On some equations concerning quantum electrodynamics coupled to quantum gravity, Mathematical...

8/8/2019 On some equations concerning quantum electrodynamics coupled to quantum gravity, Mathematical connections w

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On some equations concerning quantum electrodynamics coupled to quantum gravity, the

gravitational contributions to the gauge couplings and quantum effects in the theory of

gravitation: mathematical connections with some sector of String Theory and Number Theory

Michele Nardelli 2,1

1 Dipartimento di Scienze della Terra

Universit degli Studi di Napoli Federico II, Largo S. Marcellino, 10

80138 Napoli, Italy

2 Dipartimento di Matematica ed Applicazioni R. Caccioppoli

Universit degli Studi di Napoli Federico II Polo delle Scienze e delle Tecnologie

Monte S. Angelo, Via Cintia (Fuorigrotta), 80126 Napoli, Italy

Abstract

This paper is principally a review, a thesis, of principal results obtained from various authoritative

theoretical physicists and mathematicians in some sectors of theoretical physics and mathematics.

In this paper in the Section 1, we have described some equations concerning the quantum

electrodynamics coupled to quantum gravity. In the Section 2, we have described some equationsconcerning the gravitational contributions to the running of gauge couplings. In the Section 3, we

have described some equations concerning some quantum effects in the theory of gravitation. In the

Section 4, we have described some equations concerning the supersymmetric Yang-Mills theory

applied in string theory and some lemmas and equations concerning various gauge fields in any

non-trivial quantum field theory for the pure Yang-Mills Lagrangian. Furthermore, in conclusion, in

the Section 5, we have described various possible mathematical connections between the argument

above mentioned and some sectors of Number Theory and String Theory, principally with some

equations concerning the Ramanujans modular equations that are related to the physical vibrations

of the bosonic strings and of the superstrings, some Ramanujans identities concerning and the

zeta strings.

1. On some equations concerning the quantum electrodynamics coupled to quantumgravity. [1]

The key equations that govern the behaviour of the coupling constants in quantum field theory are

the renormalisation group Callan-Symanzik equations. If we let g denote a generic coupling

constant, then the value of g at energy scale E, the running coupling constant ( )Eg , is determinedby

( )( )gE

dE

EdgE ,= , (1)


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where ( )gE, is the renormalisation group -function. Asymptotic freedom is signaled by

( ) 0Eg as E , requiring 0


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As with the Green function (4) the divergent part of (7) comes from the 0 limit of the proper

time integral. The divergent part of L is

divp ( )

++=

2

21

2

0

44

2 ln2

1

32

1ccc EtrEtrEEtrEExdL . (9)

The lower limit on the proper time integration can be kept as 0= and the divergent part of the

effective action L contains a simple pole as 0 given by

divp ( ) = 24

216

1xtrEdL

. (10)

The general form of ij is

( ) ( ) ( )iji

j

i

j

i

j CBA ++=

(11)

for coefficients ( )ijA , ( )ijB

and ( )ijC that depend on the spacetime coordinates through the

background field. Normal coordinates are introduced at 'x with yxx += ' and all of the

coefficients in (11) are expanded about 0=y . This gives

( ) ( ) ( )

=

+=1

...0 ...1

1

n

i

j

i

j

i

jn

nyyAAA

(12)

with similar expansions for ( )ijB and ( )ijC . The Green function is Fourier expanded as usual,

( )( )

( )

= pGpedxxGi

j

yipn

n

i

j2

1', , (13)

except that the Fourier coefficient ( )pGij can also have a dependence on the origin of the coordinate

system 'x that is not indicated explicitly. If

( ) ( ) ( ) ( ) ...210 +++= pGpGpGpGi

j

i

j

i

j

i

j (14)

where ( )pG ijr

is of order rp 2 as p it is easy to see that to calculate the pole part of ( )xxG ij ,

as 4n only terms up to and including ( )pG ij2

are needed.

The gravity and gauge field contributions result in

tr ( )++++

++++=

222222

1 381232

1

32

1

8

3

2

1

8

3

8

1

4

3

8

3 vvFE . (15)

The overall result for the quadratically divergent part of the complete one-loop effective action (8)

that involves 2F is


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( )

++++=

2422

2

221

32

1

32

1

8

3

2

1

8

3

8

5

4

3

8

3

32Fxd

Ecquad

. (16)

( )

++++=

2422

2

221

32

1

32

1

8

3

2

1

8

3

8

5

4

3

8

3

216

1Fxd

Ecquad

. (16b)

If 0 , 0 , 1 are taken to obtain the gauge condition independent result, the non-zero

result

( )=

24

2

221

128Fxd

Ecquad

(17)

is found.

The net result for the divergent part of the effective action that involves 2F and therefore

contributes to charge renormalization is

divp ( )( )

+

=

242

2

22

2

2

2

221 ln

48ln

256

3

128FxdE

eE

Ecc

c

. (18)

divp ( )( )

+

=

242

2

22

2

2

2

221 ln

3ln

16

3

816

1FxdE

eE

Ecc

c

. (18b)

From this the renormalization group function in (1) that governs the running electric charge to be

calculated to be

( ) eEeeE

+=

23

3212, 2

2

2

2

3

. (19)

The first term on the right hand side of (19) is that present in the absence of gravity and results in

the electric charge increasing with the energy. The second term on the right hand side of (19)

represents the correction due to quantum gravity. For pure gravity with no cosmological constant, or

for small cosmological constant , the quantum gravity contribution to the renormalization group

-function is negative and therefore tends to result in asymptotic freedom.

2. On some equations concerning the gravitational contributions to the running of gaugecouplings. [2]

The action of Einstein-Yang-Mills theory is

=

aaggRgxdS

FF

4

112

4 (20)

where R is Ricci scalar and aF is the Yang-Mills fields strength = ,igF .

It is hard to quantize this lagrangian because of gravity-parts non-linearity and minus-dimension

coupling constant G 16= . Usually, one expands the metric tensor around a background metric


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6

++=

2

2

22

222

2

2

1 1ln16

1

c

sw

s

csc

My

MM

. (25)

Putting 1 and2 in the following equation

=

122

3g , (26)

we obtain the gravitational corrections to the gauge function

++

=

2

2

02

2

2

22 1ln16 c

sw

s

cs

My

Mg

(26b)

In general, the total function of gauge field theories including the gravitational effects may bewritten as follows

++

+=

2

2

02

2

2

223

021ln

1616

1

c

sw

s

cs

My

Mggb

. (27)

The interesting feature of gauge theory interactions is the possible gauge couplings unification at

ultra-high energy scale when the gravitational effects are absent. Where the running of gauge

coupling in the Model Standard Super Symmetric (MSSM) without gravitational contributions is

known to be

( ) ( )

M

Mee ln10

3311+=

; ( ) ( )

M

Mww ln2

111+=

; ( ) ( )

M

Mss ln2

311=

(28)

with experimental input at ZM

( ) 05.097.581 = Ze M ; ( ) 05.061.291

=

ZwM ; ( ) 22.047.81 = Zs M (29)

We note that these values are equal to the following values connected with the aurea ratio:

38,12461180 + 20,56230590 = 58,68691770; 29,12461180; 8,49844719.

