to the relative variance of ζ = t−(1+q)−1
Furthermore,
since the relative variance depends on both q and ζ = t−(1+q)−1
, it follows that the func-
tion f (U, t) generates an ensemble of returns corresponding to various values of q over
a particular time scale and also that, for a given q the distributions of returns evolves
anomalously across differing timescales.
5. Empirical Evidence
The Black Scholes model assumes lognormal distributions of stock prices. However,
deviations from such behaviour are, by now, well documented [28]. Empirical evidence
testifies that probability distributions of stock returns are negatively skewed, have fat
tails and show leptokurtosis [28]. Some of these features of empirical distributions are
modeled through Levy distributions [29-32], stochastic volatility [33] or cumulant ex-
pansions [31] around the lognormal case. Each of these models, however, attempts to
empirically attune the model parameters to fit observed data and hence, is equivalent
Electronic Journal of Theoretical Physics 4, No. 14 (2007) 69–86 79
to interpolating or extrapolating observed data in one form or the other. In contrast,
the deformed noise model preserves the analytical framework of the Black Scholes world
by retaining only one source of stochasticity and hence remaining within the domain of
complete markets. It also provides a complete form solution with enables the prediction
of option prices ab initio in lieu of parameter fitting to match observed data.
In this context, the probability distribution function of eq. (30) generates power law
distributions with consequential fat tails that are characteristic of stock price distribu-
tions. This fact is brought out explicitly by writing eq. (30), with U0 = 0, t0 = 0, in the
form:-
f(U,t)=At− 1
2(q+1) exp(1−2q)
[B
(Ut
− 12(q+1)
)2]=At
− 12(q+1)
{1+2q
[B
(Ut
− 12(q+1)
)2]} 1
2q
∼(2qA2qB)12q U
1q t
− 12q
(34)
for sufficiently large values of t.
There is an intricate yet natural relationship between the power law tails observed
in stock market data and probability distributions of the form (30) that emanates as
the solution of the nonlinear Fokker Planck equation (26). The nonlinear Fokker Planck
equation (26) is known to describe anomalous diffusion under time evolution. Empirical
results [34-37] establish that temporal changes of several financial market indices have
variances that that are shown to undergo anomalous super diffusion under time evolution.
One of the most exhaustive set of studies on stock market data in varying dimensions
has been reported in [38-42]. In [42], a phenomenological study was conducted of stock
price fluctuations of individual companies using data from two different databases cover-
ing three major US stock markets. The probability distributions of returns over varying
timescales ranging from 5 min. to 4 years were examined. It was observed that for
timescales from 5 minutes upto 16 days the tails of the distributions were well described
by a power law decay. For larger timescales results consistent with a gradual convergence
to Gaussian behaviour was observed. In another study [38] the probability distributions
of the returns on the S & P 500 were computed over varying timescales. It was, again,
seen that the distributions were consistent with an asymptotic power law behaviour with
a slow convergence to Gaussian behaviour. Similar findings were obtained on the analysis
of the NIKKEI and the Hang –Sang indices [38].
A plausible explanation of the matching of empirical behaviour referred to in the
preceding paragraphs and the probability distribution function (30) is based on the ob-
servation that if the stock prices show large deviations from the averages, then f (U)
would be small in line with the probabilities of extremal events being small. Since the
exponent qis usually negative in the region of interest, the effective volatility would be
accentuated. In terms of market behaviour, one could say that the traders would react
extremally. On the other hand, mild deviations would cause moderate reactions from
market players and hence, the effective volatility gets diminished.
80 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 69–86
6. Conclusions
Contemporary empirical research into the behavior of stock market price /return
patterns has found significant evidence that financial markets exhibit the phenomenon
of anomalous diffusion, primarily superdiffusion, wherein the variance evolves with time
according to a power law tα with α > 1.0. The standard technique for the study of su-
perdiffusive processes is through a stochastic process that evolves according to a Langevin
equation and whose probability distribution function satisfies a nonlinear Fokker Planck
equation of the form (26). The very fact that our deformed noise function satisfies the
nonlinear Fokker Planck equation is motivation enough for an adoption of this deformed
Brownian motion with statistical feedback for the modeling of financial processes.
Until recently, stock market phenomena was were assumed to result from complicated
interactions among many degrees of freedom, and thus they were analyzed as random
processes and one could go to the extent of saying that the Efficient Market Hypothesis
[43-44] was formulated with one primary objective – to create a scenario which would
justify the use of stochastic calculus [45] for the modeling of capital markets.
The Efficient Market Hypothesis contemplated a market where all assets were fairly
priced according to the information available and neither buyers nor sellers enjoy any ad-
vantage. Market prices were believed to reflect all public information, both fundamental
and price history and prices moved only as sequel to new information entering the market.
Further, the presence of large number of investors was believed ensures that all prices
are fair. Memory effects, if any at all, were assumed to be extremely short ranging and
dissipated rapidly. Feedback effects on prices was, thus, assumed to be marginal. The
investor community was assumed rational as benchmarked by the traditional concepts of
risk and return.
An immediate corollary to the Efficient Market Hypothesis was the independence of
single period returns, so that they could be modeled as a random walk and the defining
probability distribution, in the limit of the number of observations being large, would be
Gaussian.
Ever since the studies of Fama in 1964-65, evidence has been accumulating against
the validity of the Efficient Market Hypothesis – the existence of negatively skewed ob-
servations and fat tails and distortion around the mean values are but a few {28, 31-35].
Most financial returns, including stock returns have shown deviation from Gaussian be-
haviour at short time scales with the variance not scaling with the sq. root of timescale,
an attribute that is symptomatic of the possible existence of power law distributions like
the one being envisaged in this study. A useful measure of quantifying deviations from
the Gaussian distribution is the Hurst’s exponent. If a population is Gaussian, a Hurst’s
exponent of 0.5 is mandated. Empirical evidence, however, shows that the Hurst’s ex-
ponent for typical stock market data is around 0.6 for small timescales of about a day
or less and tends to approach 0.5 asymptotically with the lengthening of the timescales.
Empirical evidence also demonstrates the existence of memory effects, particularly in
stock price volatilities that show long term memory effects with lag-s autocorrelations.
Electronic Journal of Theoretical Physics 4, No. 14 (2007) 69–86 81
Further, these effects tend to fall off according to a power law rather than exponentially.
Furthermore, the access to enhanced computing power during the last decade has
enabled analysts to try refined methods like the phase space reconstruction methods
for determining the Lyapunov Exponents [46] of stock market price data, besides doing
Rescaled Analysis [47] etc. A set of several studies has indicated the existence of strong
evidence that the stock market shows chaotic behavior with fractal return structures and
positive Lyapunov exponents. Results of these studies have unambiguously established
the existence of significant nonlinearities and chaotic behavior in these time series [48-51].
As mentioned above, several studies [28,52-55] adopting largely diverse and indepen-
dent approaches have established the existence of the following characteristics in the
behavior of stock markets:-
• Long term correlation and memory effects
• Erratic markets under certain conditions and at certain times
• Fractal time series of returns
• Less reliable forecasts with increase in the horizon
thereby establishing strong evidence for the existence of chaotic behavior. In this
context, he following are conventionally accepted as the inherent characteristics of a
chaotic system [56-60]:-
• Exponential divergence of trajectories in phase space;
• Sensitive dependence on initial conditions;
• Fractal dimensions;
• Critical levels and bifurcations;
• Time dependent feedback systems;
• Far from equilibrium conditions.
This provides us with a second motivation for the adoption of this deformed Brownian
motion structure as a model for the random kicks since our model is based on a statistical
time dependent feedback into the system. This feedback may be modeled into the sys-
tem macroscopically through the explicit representation of the probability distribution
function f (U, t) and microscopically through the stochastic process U .
It need be emphasized here that the above is purely a phenomenological model for
modeling stock behavior. One could, for instance, postulate that that the statistical
feedback at the microscopic level represents the actions and interaction of the intra trader
interactions among traders constituting the market. The statistical dependency in the
noise could, further, be representing the aggregate behavior of these traders. Thus, we
could model a market with non homogeneous reactions with consequent biased return
structures
It is fair to say that the current stage of research in financial processes is dominated
by the postulation of phenomenological models that attempt to explain a limited set of
market behavior. There is a strong reason for this. A financial market consists of a huge
number of market players. Each of them is endowed with his own set of beliefs about ra-
tional behavior and it is this set of beliefs that govern his actions. The market, therefore,
invariably generates a heterogeneous response to any stimulus. Furthermore, “rational-
82 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 69–86
ity” mandates that every market player should have knowledge and understanding about
the “rationality” of all other players and should take full cognizance in modeling his re-
sponse to the market. This logic would extend to each and every market player so that
we have a situation where every market player should have knowledge about the beliefs
of every other player who should have knowledge of beliefs of every other player and so
on. We, thus, end up with an infinitely complicated problem that would defy a solution
even with the most sophisticated mathematical procedures. Additionally, unlike as there
is in physics, financial economics does not possess a basic set of postulates like General
Relativity and Quantum Mechanics that find homogeneous applicability to all systems
in their domain of validity.
Electronic Journal of Theoretical Physics 4, No. 14 (2007) 69–86 83
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86 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 69–86
EJTP 4, No. 14 (2007) 87–120 Electronic Journal of Theoretical Physics
Derivation of the Radiative Transfer Equation Insidea Moving Semi-Transparent Medium of Non Unit
Refractive Index
V. LE DEZ and H. SADAT∗
Laboratoire d’Etudes Thermiques UMR 6608 CNRS-ENSMA - 86960 FuturoscopeCedex, France
Received 26 September 2006, Accepted 6 January 2007, Published 31 March 2007
Abstract: The derivation of the radiative transfer equation inside a moving semi-transparentmedium of non unit constant refractive index has been completely achieved, leading to an exactlysimilar equation as in the case of a unit index, unless it is expressed in a particular frame withparticular time and space co-ordinates; defining first the “equivalent vacuum” and the “matter”space associated to its “matter” co-ordinates with the help of the Gordon’s metric, it is shownthat an observer at rest in vacuum perceives the isotropic moving medium as an anisotropicuniaxial medium of given optical axis, for which it is possible to derive general transmission andreflection rules for electromagnetic fields; however the exhibited refractive index characterisingthe moving medium, relatively to the observer located in vacuum, is not an effective indexbut only an apparent one without any energetic significance, and the specific intensity must beobtained relatively to a given observer at rest located inside the moving medium; finally thegeneral form of the radiative transfer equation is obtained in the moving medium.c© Electronic Journal of Theoretical Physics. All rights reserved.
Keywords: Radiation Hydrodynamics, Radiative Transfer Equation, Gordan’s Metric, OpticalPropertiesPACS (2006): 47.35.i, 47.65.d, 67.40.Hf, 02.40.k, 02.90.+p, 04.40.Nr
1. Introduction
Several years ago, Mihalas [1] proposed an elegant way to obtain the invariant radia-
tive transfer equation in a moving semi-transparent medium, noting that in some cases
it was judicious to perform energetic calculations either in a comobile frame bound to
a moving particle or in the frame bound to a given observer; then, when a radiation
participates to the energy transfer, it is necessary to be able to compute the radiative
88 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 87–120
fluxes in the appropriate frame; the energetic radiative fluxes being strongly related to
the radiative intensity whose evolution is governed by the radiative transfer equation,
one may naturally conceive to give the appropriate form of this equation either in the
comobile frame or in the observer frame. This fundamental work however is restricted
to media for which the refractive index is a unit index; this approximation is of great
interest for gases for which the refractive index is very close to 1, but dense isotropic me-
dia, liquids or solids, have generally a much higher refractive index, and we may imagine
situations where some liquids of high index are moving with a large speed: what is then
in this case the correct form of the radiative transfer equation in such media, and is it
possible to exhibit an invariant form of this equation, valid in both the comobile frame
or the observer frame?
If the optics of moving dielectric media has recently received a strong interest in the
literature [2, 3], to our knowledge no study focused on what happens from a radiative
energetic point of view in moving semi-transparent media; it is to suspect however that
the effects of high speeds may be spectacular, since some spectacular effects may arise
from an optical point of view, as described in [2, 3]; the main tool used to exhibit such
optical effects is the Gordon’s metric tensor; indeed, many decades ago, Gordon [4] had
the intuition that light in a moving dielectric medium “could see matter as a metric” in the
sense where a moving dielectric medium acts on light as an effective gravitational field, and
this is this property which is enhanced to produce in some particular conditions special
optical effects; following this basic idea, it may be interesting to see if the Gordon’s metric
is the appropriate tool to exhibit an invariant form of the radiative transfer equation.
The purpose of this paper is then the derivation of the radiative transfer equation
inside a grey (i.e. its optical properties are non frequency depending) moving semi-
transparent medium characterised by its constant refractive index different from one; to
do so, we shall first develop in section II the optical problem, that is determine the angle
and frequency transformation for a propagating radiation, between the comobile location
reference system bound to a moving particle embedded in the medium of non unit re-
fractive index and the location reference system relatively to a given observer inside the
medium: from the Gordon’s metric, we shall construct an “equivalent vacuum” and its
related “mater-light space” perceived both by the moving particle and the observer, such
that the “vacuum” bound to the particle and the “vacuum” bound to the observer are
related by a Lorentz transformation thanks to a particular rapidity different from the
usual one; this latter result will provide us in section IV, analogously to what happens
in the real vacuum, the radiation angle and frequency transformation in the refractive
medium, from which, following the work of Mihalas, one deduces the invariant form of
the radiative transfer equation. In section III, a closely related problem will be exam-
ined, which allows to interpret an uniaxial crystal (here are only studied the negative
crystals) as an isotropic moving medium, for which it is possible to derive (here only the
parallel polarisation was examined) the reflection and transmission laws for an electro-
magnetic field through an interface separating the crystal from an isotropic medium of
unit refractive index.
