EJTP · 2008. 5. 17. · some scalar ﬁeld , a non-minimal coupling between the scalar curvature...

Volume 3 Number 11

Electronic Journal of Theoretical Physics

ISSN 1729-5254

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Table of Contents

No Articles Page

1 Non-Minimal Coupling Effects of the Ultra-Light Particles on Photons Velocities in the Radiation Dominated Era of the Universe. El-Nabulsi Ahmad Rami

1

2 A Toy Model of Financial Markets J. P. Singh and S. Prabakaran

11

3 Rayleigh process and matrix elements for the one-dimensional harmonic oscillator J.H. Caltenco, J.L. López-Bonilla, and J. Morales

29

4 Identical synchronization in chaotic jerk dynamical systemsVinod Patidar and K. K. Sud

33

5 Second Order Perturbation of Heisenberg Hamiltonian for Non-Oriented Ultra-Thin Ferromagnetic Films P. Samarasekara

71

6 Frameable Processes with Stochastic Dynamics Enrico Capobianco

85

7 Ab-initio Calculations for Forbidden M1/E2 Decay Rates in Ti XIX ion A. Farrag

111

8 Some Properties of Generalized Hypergeometric Thermal Coherent States Dusan Popov

123

9 Space-Filling Curves for Quantum Control Parameters Fariel Shafee

133

10 The Spectrum of the Lagrange Velocity Autocorrelation Function in Confined Anisotropic LiquidsSakhnenko Elena I and Zatovsky Alexander V.

143

11 On the Quantum Correction of Black Hole ThermodynamicsKourosh Nozari and S. Hamid Mehdipour

151

12 A Graphic Representation of States for Quantum CopyingSara Felloni and Giuliano Strini

159

EJTP 10 (2006) 1–10 Electronic Journal of Theoretical Physics

Non-Minimal Coupling Effects of the Ultra-LightParticles on Photons Velocities in the Radiation

Dominated Era of the Universe

El-Nabulsi Ahmad Rami ∗

Plasma Application Laboratory, Department of Nuclear andEnergy Engineering and Faculty of Mechanical, Energy and Production Engineering,

Cheju National University, Ara-dong 1, Jeju 690-756,Korea

Received 6 September 2005 , Accepted 27 November 2005, Published 25 May 2006

Abstract: The effect of the ultra-light masses of the order of the Hubble constant, implementedin Einstein’s field equations from non-minimal coupling and supergravities arguments, onphotons velocities in the radiation dominated epoch of the Universe within the frameworkof non-minimal interaction of electromagnetic fields with gravity is developed and discussed indetails.c© Electronic Journal of Theoretical Physics. All rights reserved.

Keywords: Non-minimal coupling, ultra-light masses, effective cosmological constant, photonsvelocity.PACS (2006): 04.40 Nr, 98.80 -k, 95.30 SI

1. Introduction

A standard result of Einstein’s gravity is that massless particles, in particular photons,

move at light celerity ‘c’. A question worth examining is whether the velocity of massless

photons is shifted when these later propagate in exotic background filled of ultra-light

particles (ULP) of tiny masses in the order of the Hubble constant (m ≈ H). This couldhave important cosmological and astrophysical implications. In fact, the possibility of

shifting photon propagation (SPP) in gravitational fields (or non-trivial topologies) is

an interesting prediction of quantum field theory in curved space-time. It appears that

photon propagation may depend on their direction and polarisation, travel with speeds

exceeding the normal speed of light ‘c’ [1]. It is a quantum effect induced by vacuum

∗ [email protected]

2 Electronic Journal of Theoretical Physics 10 (2006) 1–10

polarisation (allowing the photons to exist as a virtual e+e−pair so that at the quantumlevel it is characterized by the Compton wavelength of the electron) and implies that

the Principle of Equivalence does not hold for interacting quantum field theories such

as QED. The propagation of photon in Schwarzschild, Robertson-Walker, gravitational

wave, de Sitter backgrounds, charged black hole were done and remarkable results were

discovered [2]. In each case (except the totally isotropic de Sitter space-time) it was

possible to find directions and polarisations for which the photon velocity exceeds ‘c’.

Generalization to neutrino propagation in a Robertson-Walker metric using the Weinberg-

Salam model was done in [3]. In a gravitational field, the photon propagation is sensitive

to an anisotropic space-time curvature and may depend on this later [4,5]. Recently,

a series of papers has appeared in which the light velocity varies in the early Universe

and this solves the horizon, monopole and the flatness problems in standard cosmology

[6,7,8,9]. In this work, we will investigate further the consequences of non-minimal

coupling on light velocity in the presence of ultra-light particles.

2. Non-minimal Coupling, Supergravities Arguments and Ein-

stein Fields Equations

We start with the non-minimal interaction of electromagnetic fields with gravity in the

following form L̃ =√

gξRμνFμα F

νμ, ξ being the coupling constant, Rμν the Riemann

tensor, g the metric scalar and Fμν = ∂μAν −∂νAμ is the electromagnetic strength (Aμ isthe vector potential). Terms like RF 2 are neglected here. In this way, the field equations

read [10,11,12]:

Dν [Fμν − 2ξ (RμαFαν − RναFαμ)] = 0

where Dν is the covariant derivative. Let us now restrict our attention to the form of the

Riemann tensor we will choose for this work. In a recent paper [13], we introduced, for

some scalar field , a non-minimal coupling between the scalar curvature and the density of

the scalar field in the following form L = −ξ√gRφ∗φ, ξ = 1/6 . R is the scalar curvatureand is the complex conjugate of . From a view point of quantum field theory in curved

space-time, it is natural to consider such a non-minimal coupling. In fact, the conformal

case results in an extension of the property of conformal invariance for massless fields,

which is attractive from physical point of view. This parameter describes the strength

of the coupling between the curvature of spacetime and the inflation. Minimal coupling

corresponds to ξ = 0. It was shown that in this case and for a particular scalar negative

complex potential field V (φφ∗) = 3/4m2(ωφ2φ∗2 − 1),ω being a tiny parameter inspiredfrom supergravity inflation theories, ultra-light masses ‘m’ are implemented naturally in

Einstein field equations (EFE), leading to a cosmological constant ‘Λ’ in accord with

observations2. In matter-free background, the scalar curvature was found to be =4Λ̄

2 It has been argued that a non-minimal coupling term-generated by quantum corrections-is to be expectedwhenever the space-time curvature is large; in most theories that describe inflationary scenarios, it turnsout that a value of ξ different from zero is unavoidable. As a matter of fact, it seems sensible to consideran explicit non-minimal coupling in the supergravities inflationary paradigm.

Electronic Journal of Theoretical Physics 10 (2006) 1–10 3

where Λ̄ = Λ−3/4m2 is the effective cosmological constant3 (in natural units, m ≈ �H/c2where ‘�’ is the Planck constant and ‘c’ being the celerity of light). As a result, one

candidate Einstein field equations is(� = c = 1):

Gμν ≡ Rμν − 12gμνR = −Λ̄gμν (1)

Λ̄ = Λ − 3m2/4 ≡ Λ1 + Λ2 is the effective cosmological constant (Λ2 = −3λ̄−2C whereλC = �/mc is the Compton wavelength in natural units). Remark that for Λ1 = 0, the

scalar curvature is negative and the space-time is not Minkowskian. Other field equations

exist also but correspond only for p = ρ/3 (radiation era):

Rμν − 12gμνR + Λgμν = −8πG

[(p + ρ +

3m2

8πG

)uμuν + pgμν

](2a)

Rμν − 12gμνR = −8πG

[(p + ρ +

3m2

8πG

)uμuν +

(p +

Λ

8πG

)gμν

](2b)

= −8πG[(p + ρ̄) uμuν +

(p +

Λ

8πG

)gμν

](2c)

≡ −8πG[Tμν (p, ρ) +

Λ

8πGgμν + tμν (m)

](2d)

≡ −8πG [Tμν (p, ρ) + Tμν (Λ,m)] (2e)≡ −8πG

∑Tμν (2f)

p and ρ are the pressure and density of matter,ρ̄ = ρ + 3m2/8πG, tμν = 3m2/8πGuμuν

and:

Tμν (p, ρ) = (p + ρ) uμuν + pgμν (3)

Tμν (Λ,m) = tμν +Λ

8πGgμν =

1

8πG

[3m2uμuν + Λgμν

](4)

Contracting equations (2) with gμν using gμνuμuν = −1 yields of course. In this way,the ultra-light masses and the cosmological constant are parts of the matter contents of

the Universe rather than geometrical entities. The radiative field equations (RFE) (2-

a,b,c,d,e,f) are identical to that of Einstein standard ones but with an additional energy

density ρm = 3m2/8πG(m ≤ H). One can also refer to equation (4) as the stress-energy

tensor of vacuum and light particles, which is a ”microscopic stress-energy tensor”. In

fact, the conservation law holds and we have:

∇ν∑

Tμν ≡ ∇νTμν (p, ρ) + ∇νTμν (Λ,m) = 0 (5)

When Tμν (p, ρ) = 0, the microscopic stress-energy tensor will behave as the macroscopic

one if we assume that:

P (Λ,m) ≡ PΛ = Λ/8πG (6)3 In [13], 8πG ≡ κ was set equal to unity.