Indeed, we have:

( ) ( ) ( ) 12461180,38370820393,12361803399,109016994,1137/77/35 =+=+ ;

( )[ ] 56230590,20385410197,637/28 = ;

( ) ( ) ( ) ( ) =++=++ 323606798,061803399,285410197,637/217/147/28

12461180,29370820394,9 = ;

( ) ( ) ( ) ( ) ( +++=+++ 055728089,0145898033,061803399,237/637/427/287/14


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) 49844719,83832815729,23013155617,0 =+ .

As usual, 61803399,12

15

+= , i.e. the value of the aurea ratio.

3. On various equations concerning some quantum effects in the theory of gravitation[3]

The general spherical static metric is given by

( ) ( ) 222122 ++= drdrrhdtrfds , (30)

where ( )rf and ( )rh are arbitrary functions of the coordinate r. The angular part of the metric isdiagonal and given by

( )

=

+=

=

2

1

2

1

222 sind

i

d

ij

jidd . (31)

Consider an interesting classical scalar field ( )x living in a spacetime with the metric (30). Thisfield has an action given by:

=

=2

2

n

n

n

dgxdS . (32)

The n are a set of arbitrary coupling constants. In particular,2

2 m gives the mass of a weakly

coupled excitation of this field. We will expand in the eigenmodes of the free, classical wave

equation such that

( ) ( )= xadx ppp , (33)

pp m 22

= . (34)

A sufficiently large set of quantum numbers p label the eigenbasis. The abstract formal expression

pd simply represents an appropriate measure over the modes under which

( ) ( ) ( ) = yxyxdd

ppp ,

and

( ) ( ) = pqqpd xxgxd .

Now we want to consider the energy density, , of a massless scalar field in a infinitely large

hypercubic blackbody cavity at temperature T. Consider a real scalar field ( )txI , , where I issome kind of p -dimensional polarization index representing p internal degrees of freedom (for

example, in a well-chosen gauge, the transverse polarization of an Abelian vector field behavesessentially like an internal index on a scalar field with 2= dp ). Further assume that the field is

sufficiently weakly coupled that each polarization component can be treated as an independent field


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obeying an action similar to eq. (32) with all interaction coefficients higher than 22 M set equal

to zero. Then each field component obeys the classical equation of motion

( ) ( )xMx II 22 = . (35)

The solutions to equation (35) may be expressed as a sum over modes labelled by a wave vector ak

obeying22

Mk = :a

a

I

k

a

a

I

k

I

kxkBxkA cossin += , (36)

for arbitrary real coefficients IkA andI

kB . We take the state to be labelled by the 1d spatial

components ofak and fix the frequency of each mode by

i

ik kkMk +=22

0

2 . We now confine

the field to live in a cubic box of side length L by demanding Dirichlet boundary conditions at

Lxi ,0= for 1...1 = di . This demands 0=I

kB and

ii mL

k

= , (37)

whereim is a spatial vector whose components are non-negative integers. The total energy in the

hypercube is given by

=

=

=

=p

I m m

mIk

d

nU1 0 01 1

... , (38)

where we understand that k is given by

=

+=1

1

2

2

22

d

i

ik mL

M

. (39)

The set of integers mIn defining the quantum state must obey the appropriate statistics for the field

I . We have described a pure quantum state of the theory. At a finite temperature T and zero

chemical potential, the system will be in a mixed state governed by the partition function

{ }

=mI

k

n

eQ

, (40)

where T/1 . This can be evaluated to give

( )

=

=

=0 01 1

1ln...lnm md

kepQ , (41)

where 1= for bosons and 1= for fermions. The overall factor ofp occurs because the energy

is independent of p , so each polarization mode contributes equally. The average occupation

number of a given momentum mode in the thermal state is then given by


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=

=

==

p

I k

mImke

pQnn

1

ln1

. (42)

The total energy in the hypercube can now be found by combining the expressions (38) and (42), or

by

=

=

=

=

==

=

0 0 0 01 1 1 1

......lnm m m m

kmk

d d

ke

pnQU

. (43)

We should pass from a state labelling in terms of quantum numbers im to a labelling in terms of

physical momenta ik , with a mode density determined by the differential limit of equation (37).

The sums over im then become integrals over ik as

( )

k

e

pLkdU k

d

dd

L

1

11

2

, (44)

where k is understood asi

ikkM +2 . The factors of 2 in the denominator of the measure arise

because the integrals over the ik run over both positive and negative values, whereas the im were

only summed over non-negative values. Equation (44) scales properly with the volume, so that even

in the infinite volume limit we can define the spectral energy density over ik modes. Using the

spherical symmetry of the infinite volume limit and defining ii

i kkkk == , eq. (44) becomes

( )( )

+

+

== 0

222

1

2

22

2 kM

d

d

d

e

kMkdk

SVolpV

U

. (45)

This defines the spectral energy density over the magnitude of the spatial momentum, via

( ) kdkuk , as

( )( )

( )

+=

+

22

222

1

2

2 kM

d

d

d

k

e

kMkSVolpku . (46)

Similarly, we can define the spectral energy density over the frequency as

( )( )

( )

( )( )

=

e

MSVolpu

d

d

d 2/3222

1

2

2, (47)

where runs over [ ],M .The total energy density can now be evaluated using either eq. (46) or eq. (47) match. Simple

analytic results can be found for the case 0=M , which will also apply when MT >> . In this case,

k= and equations (46) and (47) match. They give

( )( )

= 0

1

1

2

2 x

d

d

dd

e

x

dx

SVol

pT . (48)


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The integral can be evaluated by pulling the exponential into the numerator, performing a Taylor

series in xe , doing the integral, and resumming the Taylor series. The result for the general definite

integral is given by

( )

( )( )

=

=

b

a

d

n

bnan

dx

d

n

ebeadde

dxx 1

0

2/11

!!12

21

, (49)

where ( )s is the Riemann zeta function, which can be defined for real 1>s as

( )

=

=1

1

nsn

s . (50)

This series arises in the evaluation of the integral with the 1= , for bosons. For fermions, 1= ,

the corresponding series is

( ) ( )

=

1

1 11n

s

n

nsR . (51)

This series can be evaluated using

( ) ( ) ( )sssssss

2

2...