Electronic Journal of Theoretical Physics 4, No. 14 (2007) 87–120 89
2. The Optical Problem: Construction of an Equivalent Vac-
uum and Determination of the Fundamental Mater-Light
Space
It is well known that in a motionless medium embedded in a flat space, for which the
refractive index equals the unity, the radiative transfer equation (RTE) can be written in
Cartesian co-ordinates as
Pα
∂α I =h
c
( η
ν2− κ ν I
)=
κh
c ν2
[L
0 (T ) − L], (1)
where I = Lν3 is the specific intensity, L being the classical intensity, and
⇒P = h ν
c
(1,→Ω)
is the impulsion-energy 4-vector; κ is the absorption coefficient, L0 (T ) the black body
intensity at local thermodynamic equilibrium for a given temperature T , h the Boltzmann
constant, c the light speed in the vacuum and ν the radiation frequency inside the medium;
in absence of any relativistic event, the radiation frequency remains constant, and the
formal RTE can be rewritten under the standard form
1
c
∂L
∂t+
∂L
∂s= κ
[L
0 (T ) − L], (2)
where t is the time and s the curvilinear abscissa along a luminous trajectory, with∂L∂s
=→Ω
→grad L.
The Gordon effective gravitational field can be expressed as [2]
gμν = gμν (0) − (ε μ − 1) uμ uν ⇔ gμν = g(0)μν +
(1 − 1
ε μ
)uμ uν , (3)
where u is the mean 4-speed vector of the medium (relatively to a given observer), g(0) the
vacuum Minkowski tensor, g the effective gravitational tensor, and ε and μ the relative
dielectric and magnetic permittivity and permeability of the medium assumed hereafter
isotropic, related to its refractive index n by n2 = ε μ; we shall consider only non
magnetic media, with μ = 1, so that n2 = ε. Let us now remind a more mechanical
demonstration of this latter result: in a transparent medium where the refractive index,
hereafter assumed constant, i.e. non depending on space and/or time co-ordinates, is not
1, the proper time interval (PTI) for a photon can be written in Cartesian co-ordinates
as
d τ 2 = c2 d t2 − n2(d x2 + d y2 + d z2
)= − gμν d xμ d xν , (4)
where the contravariant co-ordinates xμ are xμ = (x0, x1, x2, x3) = (c t, x, y, z) ; this
is the most general form of the photon PTI in a motionless medium, simple translation of
the fact that light propagates at speed cn
in a dielectric for which the refractive index is
different from one, for the PTI is a light one for a photon and dτ = 0 ⇒ dsdt
= cn; hence
one deduces the covariant components of the metric tensor in Cartesian co-ordinates
g00 = − 1 gxx = n2 gyy = n2 gzz = n2, (5)
90 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 87–120
and the contravariant components, since the tensor is diagonal
g00 = − 1 gxx =1
n2gyy =
1
n2gzz =
1
n2, (6)
It has to be noticed that the PTI can be rewritten as
d τ 2 =(1 − n2
)d x02
+ n2(d x02 − dx2 − d y2 − d z2
), (7)
At a given event {xμ} in space-time, can be defined a 4-speed vector uμ = d xμ
dτrepre-
senting the mean motion of the dielectric; in a given comobile location reference system
(LRS) bound to a particle moving with a speed→β =
→vc
relatively to a “fix” (for observer)
LRS, the covariant components of the 4-speed are uμ′ = − δ0μ′ where the primes indicate
the co-ordinates relatively to the considered comobile LRS ; hence in this LRS one has
d τ 2 =(1 − n2
) (uμ′ d xμ′
)2
− n2 g(0)μ′ν′ d xμ′
d xν′= − gμ′ν′ d xμ′
d xν′, (8)
where g(0)μ′ν′ represents the vacuum metric tensor in the comobile LRS; the cinematic
transformation between the vacuum metric tensor relatively to the comobile LRS and the
vacuum metric tensor relatively to the observer LRS, and the metric tensors associated
to the dielectric is
gμν (0) =∂ xμ
∂ xμ′∂ xν
∂ xν′ gμ′ν′ (0)and gμν =
∂ xμ
∂ xμ′∂ xν
∂ xν′ gμ′ν′, (9)
Considering a motion along the→x axis, the former relations are developed for the
diagonal components
g00(0)= −
(∂ x0
∂ x0′
)2
+(
∂ x0
∂x′
)2gxx(0) = −
(∂x
∂ x0′
)2
+(
∂x∂x′)2
gyy(0) = 1 gzz(0) = 1, (10)
and for the non diagonal one
g0x(0)= −
(∂ x0
∂ x0′
) (∂x
∂ x0′
)+
(∂ x0
∂x′
) (∂x
∂x′
), (11)
since the variables x0 and x do not depend on y and z which remain unchanged; the
Lorentz transform in Cartesian co-ordinates is simply
x0′ = γ (x0 − β x)
x′ = γ (x − β x0)
y′ = y z′ = z
x0 = γ(x0′ + β x′
)x = γ
(x′ + β x0′
)y = y′ z = z′
, (12)
from which one obtains
d x0 dx =
∣∣∣∣ ∂ (x0, x)
∂ (x0′ , x′)
∣∣∣∣ d x0′ dx′ =
∣∣∣∣∣∣∣∂ x0
∂ x0′∂x
∂ x0′
∂ x0
∂x′∂x∂x′
∣∣∣∣∣∣∣ d x0′ dx′ = d x0′ dx′, (13)
Electronic Journal of Theoretical Physics 4, No. 14 (2007) 87–120 91
due to the scalar density conservation, it comes that√−Det g d x0 dx dy dz =
√−Det g′ d x0′ dx′ dy′ dz′, (14)
hence one has Det g = − n6: in Cartesian co-ordinates, the metric tensor determinant
remains unchanged; developing relations (10)-(11), one has with the help of (12) the
contravariant co-ordinates of the vacuum metric tensor relatively to the observer LRS
g00(0)= − 1 gxx(0) = 1 gyy(0) = 1 gzz(0) = 1, (15)
for the non diagonal component g0x(0)= 0 ; the tensor being diagonal, one deduces the
covariant co-ordinates as
g(0)00 = − 1 g(0)
xx = 1 g(0)yy = 1 g(0)
zz = 1, (16)
while the metric tensor associated to the dielectric in the observer LRS is gμν =∂ xμ
∂ xμ′∂ xν
∂ xν′ gμ′ν′, leading to
g00 = −γ2(n2 − β2
)n2
gxx =γ2(1 − β2 n2
)n2
gyy =1
n2gzz =
1
n2, (17)
for the diagonal components and
g0x = − β γ2 (n2 − 1)
n2, (18)
for the non diagonal one. The 4-speed vector being defined in vacuum as⇒u = γ
⎛⎜⎝ 1→β
⎞⎟⎠with γ = 1√
1−β2, one has for the contravariant co-ordinates of the 4-speed u0 = γ and
ux = β γ; hence the contravariant components of the metric tensor given by (17) and
(18) can be rewritten as
g00 = 1n2
[g00(0) − (n2 − 1) u0 u0
]gxx = 1
n2
[gxx(0) − (n2 − 1) ux ux
]g0x = 1
n2
[g0x(0) − (n2 − 1) u0 ux
] , (19)
or under a more compact form gμν = 1n2
[gμν (0) − (n2 − 1) uμ uν
], which is the Gordon
metric [4]; it is then possible to obtain the covariant components with a simple inversion
of the contravariant matrix, or more simply use Eq. (8) since uμ′ d xμ′= uμ′ ∂ xμ′
∂ xμ d xμ =
− δ0μ′
∂ xμ′
∂ xμ d xμ = − ∂ x0′
∂ xμ d xμ, from which uμ′ d xμ′= - γ (d x0 - β dx) ; hence one obtains
for the covariant components, since d x0 dx = d x0′ dx′:
g00 = − γ2(1 − β2 n2
)gxx = γ2
(n2 − β2
)gyy = n2 gzz = n2
g0x = − β γ2 (n2 − 1), (20)
92 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 87–120
which may be rewritten under the compact Gordon metric as gμν = n2 g(0)μν + (n2 − 1) uμ uν .
In such a metric the photon PTI is
d τ 2 = − gμν d xμ d xν = − g00 d x02 − gxx d x2 − 2 g0x d x0 dx− n2(d y2 + d z2
)= 0,
(21)
for a constant refractive index the luminous trajectories are straight lines, and if light
propagates along the→x axis, then dy = dz = 0, from which one deduces
gxx
(dx
d x0
)2
+ 2 g0xdx
d x0+ g00 = 0, (22)
the resolution of this equation easily leads to
dx
d x0=
1
n=
1 + β n
n + β=
dx′ + β d x0′
β dx′ + d x0′ , (23)
and if β << 1 one has for the effective refractive index 1n
= 1n
+ β(1 − 1
n2
), which is
the famous well-known Fresnel’s drag additional formula [5].
However, the physical significance of the former metric is not that obvious, since theg00 component associated to a time may be either negative or positive (or even zero),
depending on the value of β if the latter one is greater or lower than 1n
; we also make the
choice of a new metric, equivalent to the precedent one, for which the unique eigen-value
associated to a time is always negative: the calculation of the covariant metric tensor
eigen-values shows that n2 is a double eigen-value associated to the eigen-vectors→ey and
→ez, the two other eigen-values being solution of the characteristic equation
g2 − (gxx + g00) g − (g20x − gxx g00
)= g2 − (gxx + g00) g − n2 = 0, (24)
leading to
g1 = gxx + g00 −√
Δ2
= 12
[γ2(β2 + 1
)(n2 − 1) − √
Δ]
g2 = gxx + g00 +√
Δ2
= 12
[γ2(β2 + 1
)(n2 − 1) +
√Δ] , (25)
withg1 g2 = −n2 and Δ = (gxx + g00)
2 +4 (g20x − gxx g00) = (n2 +1)
2+4 β2 γ4 (n2 − 1)
2> 0
One deduces from the former result that the g1 eigen-value is strictly negative what-
ever the refractive index n and the β medium rapidity are, and that it may be associated
to a time, the g2 eigen-value being always positive and associated to a space variable: in
well chosen axis, the metric tensor relatively to the observer LRS is diagonal and can be
represented as
g =
⎛⎜⎜⎜⎜⎜⎜⎜⎝
g1 0 0 0
0 g2 0 0
0 0 n2 0
0 0 0 n2
⎞⎟⎟⎟⎟⎟⎟⎟⎠, (26)
Electronic Journal of Theoretical Physics 4, No. 14 (2007) 87–120 93
The determination of the associated Eigen-vector leads to
→E1 =
→e0 +
β γ2 (n2 − 1)γ2 (n2 − β2)− g1
→ex =
→e0 − g0x
gxx − g1
→ex
→E2 = − β γ2 (n2 − 1)
γ2 (1−β2 n2)+ g2
→e0 +
→ex = g0x
g2 − g00
→e0 +
→ex
, (27)
these two vector are orthogonal relatively to this metric and can be normed so that→e1
2= g1 and
→e2
2= g2 since (gxx − g1) (g00 − g1) = g2
0x : hence→E1
2
= g00 − 2 g20x
gxx − g1+
g20x
gxx
(gxx − g1)2= 2 g1 − g00 +
gxx (g00 − g1)gxx − g1
= g1
√Δ
gxx − g1from which
→E1
2
=→e1
2 g2 − g1gxx − g1
and→e1 =
√gxx − g1g2 − g1
(→e0 − g0x
gxx − g1
→ex
); performing the same calculation with the second
eigen-vector finally leads to the expression of the two normed eigen-vectors→e1 and
→e2
associated to the two eigen-values g1 and g2 as
→e1 = 1√
g2 − g1
(√gxx − g1
→e0 +
√g00 − g1
→ex
)→e2 = 1√
g2 − g1
(−√
g2 − gxx→e0 +
√g2 − g00
→ex
) . (28)
The 4-event vector is defined as(d⇒M)2
= − d τ 2 = gμν d xμ d xν ⇒ d⇒M = d xμ →
eμ ⇒⇒M = xμ →
eμ, hence for a given event one has
⇒M = x0 →e 0 + x
→e x + y
→e y + z
→e z = x0′ →e 0′ + x′
→e x′ + y′
→e y′ + z′
→e z′
= γ(x0′ + β x′
) →e 0 + γ
(x′ + β x0′
) →e x + y′
→e y′ + z′
→e z′
= γ(→
e 0 +β→e x
)x0′ +γ
(β→e 0 +
→e x
)x′ + y′
→e y′ +z′
→e z′ ⇒
γ(→
e 0 +β→e x
)x0′ = x0′ →e 0′
γ(β→e 0 +
→e x
)x′ = x′
→e x′
,
(29)
the variables x0′and x
′being independent; it comes then
→e 0 + β
→e x =
→e 0′γ
β→e 0 +
→e x =
→e x′γ
⇒→e 0 = γ
(→e 0′ − β
→e x′)
→e x = γ
(→e x′ − β
→e 0′) , (30)
for this event one also has⇒M = x1 →e 1 + x2 →e 2 + y
→e y + z
→e z = x0 →e 0 + x
→e x + y
→e y + z
→e z
and for the pseudo-norm⇒M
2
= g1 x12+ g2 x22
+ n2 (y2 + z2) ; but
→e1 = 1√
g2− g1
(√gxx − g1
→e0 +
√g00 − g1
→ex
)→e2 = 1√
g2− g1
(−√
g2 − gxx→e0 +
√g2 − g00
→ex
) ⇒→e0 = 1√
g2− g1
(√gxx − g1
→e1 −
√g00 − g1
→e2
)→ex = 1√
g2− g1
(√g2 − gxx
→e1 +
√g2 − g00
→e2
) ,
(31)
from which
x1 = γ√g2 − g1
[(√
gxx − g1 + β√
g2 − gxx) x0′ + (√
g2 − gxx + β√
gxx − g1) x′]
x2 = γ√g2 − g1
[(β
√g2 − g00 − √