ρ (Λ,m) =3m2 − Λ

8πG(7)

That is, in the microscopic version, if Λ = 0, P = 0 but the density is positive. While

for Λ > 3m2, the pressure is positive and the density is negative. If 0 < Λ < 3m2, then

both the pressure and the density are positive.

Before treating the non-minimal coupling scenario, we will discuss briefly the impli-

cations of equations (2) in standard cosmology. For this, we consider a homogenous and

isotropic Universe in the radiation dominated epoch described by Friedman-Robertson-

Walker line element with scale factor a (t)[14]. The radiative field equations read:

ȧ2

a2+

k

a2=

8πGρ

3+

Λ

3+ m2 (8)

ä

a= −8πGρ

3+

Λ

3− m

2

2(9)

k = −1, 0, +1 is the curvature constant for open, flat or closed space-time and dim (Λ) =dim (m2) = length−2. If the cosmological constant and the ultra-light masses are assumedto be constant with time, then from the energy conservation law: ρ ∝ a−4. For zerodensity, 2Λ > ( ( Λ, Λ̂ → 3/4m2. If for instance, we assume that m2 = β/a2and Λ = δȧ2/a2 +ηä/a, β, δ, η are constants [20,21], than from equation (10)4:

ä

a

(1 − 2η

3

)+

ȧ2

a2

(1 − 2δ

3

)+

(k − β

2

)1

a2= 0 (12)

which gives:

ȧ2 =3 (β − 2k)2 (3 − 2δ) + Da

− 2(3−2δ)3−2η , D = const. (13)

4 The fact that the two terms Λ and m2 play the role of two cosmological constant in the theory, wehave the freedom to choose Λ = δ1ȧ2

/a2 + η1ä/a + β1

/a2 and m2 = δ2ȧ2

/a2 + η2ä/a + β2

/a2where

δ1,2, η1,2, β1,2 are constants. In this work, we simplified our assumptions just to have at the beginning asimple idea about the effects of the ultra-light masses in the theory.


For D = 0 which corresponds to singular solutions, one finds for flat space-time(k = 0):

a =

√3β

2 (3 − 2δ)t (14)

where δ < 3/2, β > 0. In this way :

m2 =2 (3 − 2δ)

3t2(15)

Λ =δ

t2(16)

From equation (8), we find:

ρ =3 (δ − 1)8πGt2

(17)

with 1 < δ < 3/2. In this way, we don’t have an inflationary phase and no horizon

problem appears. From the above equations, we see that the ultra-light masses, the

cosmological constant and the density are independent of the value of β and η. The

Hubble parameter is H = ȧ/a = 1/t and the density matter of the Universe is given by

Ωr = ρ/ρc = δ − 1 < 1/2 where ρc = 3H2/8πG is the critical density. The decelerationparameter is q ≡ −äa/ȧ2 = 0. The density parameter due to vacuum contribution isΩΛ = Λ/3H2 = δ/3 and that due to ultra-light particles contributions is Ωm = m2/H2 =

2 (3 − 2δ)/3. In this way ΩTotal = Ωr + Ωm + ΩΛ = 1 as required by inflation [22]. Theultra-light particles than contribute to the total energy density and their masses decrease

as inverse to time. Note from equations (15) and (16) that Λ = 3δm2/2 (3 − 2δ) < 9m2/4.Finally, note that when the ‘Λ’ and ‘m2’ terms dominate the dynamics of equation (8)

with the assumption that the Universe undergoes a long period of evolution during which

the celerity of light changes as c = c0an, c0, n =constants [8]:

ȧ2

a2=

Λc2 (t)

3+ m2c2 (t) →

(Λ

3+ m2

)c2n0 a

2n (18)

So at large times, we have a ∝ t−1/n and it was found in [8] that for negative ”n”, thereis a solution to the quasi-lambda problem.

In order to have a very simple idea about the role of the ultra-light masses in the

theory, we suppose that the space-time is flat, that is k = 0 with the following behavior of

the ultra-light masses m2 = β/a2and the cosmological constant Λ = δ/a2, β, η=constants

(see footnote 4 ) [23,24,25]. In this case, when ‘Λ’ and ‘m2’ terms dominate at large

times the dynamics of equation (8):

ȧ2

a2=

Λc2 (t)

3+ m2c2 (t) =

(β +

δ

3

)c20a

2n−2 (19)

That is a ∝ t−1/n−1 and from [8,26], it is required that n < 0 and c = c0t−n/n−1. Insummary, m2 ∝ t2/n−1 and as a result mc ∝ 1/t. Another way to study shifting andtime-varying photons velocities is by using the non-minimal coupling of electromagnetic

fields and gravity.


3. Varying Photons Velocities From Non-Minimal Coupling

Following [10,11], we admit the existence of a surface S represented by φ (x) = 0. The

wavenumber of the photon trajectories is given by the gradient of its phase kλ = ∇λφwhere the Faraday tensor vanishes at its hypersurface, that is (Fμν)S = 0. Its derivative

defines a function φμν such that:

(∂λFμν)S = (DλFμν)S = kλφμν (20)

As a consequence, equation (1) takes the form:

[φμν − 2ξ (Rμαφαν − Rναφαμ)] kν = 0 (21)

In the radiation dominated era, it follows that:

φμνkν − 2ξ{[

φαν . (−χ){{

4ρ

3+ ρm

}uμuν +

{ρ

3+

ρm2

− Λχ

}}δμν

]

+

[−φαμ. (−χ)

{{4ρ

3+ ρm

}uνuα +

{ρ

3+

ρm2

− Λχ

}}δνα

]}kν = 0 (22)

where χ = 8πG (� = c = 1). For simplicity, we let k0 = kμuμ and we use the antisym-

metric fact of (φμν − φνμ = 2φμν) as well as Maxwell equations:

φμνkλ + ϕνλkμ + φλμkν = 0 (23)

By contracting by kλ the last equation, equation (23) reduces to:

φμνkν =−2ξχ (4ρ

3+ ρm

)1 + 4χ

(ρ3

+ ρm2− Λ

χ

)ξk0φ

μνuν ≡ (N) k0φμνuν (24)

Replacing (24) into (23), then:

φμνk2 + N (−φμνkλ − ϕλμkν − kνφμλ) k0uλ = 0 (25)

The antisymmetric of φμνeliminates all the terms in the parentheses of equation (25) and

we are left with:

φμν(k2 − Nk20

)= 0 ⇒ k2 − Nk20 = 0,∀φμν (26)

The effective photons velocity, in case allρ, ρm, Λ = 0, ∀ξ is then given by [6]:

v2 =|kiki|k20

= |1 + N | =∣∣∣∣∣∣ 1 − 4ξΛ − 4ξχ

ρ3

1 + 4ξχ(

ρ3

+ ρm3− Λ

χ

)∣∣∣∣∣∣ (27)

and the light velocities is not equal to ‘c’. Adopting equations (15), (16) and (17),

equation (27) takes the form in normal units:

v2 =

∣∣∣∣∣1 − 4 ξt2 (2δ − 1)1 + 4 ξt2

(5 − 4δ)

∣∣∣∣∣ (28)


with 1 < δ < 3/2. As a result, for ξ > () c (light celerity). Adopting

the fact c = c0an with n < 0, than the photons velocities decreases with time whatever

is the sign of the coupling constant.