6

2

4

2

2

2=+++= R , (52)

so that

( ) ( )sss

=

2

21R , (53)

which is the origin of this factor in eq. (49). So, eq. (48) becomes

( )( )( )

( )

d

ddd

Td

ddp

2/12

2/1

2

12

!1

2

21

=

. (54)

Furthermore, we can rewrite the eq. (48) also as follows:

( )( )

=

=

0

1

1

2

2

x

d

d

dd

e

xdx

SVolpT

( ) ( )( )

( )

d

ddd

Td

ddp

2/12

2/1

2

12

!1

2

21

. (54b)

Before, we have considered the energy density, , of a massless scalar field in a infinitely large

hypercubic cavity at temperature T. We now want to calculate the energy flux, , emitted from a

blackbody with this same temperature. For the flux calculation, instead of 2dSVol we will

encounter

( ) =

222

2cos dddd


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11

( ) ( )( )( )

( )( ) ==

=

=

=

2

0 0 0

2/

0

32/23

2

122321

2

1

2

22

2sincos... d

dd

i

iidddSVol

ddd

dddd

( ) ( ) ( )22

2

22

1

21

== dd SVol

d

d

dBVol

, (55)

where ( )x is the step function and nB is the n-dimensional unit ball: the compact subspace of nR

bounded by1nS (for example, 3B is the unit 3-ball of volume 3/4 bounded by the sphere 2S of

area 4 ). The fact that the expression for the energy density becomes that for the flux when

( )2dSVol is replaced by ( )2dBVol makes physical sense, since nB is the projection of nS onto nR .We are left with the relationship of flux to energy density as

( )

( )( )

2

2

2

22

2

1

=

=d

d

SVol

BVol

dd

d

. (56)

Thus, the d-dimensional Stefan-Boltzmann law is given by

( )( )( )

( )

d

ddd

Td

d

ddp

2/2

2/1

2

222

!1

2

21

=

. (57)

4. On some equations concerning the supersymmetric Yang-Mills theory applied in string

theory and some lemmas and equations concerning various gauge fields in any non-trivial

quantum field theory for the pure Yang-Mills Lagrangian. [4] [5]

The fields of the minimal 2=N supersymmetric Yang-Mills theory are the following: a gauge field

mA , fermionsi

and i& transforming as ( )2/1,0,2/1 and ( )2/1,2/1,0 under

( ) ( ) ( )IRL SUSUSU 222 , and a complex scalar B - all in the adjoint representation of the gaugegroup. Covariant derivatives are defined by

( )+= mmm iAD , (58)

and the Yang-Mills field strength is

[ ]nmmnnmmn AAAAF ,+= . (59)

The supersymmetry generators transform as ( ) ( )2/1,2/1,02/1,0,2/1 ; introducing infinitesimal

parameters i and i& , furthermore [ ]BBD ,= . The minimal Lagrangian is


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12

[ ] [ ] [ ]

+

=

M

ii

ijji

ij

m

m

i

m

m

i

mn

mn Bi

Bi

BBBDBDDiFFxTrde

L

&

&&

&,

2,

2,

2

1

4

11 242

(60)

Here Tr is an invariant quadratic form on the Lie algebra which for ( )NSUG = we can

conveniently take to be the trace in the N dimensional representation.It is possible to realize a mass term for the 1=N matter multiplet (which consists of B and2&

= ) by adding to the Lagrangian a term of the form ( ) { },...1QI + , where 1Q is the charge

corresponding to the 1 transformation. Furthermore, 1Q is the only essential symmetry. We

obtain:

( ) { } ( ) +=++=M M

BBmTrmxdemmxTrdLQILL 22

1,... 42

22

224

1 &&

&

&

&&

, (61)

with

mnklklmnm 22&&= . (62)

The mass is proportional to the holomorphic two-form . The 1=N gauge multiplet, consisting of

the gauge field mA and the gluinoi

= , remains massless.

With regard the 16 -amplitude in the type IIB description, the classical action for the operator ( )16

in the5

5 SAdS supergravity action is

[ ] ( ) ( )( ) ( ) =

+=

16

1

0

5

0

0

004165

0

00

412

12 1

;,1

5

0

p

pppSs

ig

xJxxxxKtdxdVgeJSs

. (63)

This result is in agrees with the following expression obtained in the Yang-Mills calculation:

( )( )[ ]

( )( ) =

+

+

+=

16

1

00

0

42

0

2

0

4

088

5

0

00

4

8

8

16

1122

p

A

p

A

p

ig

YMp

pp

pp

p

p

YM

YM xxxx

dddxdegxG

&

&

(64)

i.e. the correlation function in the super Yang-Mills description.

The low energy effective action for type IIB superstring theory in ten dimensions includes the

interaction (in the string frame)

( ) ( ) ( ) ( ) ( )

++++

464222104 ...69

2462323 RDeeegxdlS s

, (65)

where the involve contributions form D-instantons.

We consider the perturbative contributions to the46

RD interaction. The sum of the contributions to

the four graviton amplitude at tree level and at one loop in type II string theory compactified on 2T

is proportional to ( ) ( ) ( )( ) ( ) ( )

4

222

2222

2 24/14/14/1

4/4/4/R

+

+++

I

ultlsl

ultlsleV

sss

sss , (66)


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13

where 2V is the volume of2

T in the string frame, uts ,, are the Mandelstam variables, and I is

obtained from the one loop amplitude. The amplitude is the same for type IIA and type IIB string

theories. Now I is given by

( )

= F ,22

2

FZd

I lat (67)

where F is the fundamental domain of ( )ZSL ,2 , and 2/2 = ddd . In the above expression, the

lattice factor latZ which depends on the moduli is given by

( ) ( )( )

=

ZMatA

lat AUU

TAiTVZ

,22

2

22

22

11det2exp

. (68)

Expanding eq. (67) to sixth order in the momenta, we get that

( )[ ]213336

3

IIutsl

I s +++= , (69)

where

( ) ( ) ( ) =

=

Li

ilat

dZ

dI

F T

3

1

323121

2

2

2

2

2

1 ;;ln;ln4

, (70)

and

=2I ( )[ ]

=

Li

ilat

dZ

d

F T

3

1

3

21

2

2

2

2

2

;ln

. (71)

We have also that

( ) ( ) ( ) ++=

=L

IIIEZd

IZSL

latF

3

3

2

1

1

1

,2

32

2

2

1

3

,4

4, (72)

where 211

1, II , and 31

I are the contributions from the zero orbit, the non-degenerate orbits and the

degenerate orbits of ( )ZSL ,2 respectively. In the eq. (72), we have used the expression

( ) ( ) ( ) ( ) ( ) 12121

2

2122/5

2/5

0,0 2

12

3

2

2

3

2

,2

3 254

362,

+

+= mimmm

ZSLemmK

m

mE

. (73)

Thence, we can rewrite the eq. (72) also as follows:

( )