g00 − g1) x0′ + (√
g2 − g00 − β√
g00 − g1) x′] ,
(32)
94 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 87–120
performing the calculation of g1 x12+ g2 x22
one obtains finally
g1 x12+ g2 x22
= γ2
⎧⎪⎨⎪⎩(g00 + β2 gxx +2β g0x
)x0′2 +
(gxx + β2 g00 + 2β g0x
)x′2
+ 2[(
1 + β2)g0x +β (g00 + gxx)
]x0′ x′
⎫⎪⎬⎪⎭ = −x0′2 + n2 x′2,
(33)
hence the 4-event pseudo-norm remains unchanged, and
d x1 d x2 =
∣∣∣∣ ∂(x1,x2)∂(x0′ ,x′)
∣∣∣∣ d x0′ dx′ = γ2
g2− g1
∣∣∣∣∣∣∣√
gxx − g1 + β√
g00 − g1 β√
g2 − g00 −√
g2 − gxx
√g00 − g1 + β
√gxx − g1
√g2 − g00 − β
√g2 − gxx
∣∣∣∣∣∣∣ d x0′ dx′
= γ2
g2− g1
(1 − β2
)(g2 − g1) d x0′ dx′ = d x0′ dx′
,
(34)
from which one deduces that√−Det g d x1 d x2 dy dz =
√−Det g′ d x0′ dx′ dy′ dz′, that
is the conservation of the scalar density ; noticing furthermore that
(√
gxx − g1 − β√
g00 − g1) (β√
gxx − g1 − √g00 − g1) = − 2 β g1
(√
g2 − gxx + β√
g2 − g00) (√
g2 − g00 + β√
g2 − gxx) = 2 β g2
(√
gxx − g1 − β√
g00 − g1) (√
g2 − gxx + β√
g2 − g00) = 2 β n2
(β√
gxx − g1 − √g00 − g1) (
√g2 − g00 + β
√g2 − gxx) = 2 β
, (35)
the x1 and x2 co-ordinates can equivalently be rewritten as
x1 = γ√g2 − g1
[2β g2√
g2 − gxx +β√
g2 − g00x0′ + (
√g2 − gxx + β
√g2 − g00) x′
]x2 = γ
g2√
g2 − g1
[(√
g2 − gxx + β√
g2 − g00) x0′ + 2β g2 n2√g2 − gxx +β
√g2 − g00
x′] , (36)
with the help of the auxiliary value√
X =√
g2 − gxx + β√
g2 − g00, performing the
calculation of g1 x12+ g2 x22
, it comes for the equation verified by X that
X2 +
g2 (g2 − g1)γ2
X − 4 β2 g22 n2 = 0, (37)
the discriminant of this equation is Δ = g22
[16 β2 n2 + (g2 − g1)2
γ4
]= g2
2
(1 + β2
)2(n2 + 1)
2,
and since X is positive, one has
X = g2
2
[(1 + β2
)(n2 + 1) − (
1 − β2)
(g2 − g1)]
= g2
[g1 − g00 + β2 (gxx − g1)
]= g2
(β2 gxx − g00
)− n2(1 − β2
)= g2
(1 + β2
)− n2(1 − β2
)= g2
[1 + g1 + (1 − g1) β2
] ,
(38)
following the same steps, with the help of the auxiliary value√
X =√
g2 − g00 +
β√
g2 − gxx, and performing once again the calculation of g1 x12+ g2 x22
, it comes for
the equation verified by X
X2 − g2 (g2 − g1)
γ2 n2X − 4 β2
g22
n2= 0, (39)
Electronic Journal of Theoretical Physics 4, No. 14 (2007) 87–120 95
the discriminant of this equation is Δ =g22
n4
[16 β2 n2 + (g2 − g1)2
γ4
]=
g22 (1+β2)
2(n2 + 1)
2
n4 ,
and since X is positive, one has
X = g2
2 n2
[(1 + β2
)(n2 + 1) +
(1 − β2
)(g2 − g1)
]= g2
n2
[gxx − g1 + β2 (g1 − g00)
]= g2
n2
(gxx − β2 g00
)+ 1 − β2 = g2
(1 + β2
)+ 1 − β2 = g2 + 1 + (g2 − 1) β2
,
(40)
Finally the system (35) can be rewritten as
√g2 − gxx + β
√g2 − g00 =
√g2
[1 + g1 + (1 − g1) β2
]β√
gxx − g1 − √g00 − g1 =
√1+ g1 +(1− g1) β2
g2√gxx − g1 − β
√g00 − g1 = − g1
√g2 + 1 + (g2 − 1) β2
√g2 − g00 + β
√g2 − gxx =
√g2 + 1 + (g2 − 1) β2
, (41)
hence for the x1 and x2 co-ordinates one has
x1 = γ√− g1
[√− g1
√g2 + 1+ (g2 − 1) β2
g2 − g1x0′ + n
√1+ g1 + (1− g1) β2
g2 − g1x′]
x2 = γ√g2
[√1+ g1 +(1− g1) β2
g2 − g1x0′ + n
√− g1
√g2 + 1+ (g2 − 1) β2
g2 − g1x′] , (42)
performing then the calculation of g1 x12+ g2 x22
, it comes the useful following relation
n2 − 1 − 2 g1 + β2(n2 − 1 + 2 g1
)=
g2 − g1
γ2⇔ g1 + g2 = γ2
(1 + β2
) (n2 − 1
),
(43)
this leads for the 4-event that
⇒M = γ√− g1
[√− g1
√g2 +1+ (g2 − 1) β2
g2 − g1x0′ + n
√1+ g1 + (1− g1) β2
g2 − g1x′]→e 1
+ γ√g2
[√1+ g1 +(1− g1) β2
g2 − g1x0′ + n
√− g1
√g2 + 1+ (g2 − 1) β2
g2 − g1x′]→e 2 + y
→e y + z
→e z
= x0′ →e 0′ + x′→e x′ + y′
→e y′ + z′
→e z′
, (44)
and since the variables x0′ and x′ are independent,
→e 0′ = γ
[√g2 +1+ (g2 − 1) β2
g2 − g1
→e 1 + 1√
g2
√1+ g1 + (1− g1) β2
g2 − g1
→e 2
]→e x′ = γ
[√g2
√1+ g1 +(1− g1) β2
g2 − g1
→e 1 − g1
√g2 + 1 + (g2 − 1) β2
g2 − g1
→e 2
] , (45)
It is now comfortable to introduce symbolic variables such as
x1 = γ√− g1
(x0′ + β nx′
)x2 = γ√
g2
(β x0′ + nx′
) , (46)
96 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 87–120
since they obviously verify the relation g1 x12+ g2 x22
= − x0′2 + n2 x′2 ; however
those variables do not represent the⇒M 4-event, but it is convenient to notice the following
substitution, very useful for latter calculations
√− g1
√g2 +1+ (g2 − 1) β2
g2 − g1↔ 1√
1+ g1 +(1− g1) β2
g2 − g1↔ β
, (47)
note that this equivalence becomes a strict equality if and only if n = 1; hence for the
symbolic 4-speed one has
⇒u = γ
⎛⎜⎝ 1→β
⎞⎟⎠ = γ
( →e1√− g1
+β→e2√g2
), (48)
this is obviously an admissible 4-speed since its pseudo-norm is⇒u
2
= γ2(β2 − 1
)= − 1,
but the symbolic 4-speed is not the real 4-speed, which is given, with the help of (47) in
the(→e1,
→e2
)basis as
⇒u = γ
⎡⎣√g2 + 1 + (g2 − 1) β2
g2 − g1
→e 1 +
1√g2
√1 + g1 + (1 − g1) β2
g2 − g1
→e 2
⎤⎦ =→e 0′ , (49)
which is the case.
Let us now focus our attention on the 4-impulsion of a photon: in the former comobile
LRS its contravariant components are symbolically written⇒P = h ν′
c
⎛⎜⎝ 1→Ω′
⎞⎟⎠ where→Ω′ is
the photon propagation direction relatively to this LRS, from which⎛⎜⎜⎜⎜⎜⎜⎜⎝
P 0′
P x′
P y′
P z′
⎞⎟⎟⎟⎟⎟⎟⎟⎠= n2 h ν ′
c
⎛⎜⎜⎜⎜⎜⎜⎜⎝
1
cos Θ′n
sin Θ′ cos Φ′n
sin Θ′ sin Φ′n
⎞⎟⎟⎟⎟⎟⎟⎟⎠, (50)
the covariant components being Pμ′ = gμ′ν′ P ν′, leading naturally to the fact that its
pseudo-norm is⇒P
2
= Pμ′ P μ′= 0 , since
⇒P is a light 4-event like; then the contravariant
co-ordinates of the photon 4-impulsion vector relatively to the observer LRS are P μ =
P μ′ ∂ xμ
∂ xμ′ , leading to
P 0 = γ h ν′ nc
(n + β cos Θ′) P x = γ h ν′ nc
(nβ + cos Θ′)
P y = h ν′ nc
sin Θ′ cos Φ′ P z = h ν′ nc
sin Θ′ sin Φ′, (51)
Electronic Journal of Theoretical Physics 4, No. 14 (2007) 87–120 97
from which it is obvious that:
P0 = − γ h ν′ n2
c(1 + nβ cos Θ′) Px = γ h ν′ n2
c(β + n cos Θ′)
Py = h ν′ n3
csin Θ′ cos Φ′ Pz = h ν′ n3
csin Θ′ sin Φ′
hence a simple calculation leads to Pμ P μ = 0 which was expected; it is important
however to notice here that these components suffer from a lack of physical clear signif-
icance so as for the g metric tensor components expressed in the(→e0,
→ex,
→ey,
→ez
)basis,
since from the following tensorial relation
Pμ = P
μ′ ∂ xμ
∂ xμ′ =∂ xμ
∂ xμ′ gμ′ν′Pν′ = P
μ′ ∂ xμ
∂ xμ′∂ xμ′
∂ xσ
∂ xν′
∂ xτgστ gμ′ν′ , (52)
one has
P0′ ∂ xμ
∂ x0′
(1 − ∂ x0′
∂ xσ
∂ xν′
∂ xτgστ g0′ν′
)= P
x′ ∂ xμ
∂x′
(∂x′
∂ xσ
∂ xν′
∂ xτgστ gx′ν′ − 1
), (53)
developing the former equality leads to
1 − ∂ x0′
∂ xσ∂ xν′
∂ xτgστ g0′ν′ = 1 − γ4
n2
[n2 β4 − 2 β2 (2 n2 − 1) + n2
]∂x′∂ xσ
∂ xν′
∂ xτgστ gx′ν′ − 1 = γ4
[β4 − 2 β2 (2 − n2) + 1
] − 1, (54)
but the P 0′ and P x′variables being independent and ∂ xμ
∂ x0′ �= 0 like ∂ xμ
∂x′ �= 0, Eq. (53)
is verified if and only if
γ4
n2
[n2 β4 − 2 β2 (2 n2 − 1) + n2
]= 1
γ4[β4 − 2 β2 (2 − n2) + 1
]= 1
, that is if and only if
n = 1; hence the⇒P representation in the
(→e0,
→ex,
→ey,
→ez
)basis is not a satisfactory one;
this can be explained by the fact that there does not exist a canonical representation in
this basis of the form P μ = n2 h νc
Ωμ√|gμμ|
since the metric tensor g is not diagonal in the
former basis: the angle and frequency transformation must be then performed with the
help of the diagonal metric; noticing that the 4-impulsion is then
⇒P =
⎛⎜⎜⎜⎜⎜⎜⎜⎝
P 1
P 2
P y
P z
⎞⎟⎟⎟⎟⎟⎟⎟⎠= n2 h ν
c
⎛⎜⎜⎜⎜⎜⎜⎜⎝
1√− g1
cos Θ√g2
sin Θ cos Φn
sin Θ sin Φn
⎞⎟⎟⎟⎟⎟⎟⎟⎠, (55)
for which it is easy to verify that⇒P
2
= 0, moreover using relation (45), it comes from
(50) that
⇒P = n2 h ν ′
c
⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩γ
[√g2 +1+ (g2 − 1) β2
g2 − g1+
√g2
cos Θ′n
√1+ g1 +(1− g1) β2
g2 − g1
]→e 1
+ γ
[1√g2
√1+ g1 + (1− g1) β2
g2 − g1− g1
cos Θ′n
√g2 + 1+ (g2 − 1) β2
g2 − g1
]→e 2
+ sin Θ′n
(cos Φ′
→e y + sin Φ′
→e z
)
⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭, (56)
98 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 87–120
from which one obtains it is easy to obtain
ν = γ ν ′[√− g1
√g2 +1+ (g2 − 1) β2
g2 − g1+ cos Θ′
√1+ g1 + (1− g1) β2
g2 − g1
]ν cos Θ = γ ν ′
[√1+ g1 +(1− g1) β2
g2 − g1+
√− g1 cos Θ′√
g2 + 1+ (g2 − 1) β2
g2 − g1
]ν sin Θ cos Φ = ν ′ sin Θ′ cos Φ′
ν sin Θ sin Φ = ν ′ sin Θ′ sin Φ′
, (57)
one deduces from this result that angle Φ remains unchanged, while
ν sin Θ = ν ′ sin Θ′, (58)
noting μ = cos Θ and noticing that√[
g2 + 1 + (g2 − 1) β2] [
1 + g1 + (1 − g1) β2]
=
2 β√
g2, the two first equations of (58) can be rewritten as
ν = γ ν ′ n√
g2 +1+ (g2 − 1) β2
g2 (g2 − g1)
[1 + 2 β
√− g2
g1
μ′g2 + 1 + (g2 − 1) β2
]μ =
[g2 +1+ (g2 − 1) β2]μ′ +2β√− g2
g1
g2 +1+ (g2 − 1) β2 +2β√− g2
g1μ′
, (59)
from which it is easy to verify that for a unit refractive index one has
ν = γ ν ′ (1 + β μ′) μ =μ′ + β
1 + β μ′, (60)
which is the habitual and well-known angular aberration and frequency Doppler shift
transformation in vacuum [2]; for an isotropic and grey dielectric, that is a refractive
index independent on both the frequency and propagation direction, one has from (59)
∂μ
∂μ′= γ2
(ν ′
ν
)2
n2
[g2 + 1 + (g2 − 1) β2
]2+ 4 β2 g2
g1
g2 (g2 − g1)[g2 + 1 + (g2 − 1) β2
] ∂μ
∂ν ′= 0
∂ν
∂ν ′=
ν
ν ′, (61)
but[g2 +1+ (g2 − 1) β2]
2+4 β2 g2
g1
[g2 +1+ (g2 − 1) β2]2 =
g1 g2 + 2 g1 + 1+ (g1 g2 − 2 g1 + 1) β2
g1 [g2 +1+ (g2 − 1) β2]
= − n2 − 1− 2 g1 + (n2 − 1+2 g1) β2
g1 [g2 +1+ (g2 − 1) β2]=
g2 (g2 − g1)
γ2 n2 [g2 + 1+ (g2 − 1) β2]
, (62)
so that∂μ
∂μ′=
(ν ′
ν
)2
, (63)
hence one finally obtains
dν dμ =
∣∣∣∣∂ (ν, μ)
∂ (ν ′ μ′)
∣∣∣∣ dν ′ dμ′ =
∣∣∣∣∣∣∣νν′ 0
∂ν∂μ′
(ν′ν
)2∣∣∣∣∣∣∣ dν ′ dμ′ =
ν ′
νdν ′ dμ′, (64)
Electronic Journal of Theoretical Physics 4, No. 14 (2007) 87–120 99
the angle Φ remaining unchanged, noting dΩ = dμ dΦ the solid angle element, it comes
from (64) the final important relation valid in the medium of index n as in vacuum
ν dν dΩ = ν ′ dν ′ dΩ′. (65)
Let us then introduce the index n2 =g2 (g2 − g1)
g2 + 1 + (g2 − 1) β2 such that the co-ordinates x1 and
x2 can be rewritten:
x1 =γ√− g1
(n
nx0′ +
2 β n ng2 − g1
x′)
x2 =γ√g2
(2 β n
g2 − g1
x0′ +n2
nx′)
, (66)
note also that from the expression of n2, it is possible to obtain, after a rather difficult
calculation the following relation
dn
dβ= β γ2 n
[1 − 4 n2 (n2 + 1)
(g2 − g1)3
]⇒
dndβ
= 0 if n = 1
dndβ
= 0 if β = 0
The evolution of this index and its derivative with respect to β is plotted on the following
figure, for a refractive index n = 1.33; below a rapidity β = 0.6, n remains practically
constant so as its derivative; at β = 0.75, one has n = 1.50 and the effects of the medium
speed become appreciable; from β = 0.75 to 1, n and its derivative grow up very quickly,
and for extremely high speeds, the refractive index effects cannot be longer ignored.
