Abstract
Topological Dipole Field Theory was proposed as an extension of the Standard model of particle physics. The transition amplitude per time for a propagation of a gauge boson can be computed from this theory. This research paper shows a numerical implementation of Topological Dipole Field Theory for a simple photon propagation. For simplicity the numerical simulation is applied to a monochromatic gamma ray such that it can be executed in one space and one time direction. Effects of Topological Dipole Field Theory like the high probability for fluctuations of energy and momentum are taken into account.
Introduction
Gamma rays are used in various branches of
science and technology and are occurring during some cosmic phenomena. There
are a couple of experiments with gamma rays described in literature (Bloser
2006). An open question is whether effects predicted by TDFT can be observed. TDFT
is a quantum field theory where bosons are coupled to an additional 2-form
field 
which is governed by a topological quantum field theory (Linker
2015). The Lagrangian density of this theory contains also another 2-form field
, a Lagrange multiplier 
and a coupling constant 
. It has the form:
A calculation of a process, where a single
boson is coming in with momentum 
, energy 
and polarization 
and a single boson is coming out with momentum 
, energy 
and polarization 
can be performed analytically. The transition rate per volume 
with the length scales 
and time 
is given by the expression:
Equation (2) is difficult to implement in
numerical simulations especially due to the topological constants 
. However, if the degrees of freedom are reduced in a suitable way,
the transition rate (2) can be simplified. This research paper shows an
application of Topological Dipole Field Theory to a high-energetic gamma ray. In
this research paper a gamma ray that propagates through a vacuum (such that
interactions with matter can be neglected) in one single space direction 
is considered. The following assumptions hold for this simulation:
- 2-dimensional formulation (one space direction and one time direction)
-
There is only one direction of propagation
(along the -axis) denoted by 
and therefore the energies can always be expressed as 
.
- Only quantum electrodynamics with TDFT extension is considered (there is only one Lie algebra generator necessary)
- It is averaged over all photon polarizations (polarization dependence can be neglected in this case).
Applying these assumptions to (2) one obtains for the transition rate:
The constant 
is determined only by the coupling constant by the relation 
where 
is a constant. From the transition rate (3) the Master equation is
used in this research paper to compute the time evolution of the photon density
function 
that measures the probability to find a photon at spacetime
position 
with momentum . After this, the master equation is discretized to obtain an
algorithm on which a numerical solution of the Master equation can be computed.
Finally, the results of this numerical computation are discussed.
Theory and Results
From now one the ordinary SI-unit system is
used instead of the unit system 
and the constant is called “coupling constant”. It is assumed that a photon
propagates with constant speed of light 
(due to Einstein’s theory of special relativity), i.e. for an
initial position 
the kinematic equation for the photon has the form 
. With this equation the -dependence of the function can be omitted. This leads to the following Master equation (The
transition probability is time reversal symmetric, i.e. 
):
In (4) it is used the transition
probability per time 
and not the transition probability per time and volume 
. Denoting 
the characteristic wave number scale of the system (it is used for
practical purposes). For the numerical solution of equation (4) the following
discretizations are performed (these discretizations resolve all scattering
processes):
-
- 
The time increment per iteration step is and the wave number increment per iteration step is 
(with cutoff number 
for wave number integration), where the maximum wave number
respected in the simulation is given by 
. Equating (5a) and (5b) and substituting (3), the resulting
recursion for the solution of the Master equation has the form:
The initial condition used in this
simulation is as follows: At 
(the time where the gamma ray starts) the gamma ray is
monochromatic with the wave number 
(it is obvious to identify this wave number with the characteristic
wave number scale) such that it holds 
for 
and 
elsewhere. The following program code written in FORTRAN uses the
following parameter:
-
-
-
.
-
- Number of time iteration steps: 90.
The code which solves equation (6) is given here:
! Numerical solution of the Master equation for a gamma ray with TDFT interaction (Process: photon -> photon).