=

LlatZ

dI

F2

2

2

1

3

4

4 ( ) ( ) ( ) 121

21

2

2122/5

2/5

0,0 2

12

3

2

2

3

2 254

362

+

+ mimmm

emmKm

m

. (74)

The contribution from the zero orbit gives


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14

( ) ( ) =

=L

ZSLE

dVI

F

0,,2

32

2

2

2

1

1 . (75)

The contribution from the non-degenerate orbits gives

( ) ( )

>

++

=

=

00,0

2,2

32

2

212

2

1

2

22

2

,2

pjk

pUjkU

TiTkp

ZSLeE

ddVI

( ) ( ) ( )

=0,0

2

22/5

2/5,2

3212,2

kp

ipkTZSLepkTK

k

pUUET

. (76)

Furthermore, the contribution from the degenerate orbits gives

( ) ( ) ( ) ( ) ( ) ( )

( ) ( )

+

+=

=

2/1

2/1 0 0,0,

,2

32

2

3

23

,2

32

2

212

3

1 ,

4

5362

2,

2

22

2

pj

ZSLpUj

U

T

ZSLUUE

T

TeEd

dVI

.

(77)

Now we want to show two lemmas and equations concerning the gauge fields as described in the

Jormakkas paper Solutions to Yang-Mills equations [5]. Thence, in the next Section, we

describe some possible mathematical connections between some equations concerning this

interesting argument and some equations concerning the Ramanujans modular equations that are

related to the physical vibrations of the bosonic strings and of the superstrings, and some

Ramanujans identities concerning .

Lemma 1

Let the gauge field satisfy

( ) = =

3

1

22

321

, ,,,0 jj

ecsxxxA aRa

(78)

where j and c are as in the following expressions:

jjj ir +=

, Rjj

, ; 3211 2xxx+=

; 3212 2xxx=

; 313 2

1

xx+=

;

021 == ; 032

1x= ; ) ) )jjjjj ivurh ,, += (79)

20 =c ; 121 == cc ; 03 =c ; 20 =e ; 121 == ee ; 03 =e , (80)

and Rsa , . Then

( )( )

=

3

2/3

222

321

3 1

2,,,0

ka

a

k csxxxAxd . (81)


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15

The expression (79) describes a localized gauge field, gauge boson, which moves in the 21,xx

direction with the speed of light as a function of 0x . We select a concrete case that gives easy

calculations. Let ( ) 22 jjf = and extend it to ( )22

jj rrh = . The real and imaginary parts of

iecd += are described from (80). We evaluate the gauge potential at 00 =x and take the real

part.

We change the variables to 321 ,, yyy

32116

1

3

23 xxxy = ; 322

3

1

3

2xxy = ; 33 2xy = (82)

Then

=

++=3

1

2

3

2

2

2

1

2

j

j yyy (83)

As 2y and 3y are not functions of 1x we can change the order of integration

( ) ( ) ( ) ( )

===

= ++ ==

21

223

22

221

223

22

23

1

223

1

22 22

1

1

22

2

1

1

233

3

1 yyyyyyyedyxededxxedxedxed j

jj j

( ) ( )

+

=1

2 223

123

22

2

yyxed (84)

As 3y

is not a function of 2x we can change the order of integration

( ) ( )

+21

223

22

2 22

1

1

2

3

1 yyyedyxed

( ) ( ) ==

+

12 22

3

123

22

2

yyxed

( )( )

( )( )

===

22

223

222

223

2 22

1

23

122

1

23

1

222

3

3

122

3

1 yyyyedyedxedxedx

( )( ) ( )( ) ( ) ( ) 323

3

2

3

2

222

1

2

3

3

122

2

1

2

3

3

122

2

3

3

1 23

223

2

=== yy

edyedx . (85)

Thus

3

2

3

23 1

2

22

=

jxed

=

3

323 1

2

22

jxed

( )( )

=

3

2/3

222

321

3 1

2,,,0

ka

a

k csxxxAxd . (86)

Thence, we can conclude that the eq. (81) is true.

We can rewrite the eq. (85) also as follows:

( ) ( ) + 21

223

22

2 22

1

1

2

3

1 yyy edyxed ( ) ( ) ==

+ 12 223

123222 yyxed ( ) ( ) =3

23

2221

23

3

1


16/30

16

( ) ( )( )6

321

2

2

1

4822

3

123

22

2

=

=

+ yy

xed . (86b)

Lemma 2

Let the gauge field satisfy

( ) = =

3

1

22

321

, ,,,0 jj

ecsxxxA aRa

(87)

where j and c are as in (79), (80) and Rsa , . Then

= a aBsd 2

3

3

161

RL (88)

where in Minkowskis metric at 00 = x , we have 0=B . In the negative definite metric

( )

=

1000

0100

0010

0001

,g , (88b)

we have that 43

2

3

13++=B .

From (79) and (82) follows that

32112

1

6

1

3

1yyy += ; 3212

2

1

6

1

3

1yyy = ; 213

3

2

3

1yy = . (89)

For Minkowskis metric

( ) 3263152142332222110 yyByyByyByByByBP +++++== (90)

where 0=kB for all k. For the metric (88b)

( ) 3263152142

33

2

22

2

1132

2

3

2

2

2

1 444 yyByyByyByByByBP +++++=++= (91)

where

3

131 =B ;

3

22 =B ; 43 =B ; 2

3

44 =B ;

6

45 =B ;

3

46 =B . (92)

We do the integration with generic parameters jB . Then


17/30

17

( ) ( )( )

( ) ( )

++++

=23

22

21

223

22

21

22

2

1

32

2

1

3yyy

xedPxed

)( 3263152142

33

2

22

2

11 yyByyByyByByByB +++++ .

(93)

As 2y and 3y are not functions of 1x we can change the order of integration and change the

integration parameter 1x to 1y

( ) ( )( )

( ) ( )

++++

=23

22

21

223

22

21

22

2

1

32

2

1

3yyy

xedPxed

)( 3263152142

33

2

22

2

11 yyByyByyByByByB +++++ =

( ) ( ) ( )( )

( )

++++=

+ 2122

122

322

22

2

1

1326

2

33

2

22

22

1

2

111

22

1

2yyyy

edxyyByByBeydxBxed

( )( )

=

++

212

22

1

113524

y

eydxyByB

( ) ( ) ( ) ( ) ( ) ++++=

+ 21221223222 221

1326

2

33

2

22

2212

111

2212

3

1 yyyy edyyyByByBeydyBxed

( )( )

=

++

212

22

1

113524

y

eydyyByB

( ) ( )

( )( )

+++=

+

2

12

2

12

3

1326

2

33

2

2231

22

1

223

22

2

yyByByBBxedyy

. (94)

As 3y is not a function of 2x we can change the order of integration and change the integration

parameter 2x to 2y :

( ) ( )( )