Then for the 4-speed one has
⇒u = γ
n
n
[ →e1√− g1
+2 β n2
n (g2 − g1)
→e2√g2
], (67)
since⇒u
2= − 1 one obtains from (67) that n2 (g2 − g1)
2 − 4 β2 n4 = n2 (g2 − g1)2
γ2 and
⇒u = γ
n
n
⎡⎣ →e1√− g1
+
√1 −
(n
γ n
)2 →e2√g2
⎤⎦ , (68)
the previous expression of the 4-speed vector⇒u allows us to introduce a new rapidity
β = 2β n2
n (g2 − g1)such that β =
√1 −
(nγ n
)2
=√
1 − 1γ2 ⇔ γ = γ n
n= 1√
1− β2, with
limβ→1
β = 1 whatever n is, so that the 4-speed vector can be written under the compact
standard form⇒u = γ
( →e1√− g1
+ β
→e2√g2
), (69)
analogous to the symbolic 4-speed vector, replacing the rapidity β (expressed in vacuum)
by the rapidity β expressed in the medium relatively to the observer LRS; introducing
then the values of n and β in the angle/frequency transformations leads to
ν = γ ν ′(1 + β μ′
)μ =
μ′ + β
1 + β μ′ν dν dΩ = ν ′ dν ′ dΩ′ ν sin Θ = ν ′ sin Θ′, (70)
100 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 87–120
which has the remarkable form as the one obtained in vacuum; hence, one may
expect to find a judicious set of co-ordinates (x0, x) such that they verify the Lorentz
transform, analogous to the habitual Lorentz transform replacing β by β, that is
x0 = γ(x0′ + β x′
)x = γ
(x′ + β x0′
)y = y′ z = z′
⇔x0′ = γ
(x0 − β x
)x′ = γ
(x − β x0
)y′ = y z′ = z
, (71)
from Eq. (66) one has
x0′ = γ nn
[x1
√− g1 − 2β n2
n (g2 − g1)x2
√g2
]= γ
(x1
√− g1 − β x2√
g2
)x′ = 1
nγ nn
[x2
√g2 − 2β n2
n (g2 − g1)x1
√− g1
]= γ
n
(x2
√g2 − β x1
√− g1
) , (72)
hence with the substitution
x0′ = x0′ x′ = nx′ x0 = x1√− g1 x = x2
√g2
the Lorentz transform (72) is obtained: it has to be noticed that the fundamental variable
x′ = nx′ induces a local dilatation along x′, so that we choose for the other spatial vari-
ables the same dilatation, that is y′ = n y′ and z′ = n z′; from the 4-event vector, one
constructs then an “equivalent vacuum” completely defined from the comobile LRS by
its co-ordinates(x0′ = x0′ , x′ = nx′, y′ = n y′, z′ = n z′
)associated to the orthonor-
mal basis
(→e0′ =
→e0′ ,
→ex′ =
→ex′n
,→ey′ =
→ey′n
,→ez′ =
→ez′n
), and from the observer LRS by
its co-ordinates (x0 = x1√− g1, x = x2
√g2, y = n y, z = n z) associated to the or-
thonormal basis(→e0 =
→e1√− g1
,→ex =
→e2√g2
,→ey =
→ey
n,→ez =
→ez
n
): indeed, those light spaces
can be easily explained, reminding that the photons PTI are expressed as
d τ 2 = − gμν d xμ d xν = c2 (dt′)2 − [(n dx′)2 + (n dy′)2 + (n dz′)2]
= d x0′2 − (d x′2 + d y′2 + d z′2
)
andd τ 2 = − gμν d xμ d xν = (
√− g1 d x1)2 −
[(√
g2 d x2)2
+ (n dy)2 + (n dz)2]
= d x02 − (d x2 + d y2 + d z2)
hence the co-ordinates sets(x0′ , x′, y′, z′
)and (x0, x, y, z) associated to the two
basis(→e0′ ,
→ex′ ,
→ey′ ,
→ez′)
and(→e0,
→ex,
→ey,
→ez
)represent vacuum light co-ordinates and basis,
which are different from the mater co-ordinates and basis; from a comobile LRS point of
view, the mater co-ordinates are(x0′ , x′, y′, z′
)associated to the basis
(→e0′ ,
→ex′ ,
→ey′ ,
→ez′),
so that it is convenient analogously to introduce, from the observer LRS point of view, the
mater co-ordinates(x0 = x1
√− g1, x = x2 √g2
n, y = y, z = z
)associated to the basis
Electronic Journal of Theoretical Physics 4, No. 14 (2007) 87–120 101
(→e0 =
→e1√− g1
,→ex = n
→e2√g2
,→ey =
→ey,
→ez =
→ez
), so that mater-light metric tensors can be
expressed as
g′ =
⎛⎜⎜⎜⎜⎜⎜⎜⎝
−1 0 0 0
0 n2 0 0
0 0 n2 0
0 0 0 n2
⎞⎟⎟⎟⎟⎟⎟⎟⎠and g =
⎛⎜⎜⎜⎜⎜⎜⎜⎝
−1 0 0 0
0 n2 0 0
0 0 n2 0
0 0 0 n2
⎞⎟⎟⎟⎟⎟⎟⎟⎠, (73)
hence the two metrics, relatively to the moving particle and the observer, are strictly
equivalent, meaning that the natural mater curvilinear abscissa path is d s2 = d x2 +d y2 +d z2
relatively to the observer LRS, and that for a photon it is related to time by dt = ncds,
the latter quantities being defined relatively to the observer LRS; then one has the relation
between the two sets of co-ordinates
x0 = γ γ[(
1 − nβ β)
x0 +(n β − β
)x]
x = γ γn
[(β − nβ
)x0 +
(n − β β
)x] ⇔
x0 = γ γn
[(n − β β
)x0 −n
(n β − β
)x]
x = γ γn
[n(1 − nβ β
)x − (
β − nβ)
x0] ,
(73)
Furthermore, from the 4-event vector and the definition of the 4-speed vector, one has
⇒u = d
⇒Mdτ
= d⇒M
d x0d x0
dτ= γ
(→e0 + β
→ex
)= d x0
dτ
(→e0 + dx
d x0
→ex + dy
d x0
→ey + dz
d x0
→ez
)⇒ d x0
dτ= γ
→β = dx
d x0
→ex = dx
d x0
→ex
dyd x0 = dz
d x0 = 0, (74)
but one also has⇒u = d
⇒Mdτ
= d⇒M
dx0′d x0′
dτ=
→e0′ ⇒ d x0′
dτ= 1, so that it comes the
fundamental relation binding the proper times of a mass particle moving with speed
βg?o(defined as if it were in vacuum)
d x0 = γ d x0′ ⇔ d x0 = γ d x0′ (75)
3. Interpretation of the Fresnel’S Refractive Index as a Nega-
tive Uniaxial Anisotropy
From what precedes, it is obvious that the 4-speed is⇒u = γ
(→e0 + β
→ex
)and is natu-
rally defined in the vacuum bound to the observer located in the moving medium; it is im-
portant to remark here that the 4-speed perceived by an observer at rest located in the medium
is the same 4-speed perceived by an observer at rest but located in vacuum; for the latter
one, the 4-speed is:⇒uv = γ
(→e0 + β
→ex
)where
(→e 0,
→e x
)is the equivalent “vacuum basis” of the moving medium perceived by
the observer located in vacuum of unit refractive index, which is not the(→e0,
→ex
)basis
102 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 87–120
of the equivalent vacuum bound to the observer located inside the medium of refractive
index n; it is thus defined as follows:→e 0 = γ
(→e 0′ − β
→e x′)
= γ(→
e 0′ − βn
→e x′)
= γ2
n
[(n − β2
) →e 0 + β (n − 1)
→e x
]→e x = γ
(→e x′ − β
→e 0′)
= γ(→
e x′n
− β→e 0′)
= γ2
n
[(1 − n β2
) →e x − β (n − 1)
→e 0
]with
→e 0
2= − 1
→e x
2= 1
→e 0
→e x = 0
and equivalently:
→e 0 = γ2
[(1 − n β2
) →e 0 − β (n − 1)
→e x
]→e x = γ2
[β (n − 1)
→e 0 +
(n − β2
) →e x
]where β has to be understood as the speed of the mass particles evolving in the
medium, but evaluated as if they were evolving in vacuum; hence if n = 1 one obviously
has→e 0 =
→e 0 and
→e x =
→e x; replacing then
→e0 and
→ex by their values in terms of
→e0 and
→ex leads to
⇒uv = γ
(→e0 + β
→ex
)which was expected ; furthermore, one has:
→e 0 = γ γ
n
[(n − β β
) →e 0 +
(nβ − β
) →e x
]→e x = γ γ
n
[(β − n β
) →e 0 +
(1 − nβ β
) →e x
] so that after a simple calculation :⇒u =
γ(→e0 + β
→ex
)=
⇒uv
which proves the result; one deduces then from what precedes that:
→e 0 = γ γ
[(1 − β β
) →e 0 +
(β − β
) →e x
]→e x = γ γ
[(β − β
) →e 0 +
(1 − β β
) →e x
] ⇔→e 0 = γ γ
[(1 − β β
) →e 0 − (
β − β) →
e x
]→e x = γ γ
[− (
β − β) →
e 0 +(1 − β β
) →e x
]
this result allows to define the vacuum-like co-ordinates inside the medium for the ob-
server located in real vacuum: indeed, the existence of an observer located in vacuum
induces the existence of an interface between the moving medium and the vacuum, that
we shall assume orthogonal to the direction of motion of the moving medium; we shall
this time define a fictive observer at rest in the vacuum such that at a given instant
(hereafter designed as the initial vacuum instant) its position coincides with the position
of the interface, and that at this time one has dV =∥∥∥ →OV OVf
∥∥∥ where dV is the distance
between the reference observer OV in vacuum and the fictive observer OVfin vacuum,
this distance being evaluated in vacuum; let us introduce(x0, x, y, z
)the vacuum-like
co-ordinates of an event perceived by the reference observer at rest located in the real
vacuum,(xf
0, xf , yf , zf
)the vacuum-like co-ordinates of the same event perceived by
the fictive observer and(x0′ , x′, y′, z′
)the vacuum-like co-ordinates of this event per-
ceived by a moving particle in vacuum and bound to the moving interface; obviously, the
interface moves with the β rapidity for the observers at rest in vacuum, while it moves
with the β rapidity for observers at rest located in the moving medium, so that for the
observers in vacuum one has:
Electronic Journal of Theoretical Physics 4, No. 14 (2007) 87–120 103
xf0 = γ
(x0′ + β x′
)xf = γ
(β x0′ + x′
)yf = y′ zf = z′
from which:
x0 = γ(x0′ + β x′
)x = γ
(β x0′ + x′
)+ dV
y = y′ z = z′
since the two observers in vacuum remain at rest and have the same time perception;
similarly, one introduces a fictive observer in the medium such that at the initial vacuum
instant, its position inside the moving medium coincides with the one of the moving
interface and dM =∥∥∥ →OM OMf
∥∥∥ where dM is the distance between the reference observer
OM in the moving medium and the fictive observer OMfin the moving medium, this
distance being evaluated in the medium; note that the two fictive observers are created
only for calculation rules: indeed, it dv
(x0)
> dV where dv
(x0)
is the distance between
the reference observer in vacuum and moving interface, then the moving medium leaves
the reference observer in vacuum and goes towards the reference observer in the medium,
so that the fictive observer which was in the medium at x0 = x0 = 0 is in vacuum for
x0 > 0, while if dv
(x0)
< dV , the moving medium goes towards the reference observer
in vacuum and the fictive observer in vacuum at x0 = x0 = 0 is in the medium for
x0 > 0; then for the moving medium:
xf0 = γ
(x0′ + β x′
)xf = γ
(β x0′ + x′
)yf = y′ zf = z′
from which:
x0 = γ(x0′ + β x′
)x = γ
(β x0′ + x′
) − dM
y = y′ z = z′
with dM = n dM
note that no distinction has to be done for the basis vectors which are the same for the
fictive and reference observers; hence from a same 4-event perceived both by an observer
at rest located in the real vacuum and an observer at rest located in the moving medium,
one has:
⇒M = M
μ→eμ = xf0→e 0 + xf
→e x + yf
→e y + zf
→e z = xf
0 →e 0 +xf→e x +yf
→e y +zf
→e z
hence from what precedes, it comes:
x0 = γγ[(
1 − ββ)x0 +
(β − β
)(x + dM)
]x − dV = γγ
[(1 − ββ
)(x + dM) +
(β − β
)x0]
y = y z = z
⇔x0 = γ γ
[(1 − ββ
)x0 − (β − β
)(x − dV )
]x + dM = γγ
[(1 − ββ
)(x − dV ) − (
β − β)x0]
y = y z = z
replacing then x0′ and x′ by their values in terms of x0′ and x′, and relating x0′ and x′ tox0 and x thanks to the Lorentz transform with the vacuum-like β rapidity finally leads
to:
x0 = γ2[(
1 − nβ2)x0 +β (n − 1) (x − dV )
]x − dV = γ2
[(n − β2
)(x − dV ) − β (n − 1) x0
] ⇔x0 = γ2
n
[(n − β2
)x0 −β (n − 1) (x − dV )
]x − dV = γ2
n
[(1 − nβ2
)(x − dV ) + β (n − 1) x0
]
104 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 87–120
so that x0 = x0 and x = x if n = 1. In these conditions, the 4-impulsion vector of a
photon is, for the observer at rest located in the medium
⇒P = h ν
c
{n2
→e 0 + n
[cos Θ
→e x + sin Θ
(cos Φ
→e y + sin Φ
→e z
)]}= P μ →
eμ
= h ν n2
c
[→e 0 + cos Θ
→e x + sin Θ
(cos Φ
→e y + sin Φ
→e z
)]= E
c
⎛⎜⎝ 1→Ω
⎞⎟⎠ (76)
from the former definition of the 4-impulsion, one deduces that→Ω = cos Θ
→e x + sin Θ
(cos Φ
→e y + sin Φ
→e z
), and that E = h ν n2 = n2 E(0): it
is a well known result that the energy of a photon in a medium of index n is n2 times
its energy in vacuum, but the important fact is that this result remains valid, even for
a moving medium, in the point of view of the observer at rest located in the moving
medium.