! Subroutine for computing the gain in photon density function
subroutine gain(F,ww,Fgain)
integer N, Ntime ! Increments
parameter (N = 10, Ntime = 90)
double precision F(0:Ntime,-1*N:N) ! input density function
double precision Fgain(0:Ntime,-1*N:N) ! output density function
double precision ww(-1*N:N,-1*N:N) ! transition probability input
integer i,j,k ! Loop variables
do i = 0,Ntime
do j = -1*N,N
Fgain(i,j) = 0 ! Initialization
do k = -1*N,N
Fgain(i,j) = Fgain(i,j) + ww(k,j)*F(i,k) - ww(k,j)*F(i,j)
end do
end do
end do
end subroutine
program Gamma_Ray_TDFT
! Variable declarations
double precision kgamma ! Gamma ray wave number (is set equal to characteristic wave number scale)
double precision k ! wave number variable
double precision t ! time variable
integer ii,jj ! Loop variables
integer N ! Wave number cutoff number
integer Ntime ! Number of time steps
double precision l0 ! Time increment
double precision l1 ! Space resolution
double precision p ! Wave number increment
double precision eta ! Coupling constant
double precision c ! Velocity of light
! Assignment of parameters
parameter (kgamma = 10 ! 10^12 meter^(-1)
& , l0 = 1E-2 ! 10^(-20) second
& , l1 = 1E-1 ! 10^(-12) meter
& , p = 1E-1 ! 10^(12) meter^(-1)
& , N = 10, Ntime = 90
& , c = 3) ! 10^(8) meter/second
C kgamma = 10E12
C
C l0 = 1E-22
C l1 = 1E-13
C p = 10E11
C
C N = 100
C Ntime = 100
! Introducing photon density function (unit = 10^(-12) meter) , transfer rate, updated photon density function and assigning initial values
double precision f(0:Ntime,-1*N:N), w(-1*N:N,-1*N:N),
& fup(0:Ntime,-1*N:N) ! Photon density function f(t,k) and transfer rate w(k,k')
do ii = -1*N,N
f(0,ii) = 0
if (ii.ne.0) goto 100 ! corresponds to k = kgamma
f(0,ii) = 1/p
100 continue
end do
! Assignment of the coupling constant
print *, 'Enter a coupling constant (in 10^(-12) meter)'
read *, eta ! has dimensions 10^(-12) meter
! Assignment of transition probability density
do ii = -1*N,N
do jj = -1*N,N
if (ii.ne.jj) goto 200 ! Avoiding division by zero
w(ii,jj) = eta*p*abs(ii*p+kgamma)*abs(jj)*
& (sin((ii*p-jj*p)*c*l0)*(sin((ii*p-jj*p)
& *l1)))**2/(ii*p-jj*p)**4
200 continue
w(ii,jj) = eta*p*abs(ii*p+kgamma)*abs(jj)*
& (c*l0*l1)**2
end do
end do
! COMPUTING SOLUTION
do ii = 0,Ntime-1
do jj = -1*N,N
call gain(f,w,fup) ! Computing transition integral
f(ii + 1,jj) = f(ii,jj) + fup(ii,jj) ! Updating
end do
end do
! Output
do ii = 0,Ntime/30
print *, 'Time: ', 30*ii, '*10^(-22) seconds: '
do jj = -1*N,N
print *, 'Photon density at wavenumber',
& kgamma + jj*p, '*10^(12) meter^(-1): '
print *, f(30*ii,jj), '*10^(-12) meter'
end do
print *, '...'
end do
C open (UNIT = 100, FILE = 'Gamma_Ray_TDFT_Output.txt')
C rewind (UNIT = 100)
C do ii = 0,Ntime
C do jj = -N,N
C write (UNIT = 100, FMT = *) f(ii,jj)
C end do
C write (UNIT = 100, FMT = *) '--------------'
C end do
C close (UNIT = 100)
end program
The results are plotted for different
values of (Dimensions for : lengths in meter). Output for the choice of 
is listed here (the probability density function is captured every 
seconds):
Enter a coupling constant (in 10^(-12) meter)
1e-5
Time: 0 *10^(-22) seconds:
Photon density at wavenumber 8.9999999850988388 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 9.0999999865889549 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 9.1999999880790710 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 9.2999999895691872 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 9.3999999910593033 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 9.4999999925494194 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 9.5999999940395355 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 9.6999999955296516 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 9.7999999970197678 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 9.8999999985098839 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 10.000000000000000 *10^(12) meter^(-1):
9.9999998509883898 *10^(-12) meter
Photon density at wavenumber 10.100000001490116 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 10.200000002980232 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 10.300000004470348 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 10.400000005960464 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 10.500000007450581 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 10.600000008940697 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 10.700000010430813 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 10.800000011920929 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 10.900000013411045 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 11.000000014901161 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
...