( ) ( )

++++

=23

22

21

223

22

21

22

2

1

32

2

1

3yyy

xedPxed

)( 3263152142

33

2

22

2

11 yyByyByyByByByB +++++ =

( ) ( )

( )( ) =

+++=

+

2

12

2

12

3

1326

2

33

2

2231

22

1

223

22

2

yyByByBBxedyy

( )

( )

( ) ( )

++=

2222

222

32

22

1

2

222

22

1

231

22

1

32

1

2

12

2

3

3

1 yyyeydyBedyBedx

( ) ( )

=

++

2222

22

2

2

1

2236

2

2

1

22332

1

2

1 yyeydyyBedyyB

( )

( ) ( )=

++=

2

12

2

1

2

12

2

1

2

12

2

12

2

3

3

1 2333231

22

1

3

23

2

yBBBedxy

( )

( ) ( ) ( )=

++=

2

2

334241

22

1

32

1

2

1

2

12

2

1

2

3

3

1 232

yBBBedyy

( ) ( )( ) ( )

( )32152

3

53212

3

2

1

2

12

2

1

2

3

3

1BBBBBB ++=++=

. (95)

We can rewrite this equation also as follows:


18/30

18

( ) ( )( )

( )253

2

326315214

2

33

2

22

2

11

22

1

3

16

81

4

23

22

21

2

=

+++++

++

yyByyByyByByByBxedyyy

. (95b)

Thence, we can conclude that the eq. (88) is true. Indeed, we have that:

( ) ( )( )

( ) ( )

++++

=23

22

21

223

22

21

22

2

1

32

2

1

3yyy

xedPxed

=+++++ )( 3263152142

33

2

22

2

11 yyByyByyByByByB

( )( )3215

2

3

2

1BBB ++=

==

a

aBsd2

33

16

1

R

L . (96)

5. Ramanujans equations, zeta strings and mathematical connections

Now we describe some mathematical connections with some sectors of String Theory and Number

Theory, principally with some equations concerning the Ramanujans modular equations that are

related to the physical vibrations of the bosonic strings and of the superstrings, the Ramanujans

identities concerning and the zeta strings.

3.1Ramanujans equations [6] [7]With regard the Ramanujans modular functions, we note that the number 8, and thence the

numbers2864 = and 8232

2= , are connected with the modes that correspond to the physical

vibrations of a superstring by the following Ramanujan function:

( )

++

+

=

4

2710

4

21110log

'

142

'

cosh

'cos

log4

3

18

2

'

'4

0

'

2

2

wtitwe

dxex

txw

anti

w

wt

wx

. (97)

Furthermore, with regard the number 24 (12 = 24 / 2 and 32 = 24 + 8) this is related to the physical

vibrations of the bosonic strings by the following Ramanujan function:


19/30

19

( )

++

+

=

42710

421110log

'

142

'

cosh

'cos

log4

24

2

'

'4

0

'

2

2

wtitwe

dxex

txw

anti

w

wt

wx

. (98)

It is well-known that the series of Fibonaccis numbers exhibits a fractal character, where the forms

repeat their similarity starting from the reduction factor /1 = 0,618033 = 2

15

(Peitgen et al.

1986). Such a factor appears also in the famous fractal Ramanujan identity (Hardy 1927):

++

+=

==

q

t

dt

tf

tf

qR

0 5/45/1

5

)(

)(

5

1exp2

531

5)(

2

15/1618033,0

, (99)

and

++

+=

q

t

dt

tf

tfqR

0 5/45/1

5

)(

)(

5

1exp

2

531

5)(

20

32

, (100)

where 215 +=

.

Furthermore, we remember that arises also from the following identities (Ramanujans paper:Modular equations and approximations to Quarterly Journal of Mathematics, 45 (1914), 350-

372.):

( )( )

++=

2

13352log

130

12

, (100a) and

++

+=

4

2710

4

21110log

142

24

.

(100b)

From (100b), we have that

++

+=

4

2710

4

21110log

14224

. (100c)

Let ( )qu denote the Rogers-Ramanujan continued fraction, defined by the following equation


20/30

20

( ) ,...1111

::325/1

++++==

qqqqquu 1


21/30

21

Using (109) in (107), we find that

( )( )

( ) ( ) ( ) =+= dqvuqdqd

dqq

dqqqqq

dq

q

q/log8

5

840

5

8 5/1535

5

( ) ( ) ( ) ( ) ( ) +

+== k

kkdqqqqvudqqqq 1

1log5

24log5

840/log840

5353

, (111)

where (104) has been employed. We note that we can rewrite the eq. (111) also as follows:

( )( )

( ) ( ) +

+=k

kkdqqqq

q

dq

q

q

1

1log

5

24log

5

840

5

8 535

5

. (112)

Multiplying both sides for64

5, we obtain the following identical expression:

( )( )

( ) ( ) +

+=k

kkdqqqq

q

dq

q

q

1

1log

8

13log

8

1

8

125

8

1 535

5

, (112b)

or

( )( )

( ) ( )

+

+=

k

kkdqqqq

q

dq

q

q

1

1log3log25

8

1

8

1 535

5

. (112c)

In the Ramanujans notebook part IV in the Section Integrals are examined various results on

integrals appearing in the 100 pages at the end of the second notebook, and in the 33 pages of the

third notebook. Here, we have showed some integrals that can be related with some argumentsabove described.

( )( )( ) ( )( )

=

+

++=

01

0 442

2

442

2

2

2

414

414

8

12

kxk

na

xae

xdxa

kae

ka

adnne

; (113)

( )( ) ( )

= +++=

+0 122442

2 1

44

1

414

kx

kaaa

axae

xdxa

. (114)

Multiplying both sides for2 , we obtain the equivalent expression:

( )( ) ( )

= +++=

+

01

22

232

442

22 1

44414

kx

kaaa

axae

xdxa

. (114b)

Let 0n . Then

( )( ) ( )( )

( ) ( ) ( ){ }( ) ( ){ }

=

+

++

+=

+0 0

12

2/12cosh12

12cos12

4coscosh

2sin

k

nkk

kk

nke

xxx

dxnx

. (115)

Also here, multiplying both sides for 2 , we obtain the equivalent expression:


22/30

22

( )( ) ( )( )

( ) ( ) ( ){ }( ) ( ){ }

=

+

++

+=

+0 0

122

32

2/12cosh12

12cos12

4coscosh

2sin

k

nkk

kk

nke

xxx

dxnx

. (115b)

Now we analyze the following integral:

=

++

=1

0

2

15

2

411log

:

dxx

x

I ; (116)

Let ( ) 2/411 xu ++= , so that uux = 2 . Then integrating by parts, setting vu /1= and using the

following expression ( )( )