An interesting case is when the background is ‘free from matter ’ . From (27) we get:

v2 =

∣∣∣∣ 1 − 4ξΛ1 + 4ξ (m2 − Λ)∣∣∣∣ (29)

If m2 = 0, than v2 = 1 which is light celerity in units (� = c = 1). Assuming m2 = β/t2,

4Λ = 3m2 or Λ̄ = 0 and as a result R = 0. In this case, equation (28) gives:

v2 =

∣∣∣∣1 − 3m2ξ1 + m2ξ∣∣∣∣ =

∣∣∣∣∣1 − ξ 3βt21 + ξ βt2

∣∣∣∣∣ =∣∣∣∣t2 − 3ξβt2 + ξβ

∣∣∣∣ (30)Again, if ξ > () c (light celerity). As a result, the velocity of photons is

affected and shifted by the presence of the ultra-light tiny masses and depends on the sign

of the coupling constant. It doesn’t correspond in fact to null geodesics as in the standard

case. Positive coupling constant corresponds to friction and negative one corresponds to

superluminal case [27,28]. If we adopt the fact c = c0an, then the photons velocities not

only is shifted but also decrease with time if n < 0 and increase if n > 0.

The constancy of the speed of light is not preserved in this analysis. It depends on

how is filled the background space and how is used a coupling constant different of zero

that modifies presentation of the Einstein’s Field Equations (EFT ), with an additional

term. It is important to notice that the environment where speed of light reaches its

maximum value is the lightest one: the empty space, all because of the constancy of the

speed of light law, which in time, originates the fourth time-coordinate. In our case, the

red-shift coefficient ‘z’ varies with time according to cz = Hr combined to equations (29)

or (30) for a matter free background. ‘r’ is supposed to be the distance form the galaxy

to the earth [14]. If the coupling constant is assume to be positive, one can than have a

cosmological model based on interpretation of the red shift by decrease of the light speed

with time everywhere in the universe beginning with a certain moment of time in the

past. Of course, the agreement with the fundamental physics laws will be completed by

introducing in a future work the evolution of other fundamental constants synchronously

with the variation of the light speed [29].

Finally, we note that recently, growing amount of astrophysical data show important

evidence for statistical and apparent physical association between low-redshift galaxies

and high-redshift quasi-stellar objects suggesting noncosmological origin of their redshift

and as a result failure of classical quasar explanation [30]. The author found analytical

solution of Einstein equations describing bubbles made from axions with periodic inter-

action potential considered as one of the leading dark matter candidate. Remember that

in our model [13], the ultra-light masses implemented in Einstein field equations enabled

us to solve the ‘missing mass problem’ and as result considered as dark matter candidate.

In Minkowski space, objects at constant proper distance with respect to an observer have

zero redshift. However, in an expanding universe special relativistic concepts do not gen-

erally apply. In fact, a galaxy with zero total velocity does not have zero redshift even


in the empty universe case. This demonstrates that cosmological redshifts are not spe-

cial relativity Doppler shifts [31,32]. It was also proved that Minkowski coordinate and

the Robertson-Walker coordinates (FRW universe) are interchangeable descriptions for

an empty universe. However, velocities in the Minkowski universe are not equivalent to

velocities in the FRW universe because of the different definitions of time and distance in

these two models. A coordinate transform relates velocities in the Minkowski universe to

velocities in the FRW universe. Superluminal recession velocities in the FRW universe

do not violate special relativity because they are not in the observer’s inertial frame.

4. Conclusions:

In this work, we used the Einstein’s field equations with effective cosmological constant

inspired from non-minimal coupling and supergravities arguments to study the conse-

quences of non-minimal coupling between electromagnetic fields and gravity on light

velocity in the presence of ultra-light particles at the radiation dominated epoch of the

Universe. We showed that if the cosmological constant and the ultra-light square masses

varies as Λ = δȧ2/a2+ηä/a and m2 = β/a2, then at singular solutions and for a flat space-

time, Λ, m2 and ρ decreases with time as 1/t2, the Hubble parameter vary as H = 1/t,

the deceleration parameter is zero and ΩTotal = Ωr +Ωm+ΩΛ = 1 as required by inflation.

As a result, the ultra-light particles than contribute to the total energy density and their

masses decrease as inverse to time. When the ‘Λ’ and ‘m2’ terms dominate the dynamics

of our field equations with the assumption that the Universe undergoes a long period of

evolution during which the celerity of light changes as c = c0an, it was found that at large

times a ∝ t−1/n and that for negative ‘n’, there is a solution to the quasi-lambda problem.Finally, we studied varying light velocities from non-minimal coupling. We found that

photons velocities depends on the coupling constant and only on δ ∈ (1, 3/2)in a waythat ξ > () c with m

2 > 0.

The model described in this paper could have important implications in various sys-

tems, in particular cosmological scenarios, black hole physics and quantum interactions

[33,34,35,36,37]. It is important to discuss the impact of the assumptions established to

reach the model and the fact that a non-Minkowskian space is necessary to obtain speeds

greater than that of light, and that photon could be represented by a positron-electron

pair for characterizing it with the electron’s Compton wavelength.

To further investigate all these issues, further studies will be necessary and work is in

progress.

Acknowledgments:

The author is grateful for the referees for their useful comments and suggestions.


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EJTP 3, No. 11 (2006) 11–27 Electronic Journal of Theoretical Physics

A Toy Model of Financial Markets

J. P. Singh and S. Prabakaran ∗ †

Department of Management StudiesIndian Institute of Technology Roorkee

Roorkee 247667, India

Received 27 October 2005 , Accepted 9 January 2006, Published 25 June 2006

Abstract: Several techniques of fundamental physics like quantum mechanics, field theoryand related tools of non-commutative probability, gauge theory, path integral etc. are beingapplied for pricing of contemporary financial products and for explaining various phenomenaof financial markets like stock price patterns, critical crashes etc.. In this paper, we apply thewell entrenched methods of quantum mechanics and quantum field theory to the modeling ofthe financial markets and the behaviour of stock prices. After defining the various constituentsof the model including creation & annihilation operators and buying & selling operators forsecurities, we examine the time evolution of the financial markets and obtain the Hamiltonianfor the trading activities of the market. We finally obtain the probability distribution of stockprices in terms of the propagators of the evolution equations.c© Electronic Journal of Theoretical Physics. All rights reserved.

Keywords: Economic Physics , Financial Markets , Stock Prices, Quantum ModelsPACS (2006): 89.65.Gh , 03.65.-w, 02.50.-r, 05.30.-d, 02.30.Tb

1. Introductionn

The specialty of “physics” is the study of interactions between the various mani-

festations of matter and its constituents. The development of this subject over the last

several centuries has led to a gradual refining of our understanding of natural phenomena.

Accompanying this has been a spectacular evolution of sophisticated mathematical tools

for the modeling of complex systems. These analytical tools are versatile enough to find

application not only in point processes involving particles but also aggregates thereof

leading to field theoretic generalizations and condensed matter physics.

Furthermore, with the rapid advancements in the evolution and study of disordered

∗ [email protected]† Jatinder [email protected]

12 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 11–27

systems and the associated phenomena of nonlinearity, chaos, self organized criticality

etc., the importance of generalizations of the extant mathematical apparatus to enhance

its domain of applicability to such disordered systems is cardinal to the further develop-

ment of science.

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

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 origin of the association between physics and finance, though, can be traced way back

to the seminal works of Pareto [20] and Batchlier [21], the former being instrumental in

establishing empirically that the distribution of wealth in several nations follows a power

law with an exponent of 1.5, while the latter pioneered the modeling of speculative prices

by the random walk and Brownian motion. The cardinal contribution of physicists to the

world of finance came from Fischer Black & Myron Scholes through the option pricing

formula [22] which bears their epitaph and which won them the Nobel Prize for economics

in 1997 together with Robert Merton [23]. They obtained closed form expressions for the

pricing of financial derivatives by converting the problem to a heat equation and then

solving it for specific boundary conditions.