Time: 30 *10^(-22) seconds:
Photon density at wavenumber 8.9999999850988388 *10^(12) meter^(-1):
2.6999996074169182E-007 *10^(-12) meter
Photon density at wavenumber 9.0999999865889549 *10^(12) meter^(-1):
2.4299997132693597E-007 *10^(-12) meter
Photon density at wavenumber 9.1999999880790710 *10^(12) meter^(-1):
2.1599998043231064E-007 *10^(-12) meter
Photon density at wavenumber 9.2999999895691872 *10^(12) meter^(-1):
1.8899998805781584E-007 *10^(-12) meter
Photon density at wavenumber 9.3999999910593033 *10^(12) meter^(-1):
1.6199999420345152E-007 *10^(-12) meter
Photon density at wavenumber 9.4999999925494194 *10^(12) meter^(-1):
1.3499999886921743E-007 *10^(-12) meter
Photon density at wavenumber 9.5999999940395355 *10^(12) meter^(-1):
1.0800000205511373E-007 *10^(-12) meter
Photon density at wavenumber 9.6999999955296516 *10^(12) meter^(-1):
8.1000003761140188E-008 *10^(-12) meter
Photon density at wavenumber 9.7999999970197678 *10^(12) meter^(-1):
5.4000003987296777E-008 *10^(-12) meter
Photon density at wavenumber 9.8999999985098839 *10^(12) meter^(-1):
2.7000002733583407E-008 *10^(-12) meter
Photon density at wavenumber 10.000000000000000 *10^(12) meter^(-1):
9.9999998509883898 *10^(-12) meter
Photon density at wavenumber 10.100000001490116 *10^(12) meter^(-1):
2.7000002733583407E-008 *10^(-12) meter
Photon density at wavenumber 10.200000002980232 *10^(12) meter^(-1):
5.4000003987296777E-008 *10^(-12) meter
Photon density at wavenumber 10.300000004470348 *10^(12) meter^(-1):
8.1000003761140188E-008 *10^(-12) meter
Photon density at wavenumber 10.400000005960464 *10^(12) meter^(-1):
1.0800000205511373E-007 *10^(-12) meter
Photon density at wavenumber 10.500000007450581 *10^(12) meter^(-1):
1.3499999886921743E-007 *10^(-12) meter
Photon density at wavenumber 10.600000008940697 *10^(12) meter^(-1):
1.6199999420345152E-007 *10^(-12) meter
Photon density at wavenumber 10.700000010430813 *10^(12) meter^(-1):
1.8899998805781584E-007 *10^(-12) meter
Photon density at wavenumber 10.800000011920929 *10^(12) meter^(-1):
2.1599998043231064E-007 *10^(-12) meter
Photon density at wavenumber 10.900000013411045 *10^(12) meter^(-1):
2.4299997132693597E-007 *10^(-12) meter
Photon density at wavenumber 11.000000014901161 *10^(12) meter^(-1):
2.6999996074169182E-007 *10^(-12) meter
...
Time: 60 *10^(-22) seconds:
Photon density at wavenumber 8.9999999850988388 *10^(12) meter^(-1):
5.3999984858341691E-007 *10^(-12) meter
Photon density at wavenumber 9.0999999865889549 *10^(12) meter^(-1):
4.8599989082199138E-007 *10^(-12) meter
Photon density at wavenumber 9.1999999880790710 *10^(12) meter^(-1):
4.3199992703903049E-007 *10^(-12) meter
Photon density at wavenumber 9.2999999895691872 *10^(12) meter^(-1):
3.7799995723453371E-007 *10^(-12) meter
Photon density at wavenumber 9.3999999910593033 *10^(12) meter^(-1):
3.2399998140850072E-007 *10^(-12) meter
Photon density at wavenumber 9.4999999925494194 *10^(12) meter^(-1):
2.6999999956093058E-007 *10^(-12) meter
Photon density at wavenumber 9.5999999940395355 *10^(12) meter^(-1):
2.1600001169182257E-007 *10^(-12) meter
Photon density at wavenumber 9.6999999955296516 *10^(12) meter^(-1):
1.6200001780117623E-007 *10^(-12) meter
Photon density at wavenumber 9.7999999970197678 *10^(12) meter^(-1):
1.0800001788899073E-007 *10^(-12) meter
Photon density at wavenumber 9.8999999985098839 *10^(12) meter^(-1):
5.4000011955265574E-008 *10^(-12) meter
Photon density at wavenumber 10.000000000000000 *10^(12) meter^(-1):
9.9999998509883898 *10^(-12) meter
Photon density at wavenumber 10.100000001490116 *10^(12) meter^(-1):
5.4000011955265574E-008 *10^(-12) meter
Photon density at wavenumber 10.200000002980232 *10^(12) meter^(-1):
1.0800001788899073E-007 *10^(-12) meter
Photon density at wavenumber 10.300000004470348 *10^(12) meter^(-1):
1.6200001780117623E-007 *10^(-12) meter
Photon density at wavenumber 10.400000005960464 *10^(12) meter^(-1):
2.1600001169182257E-007 *10^(-12) meter
Photon density at wavenumber 10.500000007450581 *10^(12) meter^(-1):
2.6999999956093058E-007 *10^(-12) meter
Photon density at wavenumber 10.600000008940697 *10^(12) meter^(-1):
3.2399998140850072E-007 *10^(-12) meter
Photon density at wavenumber 10.700000010430813 *10^(12) meter^(-1):
3.7799995723453371E-007 *10^(-12) meter
Photon density at wavenumber 10.800000011920929 *10^(12) meter^(-1):
4.3199992703903049E-007 *10^(-12) meter
Photon density at wavenumber 10.900000013411045 *10^(12) meter^(-1):
4.8599989082199138E-007 *10^(-12) meter
Photon density at wavenumber 11.000000014901161 *10^(12) meter^(-1):
5.3999984858341691E-007 *10^(-12) meter
...