Replacing then→e 0 and
→e x by their values in terms of
→e 0 and
→e x, leads finally to
⇒P =
hν n
c
⎧⎪⎨⎪⎩ γγ[n − ββ +
(β − nβ
)cos Θ
]→e 0 +γγ
[nβ − β +
(1 − nββ
)cos Θ
]→e x
+ sin Θ(cos Φ
→e y + sin Φ
→e z
)⎫⎪⎬⎪⎭ = P
μ →eμ
(77)
so that introducing the values of ν and cos Θ in terms of ν ′ and cos Θ′ gives the values
of P μ obtained by Eq. (51). Replacing now→e 0 and
→e x by their values in terms of
→e0 and
→ex gives the important result
⇒P =
h ν n2
c
⎧⎪⎨⎪⎩ γ γ[1 − β β +
(β − β
)cos Θ
] →e 0 + γ γ
[β − β +
(1 − β β
)cos Θ
] →e x
+ sin Θ(cos Φ
→e y + sin Φ
→e z
)⎫⎪⎬⎪⎭
(78)
this is the 4-impulsion energy of a photon evolving in the moving medium and expressed
in the vacuum basis of an observer at rest located in the real vacuum, while this photon
4-impulsion energy perceived by an observer at rest located in the moving medium is
simply⇒P = h ν n2
c
[→e 0 + cos Θ
→e x + sin Θ
(cos Φ
→e y + sin Φ
→e z
)]; under this form,
Eq. (78) allows to define an apparent energy and direction Θ of propagation of light in
the moving medium for the observer located in vacuum, such that:
Ec
= hν n2 γγc
[1 − β β +
(β − β
)cos Θ
]cos Θ =
β−β+(1−ββ) cosΘ
1−ββ+(β−β) cosΘ
sin Θ cos Φ = sinΘ cosΦ
γγ[1−ββ+(β−β) cosΘ]
sin Θ sin Φ = sinΘ sinΦ
γγ[1−ββ+(β−β) cosΘ]
from which Φ = Φ andcos Θ =
β−β+(1−ββ) cosΘ
1−ββ+(β−β) cosΘ
sin Θ = sinΘ
γγ[1−ββ+(β−β) cosΘ]
one can easily verify that cos2 Θ + sin2 Θ = 1, so thatcos Θ =
(1−β β) cos Θ− (β− β)1−β β− (β− β) cos Θ
sin Θ = sin Θ
γ γ [1−β β− (β− β) cos Θ]
Electronic Journal of Theoretical Physics 4, No. 14 (2007) 87–120 105
hence⇒P = E
c
⎛⎜⎝ 1→Ω
⎞⎟⎠ with Ec
= h ν n2
c γ γ [1−β β + (β− β) cos Θ]
For a constant refractive index n, the light trajectories inside the medium are straight
lines; indeed, these trajectories are the light geodesics determined by the geodesics equa-
tionsdP α
dσ+ Γα
βγ P β P γ = 0 where σ is a step parameter on the trajectory defined by P α =
d xα
dσand Γα
βγare the Christoffel coefficients such that Γα
βγ= gαm
2
(∂ gmγ
∂ xβ+
∂ gmβ
∂ xγ − ∂ gβγ
∂ xm
);
obviously these coefficients are all 0 for a constant refractive index, so that the geodesics
equations lead to P α = cons tan t, or equivalently d xα
d xβ= P α
P β= cons tan t, and for light
propagating in the (x, y) plane, one has:
dx = cosΘn
d x0 dy = sinΘn
d x0 hence ds2 =[1 +
(dydx
)2]dx2 = dx2
cos2 Θ= d x02
n2
and one retrieves the obvious relation dsd x0 = 1
n, or equivalently ds
d x0 = 1; furthermore,
from what precedes, it comes that:
dx
d x0 =(β− β) d x0 + (1−β β) dx
(1−β β) d x0 + (β− β) dx=
β− β + (1−β β) cos Θ
1−β β + (β− β) cos Θ= cos Θ
dy
d x0 = dy
γ γ [(β− β) d x0 + (1−β β) dx]= sin Θ
γ γ [1−β β + (β− β) cosΘ]= sin Θ
hence ds2 =
[1 +
(dy
dx
)2]
dx2 = dx2
cos2 Θ= d x02
Note that expressed in terms of (x0, x, y) co-ordinates set, the previous relations are
equivalent to:
dx
d x0=
(n − β2
)dxd x0 − β (n − 1)
β (n − 1) dxd x0 + 1 − n β2
dy
d x0=
n dyd x0
γ2[β (n − 1) dx
d x0 + 1 − n β2]
so that when Θ = 0, one has dx
d x0 = 1 anddy
d x0 = 0, from which it comes:dyd x0 = 0 and dx
d x0 = 1+β nn+β
= 1n
which is the Fresnel’s drag additional formula
for Θ = π2, dx
d x0 = 0 anddy
d x0 = 1, from which one obtains:dxd x0 = β (n− 1)
n−β2 and dyd x0 = 1
γ2 (n−β2), hence
[(dxd x0 = 0
)and
(dyd x0 = 1
)] ⇔ (n = 1)
we retrieve the fact that it is impossible to define a physical direction for light in
the moving medium when using the sets(→
e 0,→e x,
→e y,
→e z
)and (x0, x, y, z), since the
two vectors→e 0 and
→e x are not linearly independent; let us introduce now the habitual
matter co-ordinates(x0, x, y, z
)for the observer located in vacuum, such that for an
event in vacuum the co-ordinates(x0, x, y, z
)coincide with the vacuum co-ordinates(
x0, x, y, z), for an event in the medium perceived in the
→ex direction by the observer in
vacuum they are:
x0 = x0 x = n|| x then for a light propagating in the→ex direction, one must have
dxd x0 = 1
n||, and for an event in the medium perceived in the
→ey or
→ez directions they are:
x0 = x0 y = n⊥ y z = n⊥ z where obviously n⊥ = n and y = y as z = z if
the observers in vacuum and in the medium perceive the same y and z co-ordinates; the
106 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 87–120
observer at rest in vacuum perceives naturally the time in a given “direction” which is his
fundamental time vector reference system, in the vacuum as well as for events located in
the moving medium: this implies that the time direction→e0 perceived by the observer in
vacuum for events located in the moving medium must be the time direction→e0 perceived
by this observer for events located in the vacuum, which is the case by construction of→e0; similarly, the observer at rest located in vacuum is unable to distinguish a light ray
emerging from the moving medium in the perceived→ex direction for a perceived frequency
ν, governed by dxd x0 = 1
n||, from any light travelling the vacuum for the same perceived
direction and frequency and characterised by dxd x0 , so that for the observer located in
vacuum, the light emerging from the moving medium in the perceived direction→ex will
obey to:dxd x0 = 1
n||= dx
d x0 = 1n
from which it comes 1 < n|| = n+β1+β n
< n = n⊥Then, the Fresnel’s refractive index can be interpreted as the apparent refractive index
of the moving medium in the direction of motion of this medium effectively perceived by
the observer at rest in vacuum; hence the two relations dx
d x0 = cos Θ anddy
d x0 = sin Θ
can be rewritten as:
dxd x0 = cos Θ
n||
dyd x0 = sin Θ
n⊥
⇒ d s2 =
[1 +
(dy
dx
)2] (
dx
d x0
)2
d x02=
n2⊥ cos2 Θ + n2
|| sin2 Θ
n2⊥ n2
||d x02
from which one deduces the apparent refractive index: n2e =
n2⊥ n2
||n2⊥ cos2 Θ +n2
|| sin2 Θ
This apparent refractive index is the extraordinary wave refractive index for an uni-
axial medium of optical axis→e|| =
→ey [6] : indeed, n2
e =n2⊥ n2
||n2⊥ sin2 (π
2− Θ)+n2
|| cos2 (π2− Θ)
where π2− Θ is the angle between the unit wave vector
→Ω and the optical axis
→e|| of
the medium, so that it comes→e|| =
→ey: hence, if the observer at rest located in vacuum
perceives the same y and z co-ordinates as the observer at rest located in the moving
medium, the observer located in vacuum will perceive the isotropic moving medium (in
the point of view of the observer located in the medium) as an uniaxial medium whose
optical axis→e|| is orthogonal to the perceived direction
→ex of motion of the medium and
in the plane
(→ex,
→Ω
)where
→Ω is the perceived direction of propagation of light in the
moving medium; in the fundamental orthonormal basis(→ex,
→ey,
→ez
), the dielectric tensor
of the medium will be represented as ε = ε0
⎛⎜⎜⎜⎜⎝n2⊥ 0 0
0 n2|| 0
0 0 n2⊥
⎞⎟⎟⎟⎟⎠, with 1 < n|| < n⊥, so that
the observer in vacuum will perceive the moving medium as a negative uniaxial medium,
with reference matter co-ordinates:
x0 = x0 x = ne
(Θ)
x(Θ)
y = ne
(Θ)
y(Θ)
z = ne
(Θ)
z(Θ)
Electronic Journal of Theoretical Physics 4, No. 14 (2007) 87–120 107
hence in the uniaxial medium, the only possible reference co-ordinates are vacuum-like
co-ordinates; since the two observers perceive the same y and z co-ordinates, one writes
y = n y and z = n z with y = y and z = z.