The theory of stochastic processes constitutes the “golden thread” that unites the

disciplines of physics and finance. Modeling of non relativistic quantum mechanics as

energy conserving diffusion processes is, by now, well known [24]. Unification of the

general theory of relativity and quantum mechanics to enable a consistent theory of

quantum gravity has also been attempted on “stochastic spaces” [25]. Time evolution of

stock prices has been, by suitable algebraic manipulations, shown to be equivalent to a

diffusion process [26].

Contemporary empirical research into the behavior of stock market price /return pat-

terns 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 tαwith 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 [27].

There is an intricate yet natural relationship between the power law tails observed in

stock market data and probability distributions that emanate as the solution of the Fokker

Planck equation. The Fokker Planck equation is known to describe anomalous diffusion

Electronic Journal of Theoretical Physics 3, No. 11 (2006) 11–27 13

under time evolution. Empirical results [28-31] 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 [32-36]. In [36], 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 [32] 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 [32].

Stock market phenomena are 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 [37-38] was formulated

with one primary objective – to create a scenario which would justify the use of stochastic

calculus [39] for the modeling of capital markets.

The Efficient Market Hypothesis contemplates a market where all assets are fairly

priced according to the information available and neither buyers nor sellers enjoy any

advantage. Market prices are believed to reflect all public information, both fundamental

and price history and prices move only as sequel to new information entering the market.

Further, the presence of large number of investors is believed to ensure that all prices

are fair. Memory effects, if any at all, are assumed to be extremely short ranging and

dissipate rapidly. Feedback effects on prices are, thus, assumed to be marginal. The

investor community is assumed rational as benchmarked by the traditional concepts of

risk and return.

An immediate corollary to the Efficient Market Hypothesis is the independence of

single period returns, so that they can be modeled as a random walk and the defining

probability distribution, in the limit of the number of observations being large, would be

Gaussian.

Anomalous diffusion is a hallmark of several intensively studied physical systems. It

is observed, for example, in the chaotic dynamics of fluid in rapidly rotating annulus [40],

conservative motion in a periodic potential [41], transport of fluid in a porous media [42],

percolation of gases in porous media [43], crystal growth spreading of thin films under

gravity [44], radiative heat transfer [45], systems exhibiting surface to surface growth [46]

and so on.

Several analogies between physical systems and financial processes have been explored

in the last decade, some of which have already been mentioned above. Perhaps, the most

striking one is that between financial crashes witnessed in stock markets and critical


phenomena like phase transitions is discussed here to place the main theme of this paper

in its proper perspective.

Stock market crashes are believed to exhibit log periodic oscillations which are char-

acteristic of systems exhibiting discrete scale invariance i.e. invariance through rescaling

by integral powers of some length scale like the Serpinski triangle and other similar fractal

shapes. In the years preceding the infamous crash of October 19, 1987, the S & P market

index was seen to fit the following expression exceedingly precisely [47-48],

(S & P )t = Ω + Γ (tc − t)γ {1 + Ξ cos [θ ln (tc − t) + φ]}Physicists working in solid state and condensed matter physics would immediately

recognize the analogy of the above expression with the one obtained for critical phe-

nomenon in spin model of ferromagnetism [49]. We briefly elucidate the salient features

of this model. Crystalline solids comprise of atoms arranged in a lattice. Each such

atom generates a magnetic field parallel to the direction of the atom’s spin. In the case

of substances that do not exhibit ferromagnetic character, these spin directions are ran-

domly oriented so that the aggregate magnetic field vanishes. However, in ferromagnetic

substances these spins are polarized in a particular direction resulting in a nonzero ag-

gregate field. Ferromagnetic substances usually exhibit two distinct phases. one in which

the spins orient themselves in a particular direction resulting in an aggregate magnetic

moment at temperatures below a well defined critical temperature tc and the other where

the spins are disoriented with a zero aggregate moment above the critical temperature.

At temperatures below tc, the coupling force between neighboring atoms predominates

resulting in an alignment of spins whereas above tc the additional energy manifests itself

in disorienting (randomizing) the spins.

Renormalization group theory enables us to group these atoms in blocks of spins

whose composite spins are equal to the algebraic sum of the spins of the atoms con-

stituting the block. It then provides that a model involving interactions between these

composite spins of a block can be constructed that replicates the macroscopic proper-

ties of the block and yet cannot depend on the size of the block. That is, the sys-

tem would exhibit a scaling symmetry, which is discrete, if we allow for the finite size

of the atom and continuous otherwise. The magnetic susceptibility of such a mag-

netic substance defined by χ (T ) = ∂M∂B

∣∣B=0

, where the symbols have their usual mean-

ing, obeys a power law of the form χ (T ) = Re[(T − Tc)α+iβ

]or equivalently χ (T ) =

(T − Tc)α {1 + β cos [ln (T − Tc)] + O (β2)}which is reminiscent of op cited expression forlog periodic oscillations in financial crashes.

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 [50] of stock market price data, besides doing

Rescaled Range Analysis [51] 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 [52-55].

In this paper, we attempt one such model. The objective is to apply the well en-

trenched methods of quantum mechanics and quantum field theory to the study of the

financial markets and the behaviour of stock prices. Section 2, which forms the essence of

this paper, arrives at various results for financial markets by modeling them as quantum

Hamiltonian systems. The probability distribution for stock prices in efficient markets is

also obtained. Section 3 concludes.

1.1 Quantum Model of Financial Markets

We consider an “isolated” financial market comprising of n investors and m type of

securities. The market is “isolated” in the sense that new types of securities are nei-

ther created nor are existing ones destroyed. Further, the number of investors is also

constant. The investor i, i = 1, 2, 3.......n is assumed to possess a cash balance of

xi, i = 1, 2, 3.......n (which may be negative, representing borrowings) and yij (z) , i =

1, 2, 3.....n; j = 1, 2, 3....m units of security j at a unit price of z . Obviously, yij ≥ 0,∀i, j.Towards constructing a basis for our Hilbert space representing the financial market,

we define a pure state of the system as

|Ψi〉 = |{xi, {yij (z) , j = 1, 2, ...m} , i = 1, 2, ...n}〉 (1)Thus, a pure state represents a state of the market where the entire holdings of cash

and securities of every investor are known with certainty. This represents a complete

measurement of the market and hence, is in conformity with the standard definition of

“pure state” of a system.

A basis for our Hilbert space may then be constituted by the set of all the pure states

of the type (1) i.e.

Ψ = {|{xi, {yij (z) , j = 1, 2, ...m} , i = 1, 2, ...n}〉} (2)The elements of this basis set Ψ satisfy the orthogonality condition 〈Ψi | Ψj〉 = δij withrespect to the scalar product defined in the sequel. The orthogonality condition makes

sense in the financial world – it implies that if a market is in a pure state |Ψi〉 then itcannot be in any other pure state.

However, a complete measurement of the market is, obviously, not practicable in real

life. At any point in time, we are likely to have certain information only about a fraction

of the market constituents. Hence, the instantaneous state of the market |ψ (t)〉 may berepresented by a linear combination of the pure states |Ψl (t)〉 i.e.

|ψ (t)〉 =∑

l

Cl |Ψl (t)〉 (3)

We endow our Hilbert space H with the scalar product

〈ψ (t) | ξ (t)〉 =∑l,m

C∗l Dm 〈Ψl (t) | Ψm (t)〉 =∑l,m

C∗l Dmδlm =∑

l

C∗l Dl (4)


where we have assumed the orthogonality of the pure states.

The components of the state space vector |ψ (t)〉are given byCl = 〈Ψl (t) | ψ (t)〉 andare related to the probability of finding the market in the pure state |Ψl (t)〉.