=z

dww

wzLi

02

1log, Cz , and employing the value

=

2

15log

102

152

2

2

Li , we find that

( )( ) ( )( )( )( )

+ + +

=

+=

=

=

2/15

1

2/15

1

2/15

1

2

2

1logloglog12

logdu

u

u

u

udu

u

uuduu

uu

uI

( )( )

+

=

+

+=

2/15

1

2 log1log

2

15log

2

1dv

v

vv

( )156102

15log

2

11

2

15

2

15log

2

1 222222

2 =+=

++

+= LiLi . (117)

Thence, we obtain the following equation:

++

=1

0

2

411log

dxx

x

I ( )( )

1512

log 22/15

1 2

=

=

+

duuuu

u. (118)

Multiplying both sides for 4

15, we obtain the equivalent expression:

++

=1

0

2

411log

4

15dx

x

x

I ( )( )

4154

1512

log

4

15 322/15

1 2

==

=

+

duuuu

u;

++

=1

0

2

411log

4

15dx

x

x

I ( )( )

412

log

4

15 32/15

1 2

=

=

+

duuuu

u. (118b)

In the Ramanujans paper: Some definite integrals connected with Gausss sums (Ramanujan,1915 Messanger of Mathematics, XLIV, 75-85), are described the following integrals:


23/30

23

=

20

23 3

4

1

16

1

sinh

cos

dx

x

xx , (119)

=0

23

16

1

sinh

sin

dx

x

xx , (120)

+=

022

53

2

1

64

1

1

2cos

dx

e

xxx

, (121)

+=

022

2 551

256

1

1

2cos

dx

e

xxx

. (122)

Now, we sum (119) and (120) obtaining:

+0

23

sinh

cosdx

x

xx

+=+

=

0 22

23

16

1

16

3

64

1

16

13

4

1

16

1

sinh

sin

dx

x

xx .

Now we multiply both the sides for3 and obtain:

+0

233

sinh

cos

( dxx

x

x

+=+=+=02

3233

2

3323

3416

1

1616

3

641616

3

64)sinh

sin

dxx

x

x ,

thence

+=

+

2

3

0 0

23

233 3

416

1

sinh

sin

sinh

cos

dx

x

xxdx

x

xx . (123)

Now, we sum (121) and (122) obtaining:

+

0 2 1

2cosdx

e

xx

x

+++=

0 222

2

256

5

256

5

256

1

64

5

64

3

128

1

1

2cos

dxe

xx

x

.

Now we multiply both the sides for 3 and obtain the following equivalent equation:

+++=

+

8

5

8

5

8

1

2

5

2

3

42

1

16

1

1

2cos

1

2cos 2323

0 0 2

2

2

3dx

e

xxdx

e

xxxx

. (124)

In the work of Ramanujan, [i.e. the modular functions,] the number 24 (8 x 3) appears repeatedly.

This is an example of what mathematicians call magic numbers, which continually appear where we

least expect them, for reasons that no one understands. Ramanujans function also appears in stringtheory. Modular functions are used in the mathematical analysis of Riemann surfaces. Riemann

surface theory is relevant to describing the behavior of strings as they move through space-time.

When strings move they maintain a kind of symmetry called "conformal invariance". Conformal

invariance (including "scale invariance") is related to the fact that points on the surface of a string's

world sheet need not be evaluated in a particular order. As long as all points on the surface are taken

into account in any consistent way, the physics should not change. Equations of how strings must

behave when moving involve the Ramanujan function. When a string moves in space-time by

splitting and recombining a large number of mathematical identities must be satisfied. These are the

identities of Ramanujan's modular function. The KSV loop diagrams of interacting strings can be

described using modular functions. The "Ramanujan function" (an elliptic modular function that

satisfies the need for "conformal symmetry") has 24 "modes" that correspond to the physical

vibrations of a bosonic string. When the Ramanujan function is generalized, 24 is replaced by 8 (8 +

2 = 10) for fermionic strings.


24/30

24

With regard the Section 1, we have the following mathematical connections between the eqs. (16b)

and (18b) and the eqs. (112c), (123) and (124):

( )

++++=

2422

2

221

32

1

32

1

8

3

2

1

8

3

8

5

4

3

8

3

48

1

Fxd

Ecquad

( )( )

( ) ( )

+

+=

k

kkdqqqq

q

dq

q

q

1

1log3log25

8

1

8

1 535

5

; (125)

( )

++++=

2422

2

221

32

1

32

1

8

3

2

1

8

3

8

5

4

3

8

3

216

1Fxd

Ecquad

+=

+

2

3

0 0

23

233 3

416

1

sinh

sin

sinh

cos

dx

x

xxdx

x

xx ; (126)

( )

++++=

2422

2

221

32

1

32

1

8

3

2

1

8

3

8

5

4

3

8

3

216

1Fxd

Ecquad

+++=

+

8

5

8

5

8

1

2

5

2

3

42

1

16

1

1

2cos

1

2cos 2323

0 0 2

2

2

3dx

e

xxdx

e

xxxx

; (127)

divp ( )( )

+

=

242

2

22

2

2

2

221 ln

3ln

16

3

82

1

8

1FxdE

eE

Ecc

c

( )

( )( )

( )

+

+=

k

kkdqqqq

q

dq

q

q

1

1log3log25

8

1

8

1 535

5

; (128)

divp ( )( )

+

=

242

2

22

2

2

2

221 ln

3ln

16

3

816

1FxdE

eE

Ecc

c

+=

+

2

3

0 0

23

233 3

416

1

sinh

sin

sinh

cos

dx

x

xxdx

x

xx ; (129)

divp ( )( )

+

=

242

2

22

2

2

2

221 ln

3ln

16

3

816

1FxdE

eE

Ecc

c

+++=

+

8

5

8

5

8

1

2

5

2

3

42

1

16

1

1

2cos

1

2cos 2323

0 0 2

2

2

3dx

e

xxdx

e

xxxx

. (130)

With regard the Section 2, we have the following mathematical connections between the eqs. (24)

and (27) and the eqs. (123) and (124):

( ) ( ) ( ) ( ) ( ) ( ) ( )

+++=+ 2

2

2

22 02

1162 k

R

q

RR

qkp

ed

LR IgkkgIgqqgIVdxGigT MM

+ 22 pR

Igppg M

+=

+

2

3

0 0

23

233 3

416

1

sinh

sin

sinh

cos

dx

x

xxdx

x

xx ; (131)


25/30

25

( ) ( ) ( ) ( ) ( ) ( ) ( )

+++=+ 2

2

2

22 02

1162 k

R

q

RR

qkp

ed

LR IgkkgIgqqgIVdxGigT MM

( ) ( )]+ 22 pRIgppg M

+++=

+

85

85

81

25

23

421

161

1

2cos

1

2cos 2323

0 0 2

2

2

3dx

e

xxdxe

xxxx

; (132)