Let us now examine an electromagnetic field associated to the photon propagating in
the moving medium: the co-ordinates (x0, x, y, z) and(x0′ , x′, y′, z′
)associated to the
basis(→e0,
→ex,
→ey,
→ez
)and
(→e0′ ,
→ex′ ,
→ey′ ,
→ez′)
[respectively(x0, x, y, z
)and
(x0′ , x′, y′, z′
)associated to
(→e 0,
→e x,
→e y,
→e z
)and
(→e0′ ,
→ex′ ,
→ey′ ,
→ez′)] being related thanks to a vacuum
Lorentz transform, the vacuum-like electromagnetic tensors are such that:
F μν = ∂ xμ
∂ xμ′∂ xν
∂ xν′ F μ′ν′and F μ′ν′
= ∂ xμ′
∂ xμ∂ xν′
∂ xν F μν
from which one obtains after calculation:
Ex = Ex′= Ex Ey = γ
(E y′ + c β B z′
)= γ γ
[(1 − β β
)E y + c
(β − β
)B z]
Ez = γ(E z′ − c β By′
)= γ γ
[(1 − β β
)E z − c
(β − β
)By]
Bx = Bx′= Bx By = γ
(By′ − β
c E z′)
= γ γ[(
1 − β β)
By − 1c
(β − β
)E z]
Bz = γ(B z′ + β
c E y′)
= γ γ[(
1 − β β)
B z + 1c
(β − β
)E y]
(79)
choosing a monochromatic plane wave magnetic field perceived by the reference observer
at rest located in the moving medium→B = B0 exp
{− 2 i π νc
[x0 − (x cos Θ + y sin Θ)]} →
e z,
where→Ω = cos Θ
→e x + sin Θ
→e y, leads to for the associated complex electric field (par-
allel polarisation):
→E = E0 exp
{− 2 i π ν
c
[x0 − (x cos Θ + y sin Θ)
]} (− sin Θ
→e x + cos Θ
→e y
)here
→E and
→B are vacuum-like fields so that B = E
c, from which one deduces the compo-
nents of the associated vacuum-like electromagnetic field relatively to the observer located
in vacuum:
Ex = −E sin Θ Ey = γ γ[(
1 − β β)
cos Θ + β − β]
E Ez = 0
Bx = 0 By = 0 Bz = γ γ[1 − β β +
(β − β
)cos Θ
]Ec
replacing Θ by its value in term of Θ finally leads to:
Ex = −E sin Θ
γ γ [1−β β− (β− β) cos Θ] Ey = E cos Θ
γ γ [1−β β− (β− β) cos Θ] Ez = 0
Bx = 0 By = 0 Bz = 1
γ γ [1−β β− (β− β) cos Θ]Ec
from which one obtains:→D = ε0 E
(− sin Θ
→e x + cos Θ
→e y
) →B = E
c
→e z
where E = E0 e− i Ψ, with E0 = E0
γ γ [1−β β− (β− β) cos Θ]and
Ψ = 2π νc
[x0 − (x cos Θ + y sin Θ)]
108 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 87–120
in the phase expression, the co-ordinates are the vacuum-like co-ordinates relatively
to the reference observer inside the medium,→D and
→B are the components of the vacuum-
like electromagnetic field perceived by the observer at rest in vacuum, and→Ω = cos Θ
→e x + sin Θ
→e y is the perceived unit wave vector associated to the electro-
magnetic field; hence, since the observer in vacuum perceives the moving medium as
an anisotropic uniaxial medium, the apparent electromagnetic induction field inside the
medium and perceived by this observer will obey to:→k→D = 0
→k→B = 0
→k ∧ →
E = ω→B
→k ∧
→Bμ0
= − ω→D
where→k is the wave vector inside the medium:
→k = 2π ν
cne
→Ω
hence→B = B0 exp
{− 2 i π ν
c
[x0 − ne
(x cos Θ + y sin Θ
)]} →ez = B0 e− i Ψ →
ez from
which one deduces the electric induction:
→D =
ne B0
c μ0
e− i Ψ(− sin Θ
→ex + cos Θ
→ey
)= ε
→E
then it easily comes for the electric field:→E = ne c B0 e− i Ψ
(− sin Θ
n2⊥
→ex + cos Θ
n2||
→ey
)=
E0 e− i Ψ →eE
one immediately verifies that→k ∧
→E = ω
→B and B
2
0 =n2⊥ n2
|| (n2⊥ cos2 Θ+n2
|| sin2 Θ)c2(n4⊥ cos2 Θ+n4
|| sin2 Θ) E
2
0 =
N2e
c2 E2
0, where Ne is the extraordinary ray refractive index for an uniaxial medium of op-
tical axis→e|| =
→ey [6]. Then, the apparent electromagnetic field inside the medium is for
the observer located in vacuum:→E = ne Ne E0 e− i Ψ
(− sin Θ
n2⊥
→ex + cos Θ
n2||
→ey
)→D = ε0 ne Ne E0 e− i Ψ
(− sin Θ
→ex + cos Θ
→ey
)→B = Ne
cE0 e− i Ψ →
ez
from which it comes E0 e− i ΨV =
E0 e− i ΨV ,
where ΨV and ΨV are the vacuum-like phases, that is:
E0 exp
{−2iπν
c
[x0 −
(x cos Θ + y sin Θ
)]}= E0 exp
{−2iπν
c
[x0 − (x cos Θ + y sin Θ)
]}the invariance for all y = y implies ν sin Θ = ν sin Θ, and from the transformation
formulas for the co-ordinates one has:
ν[x0−(x cosΘ+y sinΘ)]= νγγ[1−β β+(β−β) cos Θ]{x0−[(x−dV ) cos Θ+y sin Θ]}+νdM cosΘ
hence ν = ν
γ γ [1−β β− (β− β) cos Θ]is the frequency perceived by the reference observer at
rest in vacuum, depending on the propagation direction, ν being the frequency of the
Electronic Journal of Theoretical Physics 4, No. 14 (2007) 87–120 109
radiation for the reference observer located in the medium, and:
ν(x0 − x cos Θ
)= ν
(x0 −x cos Θ
)+ ν dV cos Θ + n ν dM cos Θ
on the moving interface inside the medium, x′ = 0, from which x = β x0 + dV and
x = β x0 −n dM , so that:
ν x0(1 − β cos Θ
)= ν x0
(1 − β cos Θ
)and since on the interface γ x0 = γ x0,
one has the fundamental relations:
ν γ(1 − β cos Θ
)= ν γ
(1 − β cos Θ
)ν sin Θ = ν sin Θ
with E0 exp(
2 i π νc dV cos Θ
)= E0 exp
(− 2 i π νc
n dM cos Θ)
hence the internal apparent fields perceived by the observer at rest in vacuum are:
→E = Ne ne E0 e− iΓ exp
{− 2 i π ν
c
[x0 − ne
(x cos Θ + y sin Θ
)]} (− sin Θ
n2
→e x + cos Θ
n2||
→e y
)→B =
Ne E0 e− i Γ
cexp
{− 2 i π ν
c
[x0 − ne
(x cos Θ + y sin Θ
)]} →e z
with e− iΓ = exp[− 2 i π
c
(ν dV cos Θ + n ν dM cos Θ
)], while for the observer located
in the medium, the true electromagnetic field is:
→E = E0 exp
{− 2 i π νc
[x0 −n (x cos Θ + y sin Θ)
]} (− sin Θ→e x + cos Θ
→e y
)→B = n E0
cexp
{− 2 i π νc
[x0 −n (x cos Θ + y sin Θ)
]} →e z
The observer at rest in vacuum perceives emerging electromagnetic fields (parallel polar-
isation studied here) such that:→Et = E0t e− i Ψt
(− sin Θt
→e x + cos Θt
→e y
)→Bt = E0t
ce− i Ψt
→e z
with Ψt = 2π νt
c
[x0 −
(x cos Θt + y sin Θt
)]where Θt and νt are the transmitted angle and frequency perceived by the observer
at rest in vacuum, related to the transmitted angle Θ′t and frequency ν ′ in the co-moving
LRS by ν ′ sin Θ′t = νt sin Θt and cos Θ′t = cos Θt −β
1−β cos Θt= γ νt
ν′
(cos Θt − β
); on the
moving interface, since x = x = β x0 + dV = β x0 + dV , one has for the transmitted
field on the interface perceived by the observer at rest in vacuum:
→Et = E0t exp
{− 2 i π νt
c
[x0(1 − β cos Θt
)− y sin Θt
]} (− sin Θt
→e x + cos Θt
→e y
)→Bt =
E0t
cexp
{− 2 i π νt
c
[x0(1 − β cos Θt
)− y sin Θt
]} →e z
with E0t = E0t exp(
2 i π νt
c dV cos Θt
)since Eq. (79) must be verified, one obtains the components of the transmitted field
in the co-moving LRS of the particle bound to the interface in vacuum:
Bx′
t = Bx
t = By
t = By′
t = 0 Ez′
t = Ez
t = 0
110 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 87–120
Ex′
t = Ex
t = − E0t sin Θt e− i Ψt = − E0tν ′
νt
sin Θ′t e− i Ψt
for the phase, it comes easily after calculation:
Ψt = 2π νt
c
[x0 −
(x cos Θt +y sin Θt
)]= 2πν′
c
[x0′ − (x′ cos Θ′t + y′ sin Θ′t)
]− 2π νt
c dV cos Θt
= Ψ′t −2π νt
c dV cos Θt
and: Ex′
t = − E0tν′νt
sin Θ′t e− i Ψ′t
doing so for the two other non zero components leads to:
Ey′
t = γ(E
y
t − c β Bz
t
)= γ E0t
(cos Θt − β
)e− i Ψt = E0t
ν′νt
cos Θ′t e− i Ψ′t
Bz′
t = γ(B
z
t − βc
Ey
t
)= γ E0t
c
(1 − β cos Θt
)e− i Ψt =
E0t
cν′νt
e− i Ψ′t
The electromagnetic field expressed in the co-moving LRS of a particle inside the moving
medium and on the moving plane interface has the following form:
→E ′
i = E ′0 exp
[− 2 i π ν′c
(x0′ −n y′ sin Θ′i
)]⎛⎜⎜⎜⎜⎝
− sin Θ′i
cos Θ′i
0
⎞⎟⎟⎟⎟⎠→B′
i = n E′0
cexp
[− 2 i π ν′c
(x0′ −n y′ sin Θ′i
)] →e z′
since x′ = 0 on the interface, where→E ′
i and→B′
i are the incident fields on the interface;
when the incident wave impinges the interface with incident angle Θ′i, a reflected wave
appears in the medium and a transmitted one appears in the vacuum, such that the total
fields are:
* in the medium:
→E ′
T =→E ′
i +→
E ′r = E ′
0 exp[− 2 i π ν′
c
(x0′ −n y′ sin Θ′i
)]⎡⎢⎢⎢⎢⎣− (
1 + r||)
sin Θ′i(1 − r||
)cos Θ′i
0
⎤⎥⎥⎥⎥⎦→
B′T =
→B′
i +→
B′r = n E′
0
c
(1 + r||
)exp
[− 2 i π ν′c
(x0′ −n y′ sin Θ′i
)] →e z′
* in vacuum:→E ′
t = E ′0 t|| exp
[− 2 i π ν′c
(x0′ − y′ sin Θ′t
)] (− sin Θ′t→e x′ + cos Θ′t
→e y′)
→B′
t = E′0
c t|| exp[− 2 i π ν′
c
(x0′ − y′ sin Θ′t
)] →e z′
since the reflected angle equals the incident one and where the frequency remains
unchanged threw the interface in the co-moving LRS, Θ′t is the transmitted angle, r||and t|| the parallel amplitude reflection and transmission factors, since by construction
the matter co-ordinates x0′ , y′ and associated vectors of the co-moving frame remain
unchanged from the medium to vacuum; hence the continuity relations for the fields
Electronic Journal of Theoretical Physics 4, No. 14 (2007) 87–120 111
through an interface lead to n sin Θ′i = sin Θ′t which is the classical Descartes’ law,
r|| = cos Θ′i −n cos Θ′
t
cos Θ′i +n cos Θ′
tand t|| = 2n cos Θ′
i
cos Θ′i +n cos Θ′
t; then one deduces from what precedes
that E ′0 t|| = ν′
νtE0t and the incident electromagnetic field inside the medium relatively
to the reference observer at rest in the medium is such that Eq. (79) is verified, that is
after a simple calculation:→Ei = E0 e− i Ψi
(− sin Θi
→e x + cos Θi
→e y
)→Bi = n E0
ce− i Ψi
→e z
where E0 = νi
ν′ E ′0 exp
(2 i π νi
cn dM cos Θi
)and Ψi = 2π νi
c
[x0 −n (x cos Θi + y sin Θi)
],
hence: E0 = νi
νt
E0t
t||exp
[2 i πc
(νt dV cos Θt + n νi dM cos Θi
)]while the phase continuity implies for all y that: νt sin Θt = n νi sin Θi; then from
what precedes, one has:
E0i =νi
νt
E0t
t||exp
[− 2 i π
cdV
(νi cos Θi − νt cos Θt
)]and the apparent incident field perceived by the reference observer at rest in vacuum is:
→Ei = ne
(Θi
)Ne
(Θi
)E0i e− i Ψi
(− sin Θi
n2⊥
→ex + cos Θi
n2||
→ey
)→Bi =
Ne(Θi)c
E0i e− i Ψi→ez
with Ψi = 2π νi
c
{x0 − ne
(Θi
) [x(Θi
)cos Θi + y
(Θi
)sin Θi
]}since on the moving interface x = ne
(Θi
)x(Θi
)= β x0 + dV = β x0 + dV and
ne
(Θi
)y(Θi
)= n y, one has for the apparent incident field on the interface perceived
by the observer at rest in vacuum:
→Ei = ne
(Θi
)Ne
(Θi
)E0i exp
(2 i π νi
c dV cos Θi
)× exp
{− 2 i π νi
c
[x0(1 − β cos Θi
)− n y sin Θi
]} (− sin Θi
n2⊥
→ex + cos Θi
n2||
→ey
)→Bi =
Ne(Θi)c
E0i exp(
2 i π νi
c dV cos Θi
)exp
{− 2 i π νi
c
[x0(1 − β cos Θi
)− n y sin Θi
]} →ez
the phase continuity for the fields implies νt
(1 − β cos Θt
)= νi
(1 − β cos Θi
)and
νt sin Θt = n νi sin Θi which is obviously verified, and the apparent transmission factors
are defined such that: →Et = T ||,E
→Ei
→Bt = T||,B
→Bi
with:
T||,B =
∥∥∥∥→Bt
∥∥∥∥∥∥∥∥→Bi
∥∥∥∥ = E0t
Ne(Θi) E0iexp
[− 2 i π
c dV
(νi cos Θi − νt cos Θt
)]= E0t
Ne(Θi) E0iexp
[− 2 i π dV
c β(νi − νt)
]= νt
νi
t||Ne(Θi)
112 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 87–120
and:
T ||,E =
⎛⎜⎜⎜⎜⎝T xx||,E T
xy||,E 0
Tyx||,E T
yy||,E 0
T zx||,E T
zy||,E T zz
||,E
⎞⎟⎟⎟⎟⎠ ⇒− n2
|| T xx||,E sin Θi + n2 T
xy||,E cos Θi = − n2 n2
||ne(Θi) T||,B sin Θt
− n2|| T
yx||,E sin Θi + n2 T
yy||,E cos Θi =
n2 n2||
ne(Θi) T||,B cos Θt
T zx||,E sin Θi = n2
n2||
Tzy||,E cos Θi
for the amplitudes, T||,E =
∥∥∥∥→Et
∥∥∥∥∥∥∥∥→Ei
∥∥∥∥ = Ne
(Θi
)T||,B and:
n2e
(Θi
)N2
e
(Θi
) [(− Txx
||,E sin Θi
n2 +T
xy||,E cos Θi
n2||
)2
+
(− T
yx||,E sin Θi
n2 +T
yy||,E cos Θi
n2||
)2]
= T 2||,E
⇒ n2e(Θi)n4 n4
||
⎡⎢⎣n4(T
xy||,E
2 + Tyy||,E
2)
cos2 Θi + n4||(T xx||,E
2 + Tyx||,E
2)
sin2 Θi
− 2 n2 n2||(T xx||,E T
xy||,E + T
yy||,E T
yx||,E)
sin Θi cos Θi
⎤⎥⎦ = T 2||,B
then it is efficient to choose T xx||,E T