Since our basis comprises of all possible measurable pure states, the completeness of

the basis is ensured so that

I =∑

l

|Ψl (t)〉〈Ψl (t)| (5)

In analogy with the no particle state or ground state in quantum mechanics, we can

define a ground state of our financial market as

|0〉 = |xi = 0, yij (z) = 0∀i, j, z〉 (6)

i.e. the ground state is the market state in which no investor has any cash balances nor

any securities. This state is, obviously, a pure state being fully measurable and would

also not evolve in time since no trade can take place in this market.

We define the cash and security coordinate operators x̂i& ŷij (z) by their action on

the basis state (1) to provide respectively the balances of cash and the jth security (at

price z) with the ith investor as the eigenvalues i.e.

x̂i |{xi, {yij (z) , j = 1, 2, ...m} , i = 1, 2, ...n}〉 = xi |{xi, {yij (z) , j = 1, 2, ...m} , i = 1, 2, ...n}〉(7)

ŷij (z) |{xi, {yij (z) , j = 1, 2, ...m} , i = 1, 2, ...n}〉 = yij (z) |{xi, {yij (z) , j = 1, 2, ...m} , i = 1, 2, ...n}〉(8)

A cash translation operator T̂i (z) is also defined by the following

T̂i (z′) |{xi, {yij (z) , j = 1, 2, ...m} , i = 1, 2, ...n}〉 = |{xi + z′, {yij (z) , j = 1, 2, ...m} , i = 1, 2, ...n}〉 (9)

i.e. it transfers an amount of cash z to the ith investor.

The operator T̂i (z) obviously satisfies the following properties

T̂i (z1) T̂i (z2) = T̂i (z1 + z2) (10)

T̂i (0) = Î (11)[T̂i (z) , x̂j

]= T̂i (z) x̂j − x̂jT̂i (z) = −zδijT̂i (z) (12)

T̂ ↑i (z) = T̂i (−z) (13)Towards obtaining an explicit representation of the cash translation operator, we assume

p̂i =dT̂i(z)

dz

∣∣∣z=0

as the generator of infinitesimal cash translations dz to the investor i.

Expanding T̂i (z) as a Taylor’s series and using eqs. (10), (11) we have

dT̂i (z)

dz= lim

dz→0T̂i (z + dz) − T̂i (z)

dz= lim

dz→0

�T̂i (dz) − 1

�T̂i (z)

dz= lim

dz→0

�T̂i (0) +

dT̂i(z)dz

��z=0

dz... − 1�

T̂i (z)

dz= p̂iT̂i (z)

(14)


with the solution T̂i (z) = ezp̂i . Furthermore, we have (suppressing the yij indices for the

sake of brevity)

|{xi + dz, i = 1, 2, ...n}〉 = T̂i (dz) |{xi, i = 1, 2, ...n}〉 =[T̂i (0) +

dT̂i(z)dz

∣∣∣z=0

dz...]|{xi, i = 1, 2, ...n}〉

= [I + p̂idz...] |{xi, i = 1, 2, ...n}〉(15)

Hence,

∂〈{xi,i=1,2,...n}|ψ〉∂xi

= limdz→0

〈{xi+dz,i=1,2,...n}|ψ〉−〈{xi,i=1,2,...n}|ψ〉dz

= p̂↑i 〈{xi, i = 1, 2, ...n} | ψ〉 = −p̂i 〈{xi, i = 1, 2, ...n} | ψ〉 ⇒ p̂i = − ∂∂xi(16)

so that T̂i (z) = e−z ∂

∂xi . The following commutation rule holds between x̂iand p̂i

[x̂i, p̂j] = δij (17)

The condition of an isolated market ensures that the basis and hence the Hilbert space

does not depend on time. This implies that the temporal evolution of the system is

unitary.

Creation & Annihilation Operators for Securities

We define âij (z)as the annihilation operator of the security j from the portfolio of

investor i for a price z i.e. when operator âij (z) acts on a state, the number of units of

security j is reduced by one from the portfolio of investor i for a price z.Similarly, we

define creation operators â↑ij (z) as the adjoint of the annihilation operators that increasethe number of units of security j in the portfolio of investor i for a price z.The precise

action of these operators on a state vector is defined by the following

âij(z) |{xi, {yij (z) , j = 1, 2, ...m} , i = 1, 2, ...n}〉 =√

yij(z).z |{xi, {yij (z) − 1, j = 1, 2, ...m} , i = 1, 2, ...n}〉(18)

and

â↑ij(z) |{xi, {yij (z) , j = 1, 2, ...m} , i = 1, 2, ...n}〉 =√

(yij(z) + 1)z |{xi, {yij (z) + 1, j = 1, 2, ...m} , i = 1, 2, ...n}〉(19)

where the factor ‘z’ has been introduced in the eigenvalues to ensure “scale invariance”

of the theory.

These operators satisfy the following commutation relations:-[âij(z), â

↑ij(z

′)]

= zδzz′δikδjl (20)

and

[âij(z), âkl(z′)] =

[â↑ij(z), â

↑ij(z

′)]

= 0 (21)[T̂i (z) , âjk (z

′)]

=[T̂i (z) , â

↑jk (z

′)]

= 0 (22)[T̂ ↑i (z) , âjk (z

′)]

=[T̂ ↑i (z) , â

↑jk (z

′)]

= 0 (23)


Further more

â↑ij (z) âij (z) |{xi, {yij (z) , j = 1, 2, ...m} , i = 1, 2, ...n}〉 = zyij (z) |{xi, {yij (z) , j = 1, 2, ...m} , i = 1, 2, ...n}〉(24)

which implies that the number operator would be

ŷij (z) =â↑ij (z) âij (z)

z(25)

Using the aforesaid operators we can construct an arbitrary basis state from the ground

state as follows

|{xi, {yij (z) , j = 1, 2, ...m} , i = 1, 2, ...n}〉 =n∏

i=1

T̂i (xi)m∏

j=1

∏{z,yij(z)∈N}

(â↑ij (z)

)yij(z) |0〉(26)

Buying and selling operators

The buying (selling) operation of a security is, in each case, a composite operation

consisting of the following:-

i. the creation (annihilation) of a security at the relevant price z; and

ii. the decrease (increase) in the cash balance by z of the investor undertaking the

trade.

Hence we can define the buying (selling) operator as composite of the cash translation

operator and the creation (annihilation) operators for securities as follows:-

b̂↑ij (z) = â↑ij (z) T̂

↑i (z) = â

↑ij (z) T̂i (−z) (27)

for the “buying” operation and

b̂ij (z) = âij (z) T̂i (z) (28)

for the “selling” operation. These operators satisfy the following commutation rules[b̂ij (z) , b̂

↑kl (z

′)]

= zδzz′δikδjl (29)

[b̂ij (z) , b̂kl (z

′)]

=[b̂↑ij (z) , b

↑kl (z

′)]

= 0 (30)

[b̂ij (z) , T̂k (z

′)]

=[b̂↑ij (z) , T̂k (z

′)]

= 0 (31)

[b̂↑ij (z) , x̂k

]= zδikb̂

↑ij (z) (32)[

b̂ij (z) , x̂k

]= −zδikb̂ij (z) (33)


1.2 Temporal Evolution of Financial Markets

In analogy with quantum mechanics, we mandate that the state of the market at a given

instant of time ‘t’, is represented by a vector in the Hilbert space H whose components

determine the statistical nature of the market. Hence the temporal evolution of the

market in essentially determined by the evolution of this vector with the flow of time. In

the Schrödinger picture, the time evolution of a system can be characterized through a

unitary evolution operator∧U (t, t0) in H, that acts on the initial state |ψ (t0)〉 to transform

it to |ψ (t)〉 i.e|ψ (t)〉 = Û (t, t0) |ψ (t0)〉 (34)

The assumption of the market being isolated and hence

Ψ = {|{xi, {yij (z) , j = 1, 2, ...m} , i = 1, 2, ...n}〉} being a complete basis at all times, andthe conservation of probability i.e.

∑l

|Cl (t)|2 = 1,∀t together with the group property

of∧U (t, t0) implies that the temporal evolution is unitary i.e.