++

+=

2

2

02

2

2

223

021ln

1616

1

c

sw

s

cs

My

Mggb

+=

+

2

3

0 0

23

233 3

416

1

sinh

sin

sinh

cos

dx

x

xxdx

x

xx ; (133)

+++= 2

2

02

2

2

2

2302

1ln1616

1c

sw

s

cs

MyMggb

+++=

+

8

5

8

5

8

1

2

5

2

3

42

1

16

1

1

2cos

1

2cos 2323

0 0 2

2

2

3dx

e

xxdx

e

xxxx

. (134)

Thence, mathematical connections with the Ramanujans modular equations that are related to the

physical vibrations of the superstrings and with the Ramanujans identities concerning .


and (95b), that derive from the (95), and the eqs. (114b), (115b), (118b), (123) and (124):

( ) ( )

+21

223

22

2 22

1

1

2

3

1 yyyedyxed

( ) ( ) ==

+

12 22

3

123

22

2

yyxed

( )( )

( )( )

===

22

223

222

223

2 22

1

23

122

1

23

1

222

3

3

122

3

1 yyyyedyedxedxedx

( )( ) ( )( ) ( ) ( ) 323

3

2

3

2

222

1

2

3

3

122

2

1

2

3

3

122

2

3

3

1 23

223

2

=== yy

edyedx

( ) ( )( )

=

=

+

6

321

2

2

1

4

822

3

123

22

2

yyxed

( )( ) ( )

= +++=

+

01

22

232

442

22 1

44414

kx

kaaa

axae

xdxa

; (135)

( ) ( )

+21

223

22

2 22

1

1

2

3

1 yyyedyxed

( ) ( ) ==

+

12 22

3

123

22

2

yyxed

( )( )

( )( )

===

22

223

222

223

2 22

1

23

122

1

23

1

222

3

3

122

3

1 yyyyedyedxedxedx

( )( ) ( )( ) ( ) ( )3

2

3

3

2

3

2

2221

23

3122

21

23

3122

23

31

2

3

22

3

2

=== yy edyedx


26/30

26

( ) ( )( )

=

=

+

6

321

2

2

1

4822

3

123

22

2

yyxed

( )( ) ( )( )

( ) ( ) ( ){ }( ) ( ){ }

=

+

++

+=

+

00

122

32

2/12cosh12

12cos12

4coscosh

2sin

k

nkk

kk

nke

xxx

dxnx

; (136)

( ) ( )

+21

223

22

2 22

1

1

2

3

1 yyyedyxed

( ) ( ) ==

+

12 22

3

123

22

2

yyxed

( )( )

( )( )

===

22

223

222

223

2 22

1

23

122

1

23

1

222

3

3

122

3

1 yyyyedyedxedxedx

( )( ) ( )( ) ( ) ( ) 323

3

2

3

2

222

1

2

3

3

122

2

1

2

3

3

122

2

3

3

1 23

223

2

=== yy

edyedx

( ) ( )( )

=

=

+

6

321

2

2

1

4

822

3

123

22

2

yyxed

++

=1

0

2

411log

4

15dx

x

x

I ( )( )

412

log

4

15 32/15

1 2

=

=

+

duuuu

u; (137)

( ) ( )

+21

223

22

2 22

1

1

2

3

1 yyyedyxed

( ) ( ) ==

+

12 22

3

123

22

2

yyxed

( )( )

( )( )

===

22

223

222

223

2 22

1

23

122

1

23

1

222

3

3

122

3

1 yyyyedyedxedxedx

( )( ) ( )( ) ( ) ( ) 323

3

2

3

2

222

1

2

3

3

122

2

1

2

3

3

122

2

3

3

1 23

223

2

=== yy

edyedx

( ) ( )( )

=

=

+

6

321

2

2

1

4822

3

123

22

2

yyxed

+=

+

2

3

0 0

23

233 3

416

1

sinh

sin

sinh

cos

dx

x

xxdx

x

xx ; (138)

( ) ( )

+21

22

3

2

2

2 2

2

1

1

2

3

1 yyyedyxed

( )

( ) ==

+1

2

223

123

2

2

2

yy

xed

( )( )

( )( )

===

22

223

222

223

2 22

1

23

122

1

23

1

222

3

3

122

3

1 yyyyedyedxedxedx

( )( ) ( )( ) ( ) ( ) 323

3

2

3

2

222

1

2

3

3

122

2

1

2

3

3

122

2

3

3

1 23

223

2

=== yy

edyedx

( ) ( )( )

=

=

+

6

321

2

2

1

4822

3

123

22

2

yyxed

+++=

+

8

5

8

5

8

1

2

5

2

3

42

1

16

1

1

2cos

1

2cos 2323

0 0 2

2

2

3

dxe

xxdxe

xxxx . (139)


27/30

27

( ) ( )( )

( )=

+++++

++

25

32

326315214

2

33

2

22

2

11

22

1

3

16

81

4

23

22

21

2

yyByyByyByByByBxed

yyy

( )( ) ( )

= +++=

+

01

22

232

442

22 1

44414

kx

kaaa

axae

xdxa

; (140)

( ) ( )( )

( )=

+++++

++

25

32

326315214

2

33

2

22

2

11

22

1

3

16

81

4

23

22

21

2

yyByyByyByByByBxed

yyy

( )( ) ( )( )

( ) ( ) ( ){ }( ) ( ){ }

=

+

++

+=

+

00

122

32

2/12cosh12

12cos12

4coscosh

2sin

k

nkk

kk

nke

xxx

dxnx

; (141)

( ) ( )( )

( )=

+++++

++

25

32

326315214

2

33

2

22

2

11

22

1

3

16

81

4

23

22

21

2

yyByyByyByByByBxed

yyy

++

=1

0

2

411log

4

15dx

x

x

I ( )( )

412

log

4

15 32/15

1 2

=

=

+

duuuu

u; (142)

( ) ( )( )

( )=

+++++

++

25

32

326315214

2

33

2

22

2

11

22

1

3

16

81

4

23

22

21

2

yyByyByyByByByBxed

yyy

+=

+

2

3

0 0

23

233 3

416

1

sinh

sin

sinh

cos

dx

x

xxdx

x

xx ; (143)

( ) ( )( )

( )=

+++++

++

25

32

326315214

2

33

2

22

2

11

22

1

3

16

81

4

23

22

21

2

yyByyByyByByByBxed

yyy

+++=

+

8

5

8

5

8

1

2

5

2

3

42

1

16

1

1

2cos

1

2cos 2323

0 0 2

2

2

3dx

e

xxdx

e

xxxx

. (144)

Thence, mathematical connections with the Ramanujans identities concerning , precisely4

3.