xy||,E + T
yy||,E T
yx||,E = 0 and the transmission matrix can
be diagonal, with:
Txx||,E sin Θi =
n2
ne
(Θi
) T||,B sin Θt Tyy||,E cos Θi =
n2||
ne
(Θi
) T||,B cos Θt
For the reflected fields, relatively to the two reference observers, the situation is slightly
different, since if in the co-moving LRS the reflected angle and frequency equal the
incident ones, it is not the case for the reference observer located inside the medium
and for the one located in vacuum; in the co-moving LRS, the reflected unit wave vector
is
→Ω′r =
⎛⎜⎜⎜⎜⎝− cos Θ′i
sin Θ′i
0
⎞⎟⎟⎟⎟⎠ =
⎡⎢⎢⎢⎢⎣cos (π − Θ′i)
sin (π − Θ′i)
0
⎤⎥⎥⎥⎥⎦ =
⎛⎜⎜⎜⎜⎝cos Θ′r
sin Θ′r
0
⎞⎟⎟⎟⎟⎠applying the angle and frequency transformation relatively to the observer located in the
medium leads to:
νr sin Θr = ν ′ sin Θ′r = ν ′ sin Θ′i = νi sin Θi
cos Θr = cos Θ′r + β
1+ β cos Θ′r
= cos (π−Θ′i)+ β
1+ β cos (π−Θ′i)
= β− cos Θ′i
1− β cos Θ′i
but cos Θ′i = cos Θi − β1− β cos Θi
, from which one obtains:
cos Θr =2 β− (1+ β
2) cos Θi
1+ β2 − 2 β cos Θi
�= − cos Θi ⇒ Θr �= π − Θi
Electronic Journal of Theoretical Physics 4, No. 14 (2007) 87–120 113
hence one has the fundamental result: relatively to the observer located in the moving
medium, and similarly relatively to the one located in vacuum, the reflected angle is not
the incident angle, the latter result being true if and only if Θi = 0 or β = 0; then the
relations between the reflected and incident angles and frequencies are:
cos Θr =2 β− (1+ β
2) cos Θi
1+ β2 − 2 β cos Θi
sin Θr = sin Θi
γ2 (1+ β2 − 2 β cos Θi)
νr = γ2 νi
(1 + β
2 − 2 β cos Θi
)one easily verifies that cos2 Θr + sin2 Θr = 1; relatively to the reference observer located
in vacuum, it comes:
cos Θr =β− β + (1−β β) cos Θr
1−β β + (β− β) cos Θr=
β + β− (1+β β) cos Θi
1+β β− (β + β) cos Θi=
2β− (1+β2) cos Θi
1+β2 − 2β cos Θi
sin Θr = sin Θr
γ γ [1−β β + (β− β) cos Θr]= sin Θi
γ γ [1+β β− (β + β) cos Θi]= sin Θi
γ2 (1+β2 − 2β cos Θi)
obviously the relations νi sin Θi = νr sin Θr = νi sin Θi are verified, and one ob-
tains after calculation the expected result νr = γ2 νi
(1 + β2 − 2 β cos Θi
)from which
νr sin Θr = νi sin Θi; furthermore, from the definition of the reflected angle, it is easy
to obtain :
cos Θ′r =cos Θr −β
1 − β cos Θr
= γνr
ν ′
(cos Θr − β
)=
cos Θr − β
1 − β cos Θr
= γνr
ν ′(cos Θr − β
)then the apparent reflection factors on the interface are defined such that:
→Er = R||,E
→Ei
→Br = R||,B
→Bi, with R||,B =
∥∥∥∥ →Br
∥∥∥∥∥∥∥∥→Bi
∥∥∥∥ and R||,E =
⎛⎜⎜⎜⎜⎝Rxx||,E 0 0
0 Ryy||,E 0
0 0 0
⎞⎟⎟⎟⎟⎠and the total apparent electromagnetic field on the interface is given by:
→ET = E0i exp
(2 i π νi
c dV cos Θi
)ne
(Θi
)Ne
(Θi
)e− i Ψi
(I + R||,E
) (− sin Θi
n2⊥
→ex + cos Θi
n2||
→ey
)→BT = E0i
c Ne
(Θi
)exp
(2 i π νi
c dV cos Θi
)(1 + RB
||)
e− i Ψi→ez
where Ψi = 2π νi
c
[x0(1 − β cos Θi
)− n y sin Θi
]; then the continuity of the fields
implies:
* for the magnetic field: 1 + R||,B = T||,B ⇒ 1 + R||,B =νt n (1+ r||)νi Ne(Θi)
* for the tangential component of the electric field:
Tyy||,E = 1 + R
yy||,E
(1 + R
yy||,E) cos Θi
n2||
=T||,B
ne
(Θi
) cos Θt
114 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 87–120
* for the normal component of the electric induction (when the interface is assumed free
of charges and currents):
1 + Rxx||,E =
T xx||,En2
(1 + R
xx||,E)
sin Θi =T||,B
ne
(Θi
) sin Θt
One may notice that all these expressions (for parallel polarisation) are valid for an
uniaxial negative crystal, with β =n⊥ − n||n⊥ n|| − 1
where n⊥ and n|| are the principal refractive
indices of the crystal.
It is important to note here that the refractive index such that ds =
√n2 cos2 Θ+n2
|| sin2 Θ
nn||d x0
is not an effective index but only an apparent index, and the refractive phenomena which
occur inside the moving medium have to examined from the reference observer located
inside the medium point of view; hence, when the reference observer located in vac-
uum perceives an incoming intensity emerging from the moving medium, he knows its
transmitted direction and frequency, from which he can deduce from what precedes the
apparent incident direction and frequency, directly related to the real incident direction
and frequency perceived by the reference observer at rest in the medium: then the initial
problem, that is to find an invariant form of the radiative transfer equation has to be
examined from the internal observer’s point of view.
4. Derivation of the Radiative Transfer Equation
Let us now pay attention to the derivative operator Pα ∂α along a photon path inside
the medium, using the two sets of mater co-ordinates relatively to the observer located
inside the medium; from (72), one easily obtains that
∂∂ x0′ = γ
(∂
∂ x0 + βn
∂∂x
)∂
∂x′ = γ(β n ∂
∂ x0 + ∂∂x
)∂
∂y′ = ∂∂y
∂∂z′ = ∂
∂z
, (80)
Performing the calculation of the different contravariante components of the 4-impulsion
vector leads to
P 0′ = n2 h ν′c
= n2 h γ νc
(1 − β μ
)P x′
= h ν′ nμ′c
= h ν nc
γ(μ − β
)P y′ = h ν′ n sin Θ′ cos Φ′
c= h ν n sin Θ cos Φ
c P z′ = h ν′ n sin Θ′ sin Φ′c
= h ν n sin Θ sin Φc
, (81)
from which one obtains
P 0′ ∂∂ x0′ + P x′ ∂
∂x′ = hν n2
cγ2{[
1 − βμ + β(μ − β
)]∂
∂ x0 + 1n
[β(1 − βμ
)+ μ − β
]∂∂x
}= hνn
c
(n ∂
∂ x0 + μ ∂∂x
)= P 0 ∂
∂ x0 + P x ∂∂x
,
(82)
Electronic Journal of Theoretical Physics 4, No. 14 (2007) 87–120 115
and easily deduces that
Py′ ∂
∂y′+ P
z′ ∂
∂z′=
h ν n
c
(sin Θ cos Φ
∂
∂y+ sin Θ sin Φ
∂
∂z
)= P
y ∂
∂y+ P
z ∂
∂z,
(83)
Then one has the final and important result
P0′ ∂
∂ x0′ + Px′ ∂
∂x′+ P
y′ ∂
∂y′+ P
z′ ∂
∂z′= P
0 ∂
∂ x0 + Px ∂
∂x+ P
y ∂
∂y+ P
z ∂
∂z
rewritten under the more compact form
Pα′
∂α′ = Pα
∂α, (84)
which reveals that the derivative operator along a photon path is an invariant quan-
tity, unless one uses the fundamental mater variables(x0′ , x′, y′, z′
)and
(x0, x, y, z
)associated to the fundamental basis
(→e0′ ,
→ex′ ,
→ey′ ,
→ez′)
and(→e0,
→ex,
→ey,
→ez
); note that
Pα′∂α′ = h ν′ n
c
[n ∂
∂ x0′ + cos Θ′ ∂∂x′ + sin Θ′
(cos Φ′ ∂
∂y′ + sin Φ′ ∂∂z′
)]= h ν′ n
c
(n ∂
∂ x0′ +→Ω′
→grad
)= h ν′ n
c
(n ∂
∂ x0′ + ∂∂s′
)where ∂
∂s′ is the habitual curvilinear spatial derivative, that is the propagation unit vector
so as the gradient vector (expressed with the mater co-ordinates) are given in the vacuum
basis; similarly one has
P α ∂α = h ν nc
[n ∂
∂ x0 + cos Θ ∂∂x
+ sin Θ(cos Φ ∂
∂y+ sin Φ ∂
∂z
)]= h ν n
c
(n ∂
∂ x0 +→Ω
→grad
)= h ν n
c
(n ∂
∂ x0 + ∂∂s
)hence from Eq. (83) one may rewrite the derivative operator transformation under the
useful form
ν
(n
c
∂
∂t+
∂
∂s
)= ν ′
(n
c
∂
∂t′+
∂
∂s′
), (85)
It is now time to focus on the energetic invariant quantity, namely the specific intensity; let
us first remind the obtaining of the specific intensity when the system to be considered is
vacuum, following the steps developed by Mihalas [1]: if N is the photons number passing
through a surface element perpendicular to the particle speed vector at a given frequency
and a given propagation direction, this number in the comobile LRS is expressed as
N =L′
→Ω′
→dS dν ′ dΩ′ dt′
h ν ′=
L′ cos Θ′ dν ′ dΩ′ dS dt′
h ν ′, (86)
relatively to the observer LRS, this number is
N =Ldν dΩ
→dS dt
h ν
(→Ω −
→β)
=Ldν dΩ dS dt
h ν(cos Θ − β) , (87)
116 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 87–120
from which one immediately deduces that
L′ cos Θ′ dν ′ dΩ′
ν ′= γ
L dν dΩ
ν(cos Θ − β) , (88)
since the proper times are related with dt′ = dtγ
; moreover, using the optical aberration
and frequency Doppler shift transformation, cos Θ − β = ν′ cos Θ′γ ν
, hence, using the solid
angle conservation ν dν dΩ = ν ′ dν ′ dΩ′, one obtains
L′
ν ′3=
L
ν3, (89)
which justifies that I = Lν3 is the specific intensity for an unit refractive index medium;
if the refractive index is not one, analogously to what precedes, the specific intensity
is simply L′n2 ν′3 = L
n2 ν3 since relatively to the observer at rest inside the medium, the
moving medium remains isotropic of refractive index n; then since at rest the intensity isLn2 , well-known result [7], one deduces the previous result; then the left handed side term
of Eq. (1) obeys to the following relation
ν
[n
c
∂
∂t
(L
n2 ν3
)+
∂
∂s
(L
n2 ν3
)]= ν ′
[n
c
∂
∂t′
(L′
n2 ν ′3
)+
∂
∂s′
(L′
n2 ν ′3
)], (90)
If one considers now the number of photons emitted by an elementary volume in an
elementary solid angle around a given frequency interval, this number can be expressed
as
N =η′ dν ′ dΩ′ dV ′ d x0′
c h ν ′=
η dν dΩ dV d x0
c h ν, (91)
where η is the emissive power; due do the scalar density conservation, it comes that√−Det g d x0 dV =
√−Det g′ d x0′ dV ′
⇒ d x0 dV = d x0′ dV ′, (92)
hence one has from the previous result and with the help of Eq. (70)
η′
ν ′2=
η
ν2, (93)
similarly the number of photons absorbed in the same conditions is
N =κ′ L′ dν ′ dΩ′ dV ′ d x0′
c h ν ′=
κL dν dΩ dV d x0
c h ν, (94)
where κ is the absorption coefficient; from what precedes, one deduces that κ′L′ν′2 = κL
ν2 ,
and by definition of the specific intensity, one finally obtains
κ′ ν ′ = κ ν, (95)
so that the right handed side term of Eq. (1) obeys to the following relation
1
n2 ν2(η − κL) =
1
n2 ν ′2(η′ − κ′ L′) , (96)
Electronic Journal of Theoretical Physics 4, No. 14 (2007) 87–120 117
In the vacuum, the emissive power is simply the Planck function multiplied by the ab-
sorption coefficient κ L0, while in a medium of refractive index n, it is η = n2 κ L0, from
which it comes the invariant forms of the RTE
nc
∂∂t′(
L′n2 ν′3
)+ ∂
∂s′(
L′n2 ν′3
)= κ′
ν′3(L0 − L′
n2
)nc
∂∂t
(L
n2 ν3
)+ ∂
∂s
(L
n2 ν3
)= κ
ν3
(L0 − L
n2
) , (97)
Hence, noting the total spatial derivative dds
= nc
∂∂t
+ ∂∂s
, one finally obtains the usual
form of the invariant RTE
dIds
+ κ I = κ L0
ν3
⇒ I (sf ) = I (si) exp[− ∫ sf
s= siκ (s) ds
]+∫ sf
s= si
κ(s) L0(s)ν3(s)
exp[− ∫ sf
s′ = sκ (s′) ds′
]ds
,
(98)
Then the reference observer at rest in vacuum is able to determine the (real) radiative
field inside the moving medium from the perceived emerging directional and spectral
intensity field.
5. CONCLUSION
In this paper we described a way to construct a consistent “equivalent vacuum” and
“matter” space bound to the observer after a diagonalisation of the metric tensor related
to the Gordon’s metric, due to the moving (with a constant speed) particles of the non
unit refractive index semi-transparent medium; the construction of this space relatively to
the observer allows then the calculation of the optical aberration and frequency transfor-
mation in the new fundamental co-ordinates attached to the observer space, and leads to
the determination of the invariant specific intensity and the general form of the radiative
transfer equation in this space, following the method developed by Mihalas in vacuum.
We may expect to determine a more general formulation of this work by generalisation
to the case of non constant speed moving particles in a semi-transparent medium of non
constant refractive index.