U (t, t0) U↑ (t, t0) = U↑ (t, t0) U (t, t0) = 1 (35)

Furthermore Û (t0, t0) = 1. Defining the Hamiltonian Ĥ (t) = i∂∂t

Û (t, t0)∣∣∣t=t0

as the in-

finitesimal generator of time translations (evolution) we obtain, through a Taylor’s ex-

pansion up to first order Û (t + δt, t0) = Û (t, t0) +∂Û∂t

(t + δt, t)∣∣∣δt=0

Û (t, t0) δt + ...or

∂Û (t, t0)

∂t= lim

δt→0Û (t + δt, t0) − Û (t, t0)

δt= −iĤ (t) Û (t, t0) (36)

with the immediate solution Û (t, t0) = e− � tt0 Ĥ(t)dt where time ordering of the operators

constituting the Hamiltonian is assumed.

Before progressing further with the development of the model, some observations are in

order about the theory developed thus far.

(1) In standard quantum mechanics, Ĥ (t) is usually a bounded operator and hence the

exponential series in Û (t, t0) = e− � tt0 Ĥ(t)dtconverges so that its approximation to first

order is acceptable giving i∂|ψ(t)〉∂t

= Ĥ (t) |ψ (t)〉 which is the Schrödinger equationof wave mechanics. This may not always be the case in financial markets.

(2) Since time evolution of financial market, essentially, occurs through trades in securi-

ties, it is appropriate to infer that the Hamiltonian represents the trading activities

of the market.

(3) In order that the evolution operator Û (t, t0) is well defined, we mandate that the

Hilbert space His so constructed that the kernel of Û (t, t0) is empty.

1.3 Modeling Time Value of Money

Time value of money and interest rate instruments are classically modeled through the

first order differential equation dB(t)dt

= r (t) B (t)with the solution B (t) = B (0) e�

r(t)dt.


A possible candidate for the Hamiltonian function H(in the classical picture) that would

generate this temporal development as the equations of motion is

H (x, p; t) =n∑

i=1

Hi (xi, pi; t) =n∑

i=1

ri (t) xi (t) pi (t) (37)

This Hamiltonian leads to the following equations of motion

dxi (t)

dt=

∂Hi (xi, pi; t)

∂pi= ri (t) xi (t) ,

dpi (t)

dt= −∂Hi (xi, pi; t)

∂xi= −ri (t) pi (t) (38)

While the interpretation of first of these equations is straightforward being the growth

of cash reserves of the ithinvestor with the instantaneous rate ri (t), the implications of

second equation are more subtle. To provide a financial logic to this equation, we note

that pi is the infinitesimal generator of cash translations in the classical picture and hencedpi(t)

dt= −∂Hi(xi,pi;t)

∂xi= −ri (t) pi (t) represents the rate of change of the cash translations

generator which, given a fixed rate of growth of cash, would decrease with the amount of

cash translations.

Using the Weyl formalism for transformation from the classical to the quantum pic-

ture, we require that the quantum mechanical analog of H (x, p; t) be Hermitian and

symmetric in its component operators. Hence, we postulate the ansatz

Ĥ (x̂ (t) , p̂ (t) ; t) =n∑

i=1

Ĥ (x̂i (t) , p̂i (t) ; t) =n∑

i=1

iri (t)

2(x̂ip̂i + p̂ix̂i) =

n∑i=1

iri (t) x̂i

(p̂i +

1

2Î

)(39)

for the quantum mechanical Hamiltonian representing the time value of money, so that

the time development operator is

Û (t, t0) = e

�−i

t�t0

Ĥ(t)dt

�= e

n�i=1

t�t0

ri(t)xi(p̂i+ 12 Î)dt(40)

which may be evaluated using standard methods like Green’s functions and Feynmann

propagator theory.

1.4 Representation of Trading Activity

Let us consider a deal in which an investor ‘i’ buys a security ‘j’ at a price of ‘z’ units

and immediately thereafter sells the same security to another investor ‘k’ at a price of

‘z′’ units and credits/debits the difference z′ − z to his cash account. The compositetransaction will, in our operator formalism, take the form b̂ij (z

′) b̂↑ij (z). In analogy withthis argument, we can represent the Hamiltonian for trading activity of the market as

HTr (t) =∑i,j,k,l

∞∫0

dz

z

∞∫0

dz′

z′hijkl (ξ, t) b̂

↑ij (z) b̂kl (z

′) (41)

where ξ = ln z′

zensures that the amplitudes are scale invariant.


1.5 Probability Distribution of Stock Prices

We now derive the probability distribution of stock prices in market equilibrium and show

that the prices follow a lognormal distribution, thereby vindicating the efficacy of this

model.

For this purpose, we assume that an investor i = α buys one unit of a security j = β

at time t = ti for a price z. We need to ascertain the probability PT (z′ |z )i.e. the

probability of the security j = β having a price z′ at time tf = ti + T .We assume thatduring the period tf − ti, investor α holds exactly one unit of β and that before ti andafter tf ,α holds no unit of β.

Let∣∣∣ψz

αβ(ti)

〉be the state that represents investor α holding exactly one unit of β at a

price z at time tiin the Hilbert space H. Hence, we have∣∣∣ψz

αβ(ti)

〉= b̂↑αβ (z)

∣∣∣ψαβ (ti)〉where∣∣∣ψαβ (ti)〉is the state that represents investor α not holding any unit of β. This also im-plies that b̂αβ (z)

∣∣∣ψαβ (ti)〉 = 0.Let us assume that the final state corresponding to theinitial state

∣∣∣ψzαβ

(ti)〉

is represented by∣∣∣ψz

αβ(tf )

〉so that

∣∣ψzαβ(tf )〉 = Û(ti, tf ) ∣∣ψzαβ(ti)〉 = e−itf�

ti

Ĥdt

b̂↑αβ(z)∣∣∣ψαβ(ti)〉 (42)

The amplitudes of∣∣ψz′αβ (tf )〉 are determined in the usual way by taking scalar product〈

ψz′

αβ(tf )

∣∣∣ ψzαβ

(tf )〉

and we have, for the matrix elements of the propagator

G (z′, tf ; z, ti) =〈ψαβ(tf )

∣∣∣ b̂αβ (z′) e−itf�

ti

Ĥdt

b̂↑αβ(z)∣∣∣ψαβ(ti)〉 (43)

In this case, the trading Hamiltonian will contain creation and annihilation operators

relating to the investor α and those relating to the securityβ i.e., it will be of the form

ĤTr (t) =∑k,l

∞∫0

dz

z

∞∫0

dz′

z′hαβkl (ξ, t) b̂

↑αβ (z) b̂kl (z

′) (44)

We further make the assumption that the amplitudes can be approximated by their first

two moments about ξ = 0, being sharply peaked about z′ = zsince, in the timescalesbeing considered, most trades would occur around z. Hence, we have

hαβkl ∼[Ωαβkl (t) − iξ−1Ξαβkl (t)

]δ (ξ) (45)

Noting that ξ = ln z′

z, we have ξ−1 =

(ln z

′z

)−1=(

z′z− 1)−1 = z

z′−z to first order and

δ (ξ) = δ(ln z

′z

)= δ (z′ − z)

[d�ln z

′z

�

dz′

]−1= z′δ (z′ − z). Using these results and eqs. (44)

& (45), we obtain

ĤTr (t) ∼∑k,l

∞∫0

dz

z

∞∫0

dz′

z′zδ (z′ − z)

[Ωαβkl (t) − i z

z′ − zΞαβkl (t)]b̂↑αβ (z) b̂kl (z

′)


=∑k,l

∞∫0

dz

zb̂↑αβ (z)

[Ωαβkl (t) + izΞαβkl (t)

∂

∂z

]b̂kl (z

′) (46)

We note that this expression for the Hamiltonian is linear in ∂∂z

and hence it can be

diagonalized in the “momentum space” through a Fourier transformation and we have

ĤTr (t) =1

2π

∑k,l

∞∫0

dz

z

∞∫0

dz′

z′

∞∫−∞

dpb̂↑αβ (z) [Ωαβkl (t) + iΞαβkl (t) p] b̂kl (z′) eipξ (47)