Time: 90 *10^(-22) seconds:
Photon density at wavenumber 8.9999999850988388 *10^(12) meter^(-1):
8.0999966352520873E-007 *10^(-12) meter
Photon density at wavenumber 9.0999999865889549 *10^(12) meter^(-1):
7.2899975848518545E-007 *10^(-12) meter
Photon density at wavenumber 9.1999999880790710 *10^(12) meter^(-1):
6.4799983982016848E-007 *10^(-12) meter
Photon density at wavenumber 9.2999999895691872 *10^(12) meter^(-1):
5.6699990753015508E-007 *10^(-12) meter
Photon density at wavenumber 9.3999999910593033 *10^(12) meter^(-1):
4.8599996161514545E-007 *10^(-12) meter
Photon density at wavenumber 9.4999999925494194 *10^(12) meter^(-1):
4.0500000207513473E-007 *10^(-12) meter
Photon density at wavenumber 9.5999999940395355 *10^(12) meter^(-1):
3.2400002891012164E-007 *10^(-12) meter
Photon density at wavenumber 9.6999999955296516 *10^(12) meter^(-1):
2.4300004212010396E-007 *10^(-12) meter
Photon density at wavenumber 9.7999999970197678 *10^(12) meter^(-1):
1.6200004170507921E-007 *10^(-12) meter
Photon density at wavenumber 9.8999999985098839 *10^(12) meter^(-1):
8.1000027665045354E-008 *10^(-12) meter
Photon density at wavenumber 10.000000000000000 *10^(12) meter^(-1):
9.9999998509883898 *10^(-12) meter
Photon density at wavenumber 10.100000001490116 *10^(12) meter^(-1):
8.1000027665045354E-008 *10^(-12) meter
Photon density at wavenumber 10.200000002980232 *10^(12) meter^(-1):
1.6200004170507921E-007 *10^(-12) meter
Photon density at wavenumber 10.300000004470348 *10^(12) meter^(-1):
2.4300004212010396E-007 *10^(-12) meter
Photon density at wavenumber 10.400000005960464 *10^(12) meter^(-1):
3.2400002891012164E-007 *10^(-12) meter
Photon density at wavenumber 10.500000007450581 *10^(12) meter^(-1):
4.0500000207513473E-007 *10^(-12) meter
Photon density at wavenumber 10.600000008940697 *10^(12) meter^(-1):
4.8599996161514545E-007 *10^(-12) meter
Photon density at wavenumber 10.700000010430813 *10^(12) meter^(-1):
5.6699990753015508E-007 *10^(-12) meter
Photon density at wavenumber 10.800000011920929 *10^(12) meter^(-1):
6.4799983982016848E-007 *10^(-12) meter
Photon density at wavenumber 10.900000013411045 *10^(12) meter^(-1):
7.2899975848518545E-007 *10^(-12) meter
Photon density at wavenumber 11.000000014901161 *10^(12) meter^(-1):
8.0999966352520873E-007 *10^(-12) meter
...
Also for 
one can compute the following results:
Enter a coupling constant (in 10^(-12) meter)
1e-3
Time: 0 *10^(-22) seconds:
Photon density at wavenumber 8.9999999850988388 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 9.0999999865889549 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 9.1999999880790710 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 9.2999999895691872 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 9.3999999910593033 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 9.4999999925494194 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 9.5999999940395355 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 9.6999999955296516 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 9.7999999970197678 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 9.8999999985098839 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 10.000000000000000 *10^(12) meter^(-1):
9.9999998509883898 *10^(-12) meter
Photon density at wavenumber 10.100000001490116 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 10.200000002980232 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 10.300000004470348 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 10.400000005960464 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 10.500000007450581 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 10.600000008940697 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 10.700000010430813 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 10.800000011920929 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 10.900000013411045 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 11.000000014901161 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
...