118 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 87–120
References
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120 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 87–120
EJTP 4, No. 14 (2007) 121–128 Electronic Journal of Theoretical Physics
Quantum Images and the Measurement Process
Fariel Shafee∗
Department of PhysicsPrinceton University
Princeton, NJ 08540 USA
Received 12 March 2007, Accepted 25 March 2007, Published 31 March 2007
Abstract: We argue that symmetrization of an incoming microstate with similar states in a seaof microstates contained in a macroscopic detector can produce an effective image, which doesnot contradict the no-cloning theorem, and such a combinatorial set, with conjugate quantumnumbers can form virtual bound states with the incoming microstate. This can then be usedwith first passage random walk interactions to give the right quantum mechanical weight fordifferent measured eigenvalues.c© Electronic Journal of Theoretical Physics. All rights reserved.
Keywords: Quantum Measurement, Quantum Image, Quantum Bound State, No-cloningTheoremPACS (2006): 03.65.Ta, 03.65.Ud, 03.67.Mn
1. Introduction
Random walks [1] have long been a favorite sports enjoyed by many quantum physi-
cists in search of a rationale for quantum indeterminism [2]. Different stochastic models
for transitions to collapsed states on measurement have been presented by many authors
[3, 4, 5, 6, 7]. In a previous work [8] we have presented a picture of the transition of
a superposed quantum microstate to an eigenstate of a measured operator through in-
teractions with a measuring device, which are random in the sense of the stochasticity
introduced by the large number of degrees of freedom of the macrosystem, and not due to
any intrinsic quantum indeterminism. However, in our work we made the novel departure
of using first passage walks [9] which lead to a dimensional reduction of the path in sim-
plicial complexes to simplexes of lower dimensions by turn, a possible feature also noted
very recently by Omnes [10]. In the work cited we appealed to heuristic arguments in
analogy with electrodynamic images. In the present work we try to justify the emergence
122 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 121–128
of image-like subsystems in a macrosystem from quantum symmetry principles.
2. Symmetrization and Interactions
Interactions between systems may be due to Hamiltonians connecting operators that
explicitly connect components of different systems, or they may be due to symmetriza-
tion or anti-symmetrization of the states of the systems involved. For fermions, exchange
interaction yields the exclusion principle, which may have more dominant effects than
a weak potential in a many-particle system. For bosonic systems condensation at low
temperatures indicate the creation of macro-sized quantum states. Unlike the unitary
time-dependent operators representing the explicit interactions between systems through
the Hamiltonian, (anti-)symmetrization has no explicit time involvement, and a sys-
tem includes the (anti-)symmetrization of the component subsystems ab initio, which
continues until the states change and lose their indistinguishability. Alternatively, (anti-
)symmetrization comes into action as soon as an intermediate or final state is produced
involving identical particles, even when the initial system might not have had any. The
process therefore is apparently a discrete phenomenon, going together with the abrupt
action of the creation or annihilation of particles in field theory.
In terms of first quantized quantum mechanics, we understand the permutative (anti-
)symmetry properties of identical microstates (particles) in terms of the separation of the
co-ordinates). Two identical microsystems labeled 1 and 2 in a particular states a and b
has the combined (anti)symmetric wave function
ψ(1, 2)ab =1√2[ψ(1)aψ(2)b ± ψ(2)aψ(1)b] (1)
In practice the labels 1 and 2 for the two particles usually refer to the concentration
of the two particles in two different regions of space, for example, near two attractive
potential centers. So, the labels 1 and 2 are actually also interpretable as parameters for
two different states, and [11] it is possible to combine the two sets of labels into a single
set, say α and β and demand that
ψ(α, β) = ±ψ(α′, β′) (2)
where the sign for fermionic systems depends on the number of interchanges needed
to obtain the parameter sets α′ and β′ from the unprimed sets and for the bosons it is of
course always positive.
Even with ab initio symmetry built-in, it is well-known that a state can dynamically
evolve from a nearly factorized separable product state to a fully symmetric entangled
state as the overlap becomes high from nearly zero when the two subsystems (particles)
were well-separated initially. If we know that the incoming particles labeled 1 and 2 were
in states a and b at large separations then,
Electronic Journal of Theoretical Physics 4, No. 14 (2007) 121–128 123
ψ(x1, x2, a, b) =1√2[ψ(x1, a)ψ(x2, b) ± ψ(x2, a)ψ(x1, b)]
∼ 1√2ψ(x1, a)ψ(x2, b) (3)
for |x1 − x2| large, as the second term is small .
If the states a and b are identical, then it is well-known that this exchange interaction
for bosons gives an effective attractive interaction for small |x1 − x2|, as we get simply√2 times a single wave function, whereas for fermions it becomes highly repulsive as the
antisymmetry produces the exclusion principle.
3. State of the Detector
We shall consider a detector as macrosystem which consists of a large number of
microsystems identical with the microsystem to be detected, but in all possible different
states, including the incoming state to be detected, so that initially it appears like a
neutral unbiased system with respect to the state of the incoming microsystem. This
picture is comparable to that of a sea of quarks of all flavors and colors in a quark bag,
or even the similar content of a neutral vacuum when considering vacuum polarization
contributions. To maintain the quantum number of the vacuum, i.e. to give a singlet
with respect to all possible symmetry/classification groups, all these states occur paired
with conjugate anti-states (group theoretically inverse elements):
ΨD =∑a
ψDaψDa (4)
where the label D indicates states with positional peaks inside the detector. The
expression above is the simplest spectral decomposition for our purpose. In general there
will also be simultaneous multiple state/anti-state pairs, which will introduce new numer-
ical factors from combinatorics, but will not change the relative strengths of interactions
between the incoming microstate (S) and the pairing anti-states of the detector (D),
which is the crucial part of our measurement picture.
4. System-Detector Symmetry
For a bosonic microsystem system being detected, if it is in the state ψSi, sym-
metrization with the detector states gives
ψSD =1√2N
∑j �=i
(ψSiψDj + ψSjψDi)ψDj +1√N
(ψSiψDiψDi) (5)
when there are N states uniformly distributed in the detector, including the state i.
Normalization is ensured by the orthogonality of the states, when the coefficients are as
chosen.
124 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 121–128
However, if the microsystem was well-separated from the detector and symmetrization
was not invoked, the product state in a product space would be, with the macroscopic
detector still containing a superposition of all possible states:
ΨSD0 =1√N
∑allj
(ψSiψDjψDj) (6)
Since the functions ψSi and ψDi for an identical microsystem in the same state may
both actually represent the observed incoming microsystem in Eq. 6, we can rewrite Eq.
5 as
ΨSD =1√2(ΨSD0 + ΨDS0) +
1√N
(1 −√
2)ψSiψDiψDi (7)
Here both ΨSD0 and ΨDS0 represent an incoming particle in the state ψi and its
noninteracting product with the detector. Hence, the extra term ψSiψDiψDi represents
the ’exchange interaction’ resulting from the entanglement of the microstate with the
detector.
For incoming fermionic systems the arguments are similar, but somewhat more com-
plicated. In this case anti-symmetrization gives
Ψ[SD] =1√
2(N − 1)
∑j �=i
(ψSiψDj − ψSjψDi)ψDj (8)
Since the detector includes all other states but must exclude the state ψi due to
anti-symmetrization (exclusion principle), we can actually consider the sums in Eq. 8
as involving hole-antihole pair states ψhDiψ
hDi corresponding to ψi. So we have for the
combined system of the incoming microsystem ψi and the detector:
Ψ[SD] ∼ (ψSiψhDi − ψDiψ
hSi)ψ
hDi (9)
with the definitions:
ψhDiψ
hDi =
∑j �=i
ψDjψDj
ψhSiψ
hDi =
∑j �=i
ψSjψDj (10)
In the above analysis we have not considered the eigen-basis of the detector. As
we have considered the symmetrization aspects only, the state ψi occurs as a natural
preferred vector and for the other states j �= i we can consider any set orthogonal to ψi.
5. Quantum Images and the No-cloning Theorem
The exchange interaction term due to (anti-)symmetrization contains a product of
the incoming microstate ψSi, a corresponding state ψiD in the detector, which is the same
Electronic Journal of Theoretical Physics 4, No. 14 (2007) 121–128 125
microstate for bosons, or a hole ψhDi in the case of fermions, and also associated with such
a pair is a conjugate state ψDi or ψhDi for fermions. In the case of the bosonic systems
we shall call the latter conjugate state an image of the original incoming state created
by the symmetrization process. We do not consider the symmetric identical state ψDi as
the image, because the identical state nominally in the detector is indistinguishable from
the original incoming state and when there is an overlap of functions they may represent
the same physical entity. In the case of fermionic systems the incoming state ψiS and
the corresponding hole state ψhDi or its conjugate ψh
Di are in general all nonidentical
systems. Sine the incoming state is definitely not ψhSi, we can neglect the second term in
Eq. 9. Hence the effect of the antisymmetrization effectively gives a simple product as
for a bosonic system:
ΨSDferm= ψSiψ
hDiψ
hDi (11)
However, since the hole is more like a conjugate and the conjugate of the hole is more
like the original incoming microsystem, we can expect that both ψSi and ψhDi interact
in a similar manner with ψhDi.
There is no conflict with the no-cloning theorem [12] when (anti-)symmetrization
produces such quantum images, which, as we have seen, are either extensions of the
original functions, or are conjugate states. Though there is a one-to-one correspondence
with the incoming state, the states in the detector simply extend the original state by
(anti-)symmetry or produce a state which is conjugate to the original state, and is not
producible by a unitary operator assumed in the no-cloning theorem. In other words,
(anti-)symmetrization and the consequent exchange interactions are not producible by
the linear unitary operators and the simple and elegant proof of the no-cloning theorem
is inappropriate for quantum images of the kind described above.
6. Measurement and Eigenstates
Quantum images, formed by invoking symmetrization properties of the combined
system, do not depend on the operator involved in the measurement process associated
with the detector. The quantity measured is represented by a unitary operator in quan-
tum mechanics, and, if the microsystem is an eigenstate, it remains in the same state even
after measurement, but if it is a mixture of eigenstates of the operator, then it is taken as
a postulate of quantum mechanics that the emerging state after measurement is one of
the eigenstates and the detector too carries off the information of the final state to which
it collapses. We have shown recently [8] how a first passage random walk model reduces
an arbitrary linear combination of eigenstates to one of the component eigenstates with a
probability proportional to the square of the absolute magnitude of the coefficient of that
component. In that work we appealed to an electrostatic analogy for the formation of
the image in the detector which interacts with the incoming microsystem in steps, both
changing simultaneously till an eigenstate is reached.
If the state ψSi is expressed in terms of the eigenstates in a simple two-state system
126 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 121–128
ψSi = ai|α〉S + bi|β〉S (12)
then we get
ψDi = ai|α〉D + bi|β〉D (13)
and
ψDi = a∗i |α〉D + b∗i |β〉D (14)
and similarly for the hole states in the case of the fermionic systems.
This shows how the complex conjugate of the co-efficients occur in a natural way in
the image, which is not possible by cloning with a unitary operator.
Here we also see that the conjugate can interact interchangeably with the incoming
state or its indistinguishable extension in the detector and form virtual bound states
|SD〉i ∼ |ai|2|α〉S|α〉D + |bi|2|β〉S|β〉D|DD〉i ∼ |ai|2|α〉D|α〉D + |bi|2|β〉D|β〉D (15)
We can now think of the initial state of the virtual bound (SD) system to be a point in
a real space ( {xp = |ap(i)|2}, where we have now the running index p, in place of the α and
β for the 2-dimensional case, to indicate the label of the eigenvalue) of n-dimensions, if
the microsystem can have n different eigenvalues of the operator representing the quantity
to be observed.
|ψ〉i =n∑
p=1
xp(i)|S >p |D >p (16)
The process of interaction between the detector and the microsystem proceeds as a
first passage random walk in this x space, with the constraint
∑p
xp(t) = 1 (17)
which describes a n-dimensional plane restricted to the sector 0 ≤ xp(t) ≤ 1. We are
also now using the notation of time or step t, with xp beginning at the initial values of
the co-ordinates.
The random walk can be described [9] by a diffusion equation for small steps. The
concentration of path points, i.e. the probability c of finding the system at x at time t,
can be found [8, 9] from an integrable Green’s function of the corresponding equation for
the Laplace transform c, with D a diffusion constant:
∇2c(x, s) − (s/D)c(x; s) = −c(x, t = 0)/D (18)
Electronic Journal of Theoretical Physics 4, No. 14 (2007) 121–128 127
The interesting thing about this random walk is that, whenever a path reaches a
co-ordinate at an edge of the plane of motion, with say xq = 0, the walk continues in the
lower dimensional sub-simplex confined to this fixed value of xq. Eventually, the path
ends at a vertex, say, xf = 1, with all other xp �=f = 0, and the probability for reaching it
is obtained [8] from the gradient of c at the vertex.
pf = k|af (t = 0)|2. (19)
Usual quantum mechanics postulates this relation, and considers a derivation impossi-
ble. Here k represents the detector efficiency, which includes the strength of the coupling
between S and D.
7. Conclusions
We have shown above that if the detector is a macroscopic system and is initially
neutral with respect to the measured quantity, which we have expressed as the sum
of microstates with all different states, then symmetrization with the measured system
for bosonic systems or anti-symmetrization for fermionic systems breaks the neutrality
in a unique way which may be regarded as the formation of a quantum image of the
measured microsystem in the detector. These images are conjugates of the incoming
microsystems, or hole-type states equivalent to conjugate states, and since the process is
not a linear unitary operation, the no-cloning theorem does not pose a problem. That the
interaction between the incoming state and these images can be modeled by first passage
random walks to give the probabilities for different eigenstates as final states of both the
incoming state and the detector’s microstate component has been shown in [8]. We shall
later examine the question of measurement of entangled systems in spatially separated
detectors.
Acknowledgements
The author would like to thank Professor R. Omnes of Universite de Paris, Orsay,
for useful feedback from the earlier work.
128 Electronic Journal of Theoretical Physics 4, No. 14 (2007) 121–128
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