The assumption of market equilibrium implies that the Hamiltonian should be inde-

pendent of time over the relevant timescales that would be much smaller than those

determining aggregate market behaviour so that we may write eq. (43) as

G (z′, tf ; z, ti) =〈ψ

αβ(ti)

∣∣∣ b̂αβ (z′) e−iĤ(ti)T b̂↑αβ (z) ∣∣∣ψαβ (ti)〉 (48)Because of the Hamiltonian being diagonal in momentum space, it is more convenient to

work in momentum space for evaluating the propagators and we have, for the equivalent

of eq. (48) in momentum space as

G̃ (p′, p;T, ti) =〈ψ

αβ(ti)

∣∣∣⎡⎣ ∞∫

0

dz′

z′eip

′ln(z′/κ)b̂αβ (z′)

⎤⎦ e−iĤ(ti)T⎡⎣ ∞∫

0

dz

ze−ipln(z/κ)b̂↑αβ (z)

⎤⎦ b̂↑αβ (z) ∣∣∣ψαβ (ti)〉(49)

To solve the problem further, we make use of second order perturbation theory. The first

step is to split the Hamiltonian into components as follows

Ĥ (t) =∑

l

∞∫0

dz

zb̂↑αβ (z) [Ωαβαl (t) + Ξαβαl (t) p] b̂αβ (z)+

∑k,l,k �=α

∞∫0

dz

zb̂↑αβ (z) [Ωαβkl (t) + iΞαβkl (t) p] b̂kl (z

′)

(50)

Let Eibe the energy eigenstate of the unperturbed Hamiltonian i.e. of the state of the

market before the purchase of security β by the investor α , then the energy eigenstate

of the Hamiltonian Ĥ (ti)i.e. after the purchase of security β by the investor α will be of

the form Ep = Ei +∑l

[Ωαβαl (ti) + iΞαβαl (ti) p]− ip2σ2where the second term representsthe impact on the energy eigenstates of the transactions involving investor α or security

β and the last term is the second order perturbation term due to the overall fluctuations

of the market. Substituting this value of Ep in eq. (49) and noting that the Hamiltonian

and hence the propagator G̃ (p′, p; T, ti)is also diagonal in “momentum space”, we have

G̃ (p′, p; T, ti) ∼ 2πδ (p′ − p) e−iTEp = 2πδ (p′ − p) e−iT [Ei+Ω(ti)+iΞ(ti)p−ip2σ2](51)

where∑l

Ωαβαl (ti) = Ω (ti) ,∑l

Ξαβαl (ti) = Ξ (ti) .

Inverting back to “coordinate space”, we obtain

G (z′, tf ; z, ti) =1

2π

∞∫−∞

dpe−iT [Ei+Ω(ti)+iΞ(ti)p−ip2σ2]−ip ln z′/z ∼ e

iT (Ei+Ω)

2σ√

πTe

�(ln(z′/z)+ΞT)2

4σ2T

�

(52)


The probability PT (z′ |z )i.e. the probability of the security j = β having a price z′ at

time tf = ti + T will then be proportional to the square of the above amplitude and

hence, we finally obtain

PT (z′ |z ) α |G (z′, tf ; z, ti)|2 =

(4πσ2T

)−1e

�(In(z′/z)+ΞT)2

2σ2T

�(53)

which agrees perfectly with the standard stochastic theory of finance wherein stock re-

turns are modeled extensively through lognormal distributions.

2. Conclusions

The following interesting observations emanate from the above analysis:-

(1) Eq. (53), on comparison with the standard expression for probability distribution

of stock price in the conventional stochastic calculus based approach to the Black Scholes

formula, identifies Ξ with the expected return on stock. This return is independent of

the eigenvalue Eiand hence, the state of the market. A financial interpretation of this

could be that the stock returns are dependent on the performance of the company and

independent of market dynamics.

(2) Independence of stock returns of the market dynamics would, however, mandate

that the stock volatility measured by the standard deviation σ is related to the stock

market dynamics which seems justified since higher trading volumes would imply greater

volatility and vice versa.

(3) If we define the uncertainty of measurement of a random variate by its standard

deviation, then, from eqs. (52) & (53), we have the uncertainty for the stock price process

zand its Fourier conjugate p, after a time T , as σz = σ√

Tand σp =1

2σ√

Tso that σzσp =

12

as it should be, since the distribution of z is assumed Gaussian in the aforesaid analysis.

(4) Furthermore, σzσp

= 2σ2T which enables the identification of σ2 as the reciprocal

of the mass and hence, the inertia of the stock price process. It is intriguing to note that

the same analogy follows through another completely independent analysis i.e. the Black

Scholes equation for the option price in its standard form is given by ∂C∂t

= −rSt ∂C∂S −12σ2S2t

∂2C∂S2

+ rC, where C (St, t) denotes 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. Making the substitution St = exwe obtain ∂C

∂t= −r ∂C

∂x− 1

2σ2 ∂

2C∂x2

+ 12σ2 ∂C

∂x+ rC

which, when compared with the standard quantum mechanical Hamiltonian in one degree

of freedom identifies σ2 as the reciprocal of the mass of the underlying system.

Contemporary quantitative finance is dominated by stochastic modeling of market

behaviour. These models are essentially in the nature of tools of data analysis that aim

to predict future events by applying probabilistic methods to historical data. Empirical

evidence testifies that probability distributions of stock returns are negatively skewed,

have fat tails and show leptokurtosis [56]. Some of the features of empirical distributions

of stock prices are modeled through Levy distributions [57-60], stochastic volatility [61]

or cumulant expansions around the lognormal case. Each of these models, however,


attempts to empirically attune the model parameters to fit observed data and hence, is

equivalent to interpolating or extrapolating observed data in one form or the other.

Hence, stochastic models fail to take cognizance of causal factors that get submerged

in the superficial patterns exhibited by the avalanche of data being analysed. In actual

fact, every new price determination of a security and hence, the fluctuation of prices is

attributable to a new trade in the relevant security at that price. The “trading process”

therefore manifests itself as a price history of a security. The fundamental limitation of

stochastic tools in simulating extended memory effects is circumvented by this approach.

An attempt has also been made through this “toy model” to establish that a quantum

mechanical version of financial markets results in a temporal evolution of the probabil-

ity distribution analogous to that of simple stochastic systems. Stochastic models also

lack ability to accommodate collective effects like phase coherency in lasers that could,

possibly, be built into this quantized description.

It need be emphasized here that the above is purely a phenomenological model for

modeling stock behavior. 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 rational behavior and it is this set of beliefs that govern

his actions. The market, therefore, invariably generates a heterogeneous response to

any stimulus. Furthermore, “rationality” 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 response 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.


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Rayleigh process and matrix elements for theone-dimensional harmonic oscillator

J.H. Caltenco1∗, J.L. López-Bonilla1†, J. Morales2‡

1Escuela Superior de Ingenieŕıa Mecánica y EléctricaInstituto Politécnico Nacional

Edif. Z, Acc. 3-3er Piso Col. Lindavista C.P. 07738 México D.F.2Area de F́ısica AMA, CBI

Universidad Autónoma Metropolitana-Azc.Apdo. Postal 16-306, CP 02200 México DF

Received 21 December 2005 , Accepted 2 February 2005, Published 25 June 2006

Abstract: We show that, the matrix elements 〈m |e−γ x |n 〉 for the one-dimensional harmonicoscillator have application in Markov process theory, permitting thus to resolve the Fokker-Planck equation for the two-dimensional probability density corresponding to Rayleigh case.c© Electronic Journal of Theoretical Physics. All rights reserved.

Keywords: Matrix elements, One-dimensional harmonic oscillator, Markov process theory,Fokker-Planck equationPACS (2006): 03.65.Ta , 03.65.Ge ,02.50.Ga, 46.25.Cc, 02.70.Ns

1. Introduction

In [1-4] were calculated the matrix elements:

f(γ) = 〈m| e−γx |n〉 =∫ ∝−∝

ψ∗m(x) e−γ xψn(x)dx (1)

for the harmonic oscillator in one dimension, where γ ≥ 0 is an arbitrary parameter.Then, it was deduced the following result for m ≥ n:

f(γ) =

√n !

m !