Time: 30 *10^(-22) seconds:
Photon density at wavenumber 8.9999999850988388 *10^(12) meter^(-1):
2.6999647252684660E-005 *10^(-12) meter
Photon density at wavenumber 9.0999999865889549 *10^(12) meter^(-1):
2.4299749119948591E-005 *10^(-12) meter
Photon density at wavenumber 9.1999999880790710 *10^(12) meter^(-1):
2.1599836189080815E-005 *10^(-12) meter
Photon density at wavenumber 9.2999999895691872 *10^(12) meter^(-1):
1.8899908460003031E-005 *10^(-12) meter
Photon density at wavenumber 9.3999999910593033 *10^(12) meter^(-1):
1.6199965932636931E-005 *10^(-12) meter
Photon density at wavenumber 9.4999999925494194 *10^(12) meter^(-1):
1.3500008606904190E-005 *10^(-12) meter
Photon density at wavenumber 9.5999999940395355 *10^(12) meter^(-1):
1.0800036482726504E-005 *10^(-12) meter
Photon density at wavenumber 9.6999999955296516 *10^(12) meter^(-1):
8.1000495600255564E-006 *10^(-12) meter
Photon density at wavenumber 9.7999999970197678 *10^(12) meter^(-1):
5.4000478387230308E-006 *10^(-12) meter
Photon density at wavenumber 9.8999999985098839 *10^(12) meter^(-1):
2.7000313187406180E-006 *10^(-12) meter
Photon density at wavenumber 10.000000000000000 *10^(12) meter^(-1):
9.9999998509883898 *10^(-12) meter
Photon density at wavenumber 10.100000001490116 *10^(12) meter^(-1):
2.7000313187406180E-006 *10^(-12) meter
Photon density at wavenumber 10.200000002980232 *10^(12) meter^(-1):
5.4000478387230308E-006 *10^(-12) meter
Photon density at wavenumber 10.300000004470348 *10^(12) meter^(-1):
8.1000495600255564E-006 *10^(-12) meter
Photon density at wavenumber 10.400000005960464 *10^(12) meter^(-1):
1.0800036482726504E-005 *10^(-12) meter
Photon density at wavenumber 10.500000007450581 *10^(12) meter^(-1):
1.3500008606904190E-005 *10^(-12) meter
Photon density at wavenumber 10.600000008940697 *10^(12) meter^(-1):
1.6199965932636931E-005 *10^(-12) meter
Photon density at wavenumber 10.700000010430813 *10^(12) meter^(-1):
1.8899908460003031E-005 *10^(-12) meter
Photon density at wavenumber 10.800000011920929 *10^(12) meter^(-1):
2.1599836189080815E-005 *10^(-12) meter
Photon density at wavenumber 10.900000013411045 *10^(12) meter^(-1):
2.4299749119948591E-005 *10^(-12) meter
Photon density at wavenumber 11.000000014901161 *10^(12) meter^(-1):
2.6999647252684660E-005 *10^(-12) meter
...
Time: 60 *10^(-22) seconds:
Photon density at wavenumber 8.9999999850988388 *10^(12) meter^(-1):
5.3998565537591834E-005 *10^(-12) meter
Photon density at wavenumber 9.0999999865889549 *10^(12) meter^(-1):
4.8598979939503103E-005 *10^(-12) meter
Photon density at wavenumber 9.1999999880790710 *10^(12) meter^(-1):
4.3199334130803244E-005 *10^(-12) meter
Photon density at wavenumber 9.2999999895691872 *10^(12) meter^(-1):
3.7799628110832282E-005 *10^(-12) meter
Photon density at wavenumber 9.3999999910593033 *10^(12) meter^(-1):
3.2399861878930156E-005 *10^(-12) meter
Photon density at wavenumber 9.4999999925494194 *10^(12) meter^(-1):
2.7000035434436813E-005 *10^(-12) meter
Photon density at wavenumber 9.5999999940395355 *10^(12) meter^(-1):
2.1600148776692226E-005 *10^(-12) meter
Photon density at wavenumber 9.6999999955296516 *10^(12) meter^(-1):
1.6200201905036311E-005 *10^(-12) meter
Photon density at wavenumber 9.7999999970197678 *10^(12) meter^(-1):
1.0800194818809021E-005 *10^(-12) meter
Photon density at wavenumber 9.8999999985098839 *10^(12) meter^(-1):
5.4001275173502786E-006 *10^(-12) meter
Photon density at wavenumber 10.000000000000000 *10^(12) meter^(-1):
9.9999998509883898 *10^(-12) meter
Photon density at wavenumber 10.100000001490116 *10^(12) meter^(-1):
5.4001275173502786E-006 *10^(-12) meter
Photon density at wavenumber 10.200000002980232 *10^(12) meter^(-1):
1.0800194818809021E-005 *10^(-12) meter
Photon density at wavenumber 10.300000004470348 *10^(12) meter^(-1):
1.6200201905036311E-005 *10^(-12) meter
Photon density at wavenumber 10.400000005960464 *10^(12) meter^(-1):
2.1600148776692226E-005 *10^(-12) meter
Photon density at wavenumber 10.500000007450581 *10^(12) meter^(-1):
2.7000035434436813E-005 *10^(-12) meter
Photon density at wavenumber 10.600000008940697 *10^(12) meter^(-1):
3.2399861878930156E-005 *10^(-12) meter
Photon density at wavenumber 10.700000010430813 *10^(12) meter^(-1):
3.7799628110832282E-005 *10^(-12) meter
Photon density at wavenumber 10.800000011920929 *10^(12) meter^(-1):
4.3199334130803244E-005 *10^(-12) meter
Photon density at wavenumber 10.900000013411045 *10^(12) meter^(-1):
4.8598979939503103E-005 *10^(-12) meter
Photon density at wavenumber 11.000000014901161 *10^(12) meter^(-1):
5.3998565537591834E-005 *10^(-12) meter
...