3.2Zeta Strings [8]The exact tree-level Lagrangian for effective scalar field which describes open p-adic stringtachyon is

++

=

+

12

2

2 1

1

2

1

1

1 pp

pp

p

p

g

L

, (145)

wherep

is any prime number,

22+= t

is the D-dimensional dAlambertian and we adoptmetric with signature ( )++ ... . Now, we want to show a model which incorporates the p-adicstring Lagrangians in a restricted adelic way. Let us take the following Lagrangian


28/30

28

+

++=

==

1 1 1 1

1222 1

1

2

111

n n n n

n

nnnn

ngn

nCL

LL

. (146)

Recall that the Riemann zeta function is defined as

( )

==1 1

11

n pss

pns

, is += , 1> . (147)

Employing usual expansion for the logarithmic function and definition (147) we can rewrite (146)

in the form

( )

++

= 1ln

22

112

gL

, (148)

where1= 2220

2kkkr

, (149)

where( ) ( ) ( )dxxek ikx

=

~

is the Fourier transform of ( )x .

Dynamics of this field

is encoded in the (pseudo)differential form of the Riemann zeta function.When the dAlambertian is an argument of the Riemann zeta function we shall call such string a

zeta string. Consequently, the above is an open scalar zeta string. The equation of motion for

the zeta string is

( )( ) +> =

=

2

2

220 1

~

22

1

2 kkixk

Ddkk

ker

(150)

which has an evident solution 0= .

For the case of time dependent spatially homogeneous solutions, we have the following equation ofmotion

( )( )

( )( )

( )tt

dkkk

etk

tikt

=

=

+>

1

~

22

1

200

2

0

2

2

0

0

. (151)

With regard the open and closed scalar zeta strings, the equations of motion are

( )( )

( )

=

=

n

n

nn

ixk

Ddkk

ke

1

2

12 ~

22

1

2

, (152)


29/30

29

( )( )

( )( )

( )

( )

+

+

+=

=

1

11

2

12

112

1~

42

1

4

2

n

n

nn

nixk

Dn

nndkk

ke

, (153)

and one can easily see trivial solution 0== .

With regard the Section 3, we have the following mathematical connection between the eq. (54b)

and the eq. (150):

( )( ) +> =

=

2

2

220 1

~

22

1

2 kkixk

Ddkk

ker

( )( )

=

=

0

1

1

2

2

x

d

d

dd

e

xdx

SVolpT

( )( )( )

( )

d

ddd

Td

ddp

2/12

2/1

2

12

!1

2

21

. (154)


and (77) and the eqs. (150):

( )

=

LlatZ

dI

F2

2

2

1

3

4

4 ( ) ( ) ( ) +

+

12121

2

2122/5

2/5

0,0 2

12

3

2

2

3

2 254

362

mim

mm

emmKm

m

( )

( ) +> =

=

2

2

220 1

~

22

1

2 kkixk

Ddkk

ker

; (155)

( ) ( ) ( ) ( ) ( ) ( )( ) ( )

+=

=

+

2/1

2/1 00,0,

,2

32

2

3

23

,2

32

2

212

3

1 ,4

5362

2,

2

22

2

pj

ZSLpUj

U

T

ZSLUUE

TTeE

ddVI

( )

( ) +> =

=

2

2

220 1

~

22

1

2 kkixk

Ddkk

ker

. (156)

In conclusion, multiplying for 6

8

1 the eq. (77), we obtain the following equivalent equation:

( )( )

( )

( )

( )( )

( ) ( )

+

+=

=

2/1

2/1 00,0,

,2

322

3

2

3,2

322

2

12

63

1 ,4

53

624,8

12

22

2

pj

ZSLpUj

U

T

ZSL

UUETTeE

d

dVI

(157)

This equation can be related with the expression (138) and with the eq. (150), giving the following

interesting complete mathematical connections:

( ) ( ) ( ) ( ) ( ) ( )( ) ( )

+=

=

+

2/1

2/1 00,0,

,2

32

2

3

2

3,2

32

2

212

63

1 ,4

5362

4,

8

12

22

2

pj

ZSLpUj

U

T

ZSLUUE

TTeE

ddVI

( ) ( )

+21

223

22

2 22

1

1

2

3

1 yyyedyxed

( ) ( ) ==

+

12 22

3

123

22

2

yyxed

( )( )

( )( )

===

22

223

222

223

2 22

1

23

122

1

23

1

222

3

3

122

3

1 yyyyedyedxedxedx


30/30

( )( ) ( )( ) ( ) ( ) 323

3

2

3

2

222

1

2

3

3

122

2

1

2

3

3

122

2

3

3

1 23

223

2

=== yy

edyedx

( ) ( )( )

=

=

+

6

321

2

2

1

4822

3

123

22

2

yyxed

+=

+

2

3

0 0

23

233 3

416

1

sinh

sin

sinh

cos

dx

x

xxdx

x

xx

( )( ) +> =

=

2

2

220 1

~

22

1

2 kkixk

Ddkk

ker

. (158)

Acknowledgments

I would like to thank Prof. Branko Dragovich of Institute of Physics of Belgrade (Serbia) for his

availability and friendship. Ild like to thank Prof. Jorma Jormakka of Department of Military

Technology, National Defence University, Helsinki, for his availability. In conclusion, Ild like to

thank also Dr. Pasquale Cutolo for his important and useful work of review and his kind advices.

References

[1] David J. Toms Quantum gravitational contributions to quantum electrodynamics arXiv:

1010.0793v1 [hep-th] 5 Oct 2010

[2] Yong Tang and Yue-Liang Wu Gravitational Contributions to the Running of Gauge

Couplings arXiv:0807.0331v2 [hep-ph] 10 Dec 2008

[3] Sean Patrick Robinson Two Quantum Effects in the Theory of Gravitation Massachusetts

Institute of Technology June 2005

[4] Edward Witten Supersymmetric Yang-Mills Theory on a four-manifold arXiv:hep-th/

9403195v1 31 Mar 1994;

Massimo Bianchi and Stefano Kovacs Yang-Mills Instantons vs. type IIB D-instantons

arXiv:hep-th/9811060v2 27 Nov 1998;Anirbau Basu The

46RD term in type IIB string theory on

2T and U-duality arXiv:0712.

1252v1 [hep-th] 8 Dec 2007

[5] Jorma Jormakka Solutions to Yang-Mills Equations arXiv:1011.3962v2 [math.GM]

15 Nov 2010 - http://arxiv.org/PS_cache/arxiv/pdf/1011/1011.3962v2.pdf

[6] Bruce C. Berndt, Heng Huat Chan, Sen-Shan Huang Incomplete Elliptic Integrals in

Ramanujans Lost Notebook - American Mathematical Society, Providence, RI, 2000, pp. 79-

126.

[7] Bruce C. Berndt Ramanujans Notebook Part IV Springer-Verlag (1994).

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