(− γ√

2

)m−neγ2/4Lm−nn

(−γ

2

2

)(2)

∗ [email protected]† jlopezb@ipn,mx, [email protected]‡ [email protected]


in terms of the associated Laguerre polynomials Lqn.

It is interesting to observe that the 2nd order differential equation [5-8] defining to

Lpq permits to show, via (2), that f(γ)satisfies the equation:

d2f

dγ2+

1

γ

df

dγ− 1

4γ2(γ4 + 4Aγ2 + 4Q

)f = 0 (3)

where A = m + n + 1 and Q = (m − n)2. That is, (2) is solution of (3), with which it ispossible to obtain [9] the Morse’s radial wave function [10].

In Sec. 2, f(γ)is employed to resolve the nonstationary stochastic Fokker-Planck

equation (FPE) [11] for the Rayleigh distribution.

2. Two-Dimensional Probability Density Associated to Rayleigh

Process.

The equation (3) has the structure:

D1(γ)f =

(γ2

4+ A +

Q

γ2

)f (4)

where it appears the important Bessel’s operator [12]:

DC(γ) =d2

dγ2+

C

γ

d

dγ(5)

for the case C = 1. The operator (5) has interesting applications in hydrodynamics,

the theory of subharmonic functions, electrostatics, the Euler-Poisson-Darboux equation,

elasticity, the generalized radiation problem, quantum mechanics and generalized axially

symmetric potential. Here we shall show that, the Bessel’s operator D1 is useful for to

determine the probability density ω associated to Rayleigh process in two dimensions,

because the FPE can adopt a form similar to (4).

In fact, the nonstationary stochastic FPE for the Rayleigh distribution is given by

[11] p. 73:

ω̇ =∂

∂ x

[(β x − k

2x

)ω

]+

k

2

∂2ω

∂ x2(6)

being k and β positive parameters, then the corresponding eigenfunctions X (x) are

solutions of:

σ2d2X

d x2+

(x − σ

2

x

)d X

d x+

(1 +

σ2

x2+

λ

β

)X = 0 (7)

with σ2 = k2β

. The operator D1 participates when in (7) we make the following change

of functions:

F (x) = x−1ex2

4σ2 X (8)

obtaining thus the relation:

D1(x)F =

(x2

4σ4− 1

σ2− λ

σ2β

)F (9)


On the other hand, in (4) it is possible to realize the formal change of variable:

γ =i

σx , i =

√−1 (10)

then it results the equation:

D1(x)f =

(x2

4σ4+

Q

x2− A

σ2

)f , (11)

with the same structure as (9); therefore Q = 0, that is m = n, and A = 2n+ 1 = 1 + λβ

,

then λ = 2nβ , n= 0,1,2,. . . Besides, F is proportional to f given by (2) with the change

(10):

F ∝ e− x2

4σ2 Ln

(x2

2σ2

)(12)

then (8), (12) and factors of normalization lead us to the eigenfunctions:

Xn(x) =x

n!σ2e−

x2

2σ2 Ln

(x2

2σ2

)(13)

which are solutions of (7) for λ = 2nβ. The result (13) is our principal aim because

it permits to write immediately the two-dimensional probability density associated to

Rayleigh case, to see [11].


References

[1] A. E. Glassgold and D. Holliday, Phys. Rev. A139 (1965) 1717.

[2] J. Morales, J. López-Bonilla and A. Palma, J. Math. Phys. 28 (1987) 1032.

[3] J. Morales and A. Flores-Riveros, J. Math. Phys. 30 (1989) 393.

[4] J. López-Bonilla and G. Ovando, Bull. Irish Math. Soc. N.44 (2000) 61.

[5] C. Lanczos, Linear differential operators, D. Van Nostrand Co., London (1961).

[6] J. D. Talman, Special functions, W. A. Benjamin Inc. New York (1968).

[7] H. Hochstadt, The functions of Mathematical Physics, Dover, New York (1971).

[8] M. Abramowitz and I. A. Stegun, Handbook of mathematical functions, John Wileyand Sons, New York (1972).

[9] J. H. Caltenco, J. López-Bonilla and R. Peña-Rivero, J. Sci. Res. 50 (2002) 125.

[10] Ch. S. Johnson Jr. and L. G. Pedersen, Problems and solutions in QuantumChemistry and Physics, Dover, New York (1986).

[11] R. L. Stratonovich, Topics in the theory of random noise. Vol. I, Gordon and Breach,New York (1963).

[12] A. Weinstein, Ann. Mat. Pura Appl. 49 (1960) 359.


Identical synchronization in chaoticjerk dynamical systems

Vinod Patidar1∗ and K. K. Sud2

1Department of PhysicsBanasthali Vidyapith Deemed UniversityBanasthali - 304022, Rajasthan, INDIA

2Department of PhysicsCollege of Science Campus

M. L. S. University, Udaipur – 313002, INDIA

Received 23 December 2005 , Accepted 24 February 2006, Published 25 June 2006

Abstract: It has been recently investigated that the jerk dynamical systems are the simplest ever systems, which possessvariety of dynamical behaviours including chaotic motion. Interestingly, the jerk dynamical systems also describe various

phenomena in physics and engineering such as electrical circuits, mechanical oscillators, laser physics, solar wind driven

magnetosphere ionosphere (WINDMI) model, damped harmonic oscillator driven by nonlinear memory term, biological

systems etc. In many practical situations chaos is undesirable phenomenon, which may lead to irregular operations in

physical systems. Thus from a practical point of view, one would like to convert chaotic solutions into periodic limit cycle

or fixed point solutions. On the other hand, there has been growing interest to use chaos profitably by synchronizing chaotic

systems due to its potential applications in secure communication. In this paper, we have made a thorough investigation

of synchronization of identical chaotic jerk dynamical systems by implementing three well-known techniques: (i) Pecora-

Carroll (PC) technique, (ii) Feedback (FB) technique and (iii) Active Passive decomposition (APD). We have given a

detailed review of these techniques followed by the results of our investigations of identical synchronization of chaos in

jerk dynamical systems. The stability of identical synchronization in all the aforesaid methods has also been discussed

through the transversal stability analysis. Our extensive numerical calculation results reveal that in PC and FB techniques

the x-drive configuration is able to produce the stable identical synchronization in all the chaotic jerk dynamical systems

considered by us (except for a few cases), however y-drive and z-drive configurations do not lead to the stable identical

synchronization. For the APD approach, we have suggested a generalized active passive decomposition, which leads to the

stable identical synchronization without being bothered about the specific form of the jerk dynamical system. Several other

active passive decompositions have also been listed with their corresponding conditional Lyapunov exponents to achieve

the stable identical synchronization in various chaotic jerk dynamical systems.

c© Electronic Journal of Theoretical Physics. All rights reserved.

Keywords: Chaos, Jerk dynamical systems, Identical synchronization of chaos, synchronizedchaosPACS (2006): 05.45.+b; 47.52.+j; 05.45.-a

∗ vinod r [email protected]


1. Introduction

Various studies of nonlinear dynamical systems in the last four decades have signif-

icantly extended the notion of oscillations in these systems. It has been shown that the

post-transient oscillations in dynamical systems can be associated not only with the regu-

lar behavior such as periodic or quasiperiodic oscillations, but also with chaotic behavior

[1-5].

Chaos has long-term unpredictable behavior, which is usually couched mathematically

as sensitivity to initial conditions i.e., where the system’s dynamics takes it, is hard to

predict from the starting point. One way to demonstrate this is to run two identical

chaotic systems side by side, starting both at very close, but not exactly equal initial

conditions. The systems soon diverge from each other, but both retain the same attractor

pattern. An interesting question to ask is: Can we force the two chaotic systems to follow

the same path on the attractor? i.e., Can chaotic systems be made synchronized? The

affirmative answer is possible to this question. It has been shown that some of the ideas of

synchronization can also be extended for the description of particular type of behaviour

in c

EJTP · 2008. 5. 17. · some scalar ﬁeld , a non-minimal coupling between the scalar curvature...

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