Time: 90 *10^(-22) seconds:
Photon density at wavenumber 8.9999999850988388 *10^(12) meter^(-1):
8.0996754888042465E-005 *10^(-12) meter
Photon density at wavenumber 9.0999999865889549 *10^(12) meter^(-1):
7.2897692477902362E-005 *10^(-12) meter
Photon density at wavenumber 9.1999999880790710 *10^(12) meter^(-1):
6.4798493834101472E-005 *10^(-12) meter
Photon density at wavenumber 9.2999999895691872 *10^(12) meter^(-1):
5.6699158954374105E-005 *10^(-12) meter
Photon density at wavenumber 9.3999999910593033 *10^(12) meter^(-1):
4.8599687836454250E-005 *10^(-12) meter
Photon density at wavenumber 9.4999999925494194 *10^(12) meter^(-1):
4.0500080478076055E-005 *10^(-12) meter
Photon density at wavenumber 9.5999999940395355 *10^(12) meter^(-1):
3.2400336876973543E-005 *10^(-12) meter
Photon density at wavenumber 9.6999999955296516 *10^(12) meter^(-1):
2.4300457030880739E-005 *10^(-12) meter
Photon density at wavenumber 9.7999999970197678 *10^(12) meter^(-1):
1.6200440937531588E-005 *10^(-12) meter
Photon density at wavenumber 9.8999999985098839 *10^(12) meter^(-1):
8.1002885946600514E-006 *10^(-12) meter
Photon density at wavenumber 10.000000000000000 *10^(12) meter^(-1):
9.9999998509883898 *10^(-12) meter
Photon density at wavenumber 10.100000001490116 *10^(12) meter^(-1):
8.1002885946600514E-006 *10^(-12) meter
Photon density at wavenumber 10.200000002980232 *10^(12) meter^(-1):
1.6200440937531588E-005 *10^(-12) meter
Photon density at wavenumber 10.300000004470348 *10^(12) meter^(-1):
2.4300457030880739E-005 *10^(-12) meter
Photon density at wavenumber 10.400000005960464 *10^(12) meter^(-1):
3.2400336876973543E-005 *10^(-12) meter
Photon density at wavenumber 10.500000007450581 *10^(12) meter^(-1):
4.0500080478076055E-005 *10^(-12) meter
Photon density at wavenumber 10.600000008940697 *10^(12) meter^(-1):
4.8599687836454250E-005 *10^(-12) meter
Photon density at wavenumber 10.700000010430813 *10^(12) meter^(-1):
5.6699158954374105E-005 *10^(-12) meter
Photon density at wavenumber 10.800000011920929 *10^(12) meter^(-1):
6.4798493834101472E-005 *10^(-12) meter
Photon density at wavenumber 10.900000013411045 *10^(12) meter^(-1):
7.2897692477902362E-005 *10^(-12) meter
Photon density at wavenumber 11.000000014901161 *10^(12) meter^(-1):
8.0996754888042465E-005 *10^(-12) meter
...
Finally, for 
one has the results:
Enter a coupling constant (in 10^(-12) meter)
1e-1
Time: 0 *10^(-22) seconds:
Photon density at wavenumber 8.9999999850988388 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 9.0999999865889549 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 9.1999999880790710 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 9.2999999895691872 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 9.3999999910593033 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 9.4999999925494194 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 9.5999999940395355 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 9.6999999955296516 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 9.7999999970197678 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 9.8999999985098839 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 10.000000000000000 *10^(12) meter^(-1):
9.9999998509883898 *10^(-12) meter
Photon density at wavenumber 10.100000001490116 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 10.200000002980232 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 10.300000004470348 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 10.400000005960464 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 10.500000007450581 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 10.600000008940697 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 10.700000010430813 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 10.800000011920929 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 10.900000013411045 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
Photon density at wavenumber 11.000000014901161 *10^(12) meter^(-1):
0.0000000000000000 *10^(-12) meter
...
Time: 30 *10^(-22) seconds:
Photon density at wavenumber 8.9999999850988388 *10^(12) meter^(-1):
2.6964814650656211E-003 *10^(-12) meter
Photon density at wavenumber 9.0999999865889549 *10^(12) meter^(-1):
2.4274976458384485E-003 *10^(-12) meter
Photon density at wavenumber 9.1999999880790710 *10^(12) meter^(-1):
2.1583664065873017E-003 *10^(-12) meter
Photon density at wavenumber 9.2999999895691872 *10^(12) meter^(-1):
1.8890876692842968E-003 *10^(-12) meter
Photon density at wavenumber 9.3999999910593033 *10^(12) meter^(-1):
1.6196613558617187E-003 *10^(-12) meter
Photon density at wavenumber 9.4999999925494194 *10^(12) meter^(-1):
1.3500873882120058E-003 *10^(-12) meter
Photon density at wavenumber 9.5999999940395355 *10^(12) meter^(-1):
1.0803656881877306E-003 *10^(-12) meter
Photon density at wavenumber 9.6999999955296516 *10^(12) meter^(-1):
8.1049617760157761E-004 *10^(-12) meter
Photon density at wavenumber 9.7999999970197678 *10^(12) meter^(-1):
5.4047877822632675E-004 *10^(-12) meter
Photon density at wavenumber 9.8999999985098839 *10^(12) meter^(-1):
2.7031341179483136E-004 *10^(-12) meter
Photon density at wavenumber 10.000000000000000 *10^(12) meter^(-1):
9.9999998509883898 *10^(-12) meter
Photon density at wavenumber 10.100000001490116 *10^(12) meter^(-1):
2.7031341179483136E-004 *10^(-12) meter
Photon density at wavenumber 10.200000002980232 *10^(12) meter^(-1):
5.4047877822632675E-004 *10^(-12) meter
Photon density at wavenumber 10.300000004470348 *10^(12) meter^(-1):
8.1049617760157761E-004 *10^(-12) meter
Photon density at wavenumber 10.400000005960464 *10^(12) meter^(-1):
1.0803656881877306E-003 *10^(-12) meter
Photon density at wavenumber 10.500000007450581 *10^(12) meter^(-1):
1.3500873882120058E-003 *10^(-12) meter
Photon density at wavenumber 10.600000008940697 *10^(12) meter^(-1):
1.6196613558617187E-003 *10^(-12) meter
Photon density at wavenumber 10.700000010430813 *10^(12) meter^(-1):
1.8890876692842968E-003 *10^(-12) meter
Photon density at wavenumber 10.800000011920929 *10^(12) meter^(-1):
2.1583664065873017E-003 *10^(-12) meter
Photon density at wavenumber 10.900000013411045 *10^(12) meter^(-1):
2.4274976458384485E-003 *10^(-12) meter
Photon density at wavenumber 11.000000014901161 *10^(12) meter^(-1):
2.6964814650656211E-003 *10^(-12) meter
...
Time: 60 *10^(-22) seconds:
Photon density at wavenumber 8.9999999850988388 *10^(12) meter^(-1):
5.3857050545415754E-003 *10^(-12) meter
Photon density at wavenumber 9.0999999865889549 *10^(12) meter^(-1):
4.8498306592493733E-003 *10^(-12) meter
Photon density at wavenumber 9.1999999880790710 *10^(12) meter^(-1):
4.3133588782991754E-003 *10^(-12) meter
Photon density at wavenumber 9.2999999895691872 *10^(12) meter^(-1):
3.7762890567027800E-003 *10^(-12) meter
Photon density at wavenumber 9.3999999910593033 *10^(12) meter^(-1):
3.2386205387660997E-003 *10^(-12) meter
Photon density at wavenumber 9.4999999925494194 *10^(12) meter^(-1):
2.7003526680884037E-003 *10^(-12) meter
Photon density at wavenumber 9.5999999940395355 *10^(12) meter^(-1):
2.1614847875615737E-003 *10^(-12) meter
Photon density at wavenumber 9.6999999955296516 *10^(12) meter^(-1):
1.6220162393693520E-003 *10^(-12) meter
Photon density at wavenumber 9.7999999970197678 *10^(12) meter^(-1):
1.0819463649866014E-003 *10^(-12) meter
Photon density at wavenumber 9.8999999985098839 *10^(12) meter^(-1):
5.4127450517853985E-004 *10^(-12) meter
Photon density at wavenumber 10.000000000000000 *10^(12) meter^(-1):
9.9999998509883898 *10^(-12) meter
Photon density at wavenumber 10.100000001490116 *10^(12) meter^(-1):
5.4127450517853985E-004 *10^(-12) meter
Photon density at wavenumber 10.200000002980232 *10^(12) meter^(-1):
1.0819463649866014E-003 *10^(-12) meter
Photon density at wavenumber 10.300000004470348 *10^(12) meter^(-1):
1.6220162393693520E-003 *10^(-12) meter
Photon density at wavenumber 10.400000005960464 *10^(12) meter^(-1):
2.1614847875615737E-003 *10^(-12) meter
Photon density at wavenumber 10.500000007450581 *10^(12) meter^(-1):
2.7003526680884037E-003 *10^(-12) meter
Photon density at wavenumber 10.600000008940697 *10^(12) meter^(-1):
3.2386205387660997E-003 *10^(-12) meter
Photon density at wavenumber 10.700000010430813 *10^(12) meter^(-1):
3.7762890567027800E-003 *10^(-12) meter
Photon density at wavenumber 10.800000011920929 *10^(12) meter^(-1):
4.3133588782991754E-003 *10^(-12) meter
Photon density at wavenumber 10.900000013411045 *10^(12) meter^(-1):
4.8498306592493733E-003 *10^(-12) meter
Photon density at wavenumber 11.000000014901161 *10^(12) meter^(-1):
5.3857050545415754E-003 *10^(-12) meter
...
Time: 90 *10^(-22) seconds:
Photon density at wavenumber 8.9999999850988388 *10^(12) meter^(-1):
8.0677038527882921E-003 *10^(-12) meter
Photon density at wavenumber 9.0999999865889549 *10^(12) meter^(-1):
7.2670181687958177E-003 *10^(-12) meter
Photon density at wavenumber 9.1999999880790710 *10^(12) meter^(-1):
6.4649863181379203E-003 *10^(-12) meter
Photon density at wavenumber 9.2999999895691872 *10^(12) meter^(-1):
5.6616060613337285E-003 *10^(-12) meter
Photon density at wavenumber 9.3999999910593033 *10^(12) meter^(-1):
4.8568751552180737E-003 *10^(-12) meter
Photon density at wavenumber 9.4999999925494194 *10^(12) meter^(-1):
4.0507913529355000E-003 *10^(-12) meter
Photon density at wavenumber 9.5999999940395355 *10^(12) meter^(-1):
3.2433524039342460E-003 *10^(-12) meter
Photon density at wavenumber 9.6999999955296516 *10^(12) meter^(-1):
2.4345560539602447E-003 *10^(-12) meter
Photon density at wavenumber 9.7999999970197678 *10^(12) meter^(-1):
1.6244000450511054E-003 *10^(-12) meter
Photon density at wavenumber 9.8999999985098839 *10^(12) meter^(-1):
8.1288211553007521E-004 *10^(-12) meter
Photon density at wavenumber 10.000000000000000 *10^(12) meter^(-1):
9.9999998509883898 *10^(-12) meter
Photon density at wavenumber 10.100000001490116 *10^(12) meter^(-1):
8.1288211553007521E-004 *10^(-12) meter
Photon density at wavenumber 10.200000002980232 *10^(12) meter^(-1):
1.6244000450511054E-003 *10^(-12) meter
Photon density at wavenumber 10.300000004470348 *10^(12) meter^(-1):
2.4345560539602447E-003 *10^(-12) meter
Photon density at wavenumber 10.400000005960464 *10^(12) meter^(-1):
3.2433524039342460E-003 *10^(-12) meter
Photon density at wavenumber 10.500000007450581 *10^(12) meter^(-1):
4.0507913529355000E-003 *10^(-12) meter
Photon density at wavenumber 10.600000008940697 *10^(12) meter^(-1):
4.8568751552180737E-003 *10^(-12) meter
Photon density at wavenumber 10.700000010430813 *10^(12) meter^(-1):
5.6616060613337285E-003 *10^(-12) meter
Photon density at wavenumber 10.800000011920929 *10^(12) meter^(-1):
6.4649863181379203E-003 *10^(-12) meter
Photon density at wavenumber 10.900000013411045 *10^(12) meter^(-1):
7.2670181687958177E-003 *10^(-12) meter
Photon density at wavenumber 11.000000014901161 *10^(12) meter^(-1):
8.0677038527882921E-003 *10^(-12) meter
...
From these results it can be seen that all
photon density functions look very similar to photon density functions on . As expected, only a small deviation in the wavenumber distribution
due to TDFT scattering processes can be observed. Interactions predicted by
TDFT become significant after approximate
3 zeptoseconds. These TDFT correction clearly
increases if the TDFT coupling constant increases. In the neighborhood of the
characteristic wave number that has the value 
one can observe values in the wavenumber density which are much
smaller than the wavenumber density at the characteristic wave number; these
values are oscillating, but remain on the same magnitude scale over the
remaining observation time. There is a cloud of other photons with lower
intensity around the dominating photon (which has the characteristic wave
number).
Conclusions
From the simplified version of the transition rate there can be obtained statements about a system that is governed by interactions predicted by TDFT. When plotting the results diagrammatically one cannot see the deviations in wavenumber distribution, because these are very small. The numerical results have shown that a monochromatic photon governed by TDFT interactions generates some other photon rays with distinct frequency with much smaller intensity. For the experimental validation of TDFT one has to check whether these photon rays with distinct frequencies exist even if a monochromatic photon is emitted to the vacuum.
References
Bloser, P. F. et al. "GRAPE-A Balloon-Borne Gamma-Ray Polarimeter Experiment." Chinese Journal of Astronomy and Astrophysics, 2006, 6 (1): 393.
Linker, P. "Topological Dipole Field Theory." The Winnower, 2015, 2: e144311.19292. doi: 10.15200/winn.144311.19292
Linker, P. "Nonabelian Generalization of Topological Dipole Field Theory." The Winnower, 2015, 2: e144564.43935. doi: 10.15200/winn.144564.43935
Linker, P. "Scattering processes predicted by Topological Dipole Field Theory." The Winnower, 2015, 2: e144607.74141. doi: 10.15200/winn.144607.74141
Showing 2 Reviews
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1It was a good paper. I think Black Holes could use your idea of spotting Gamma Rays in them. Which correlate with your TDFT and Quantum Gravity.
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1A really great work by Patrick Linker.
License
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Thanks for the good review! If you are celebrating christmas:
Merry christmas!