From: sarfatti@well.sf.ca.us (Jack Sarfatti)
Newsgroups: sci.physics,alt.sci.physics.new-theories
Subject: Feynman 38 - boundary conditions on propagators (include tachyons
Date: 20 Dec 92 19:19:35 GMT
Organization: Whole Earth 'Lectronic Link


Feynman 38
6. Energy-momentum representation.
"The practical evaluation of matrix elements ...  K+(2,1) is fairly
complicated but ... its Fourier transform is very simple , namely

K+(2,1) = (i/4pi^2)S^4[(p/ - m)^-1e^ipx21 d^4p]    (31)

where p/ is Lorentz inner product of 4-vector p with gamma matrices. x21 =
x2-x1 etc.

K+(2,1) = i(igrad(2) + m)I+(2,1)

I+(2,1) = (2pi)^-2S[(p/^2 - m^2)^-1 e^-ipx21 d^4p]   (32)

I+ is the solution of inhomogeneous Klein-Gordon equation with a 4D Dirac
delta function source.

WAVE(2)I+(2,1) - m^2I+(2,1) = Dirac(2,1)          (33)

WAVE = d^2/dt^2 - Grad^2

Feynman uses + - - - signature of flat Lorentzian metric.
"The integrals (31) and (32) are not yet completely defined for there are
poles in the integrand when p/^2 - m^2 = 0.  We can define how these poles
are to be evaluated by the rule that m is considered to have an
infinitesimal negative imaginary part.  That is, m, is replaced by m -
idelta and the limit taken as delta -> from above. This can be seen by
imagining that we calculate K+ by integrating on p4 first. If we call
E = +(m^2 + p1^2 + p2^2 +p3^2)^1/2 then the integrals involve p4
essentially as S[e^-ip4(t2 - t1) dp4 (p4^2 - E^2)^-1 which has poles at p4
= +E and p4 = -E.  The replacement of m by m - idelta means that E has a
small negative imaginary part; the first pole is below, the second above
the real axis.  Now if t2 - t1 > 0 the contour can be completed around the
semicircle below the real axis thus giving a residue from the p4 = +E pole,
or -(2E)^-1e^-iE(t2 - t1).  If t2 - t1 < 0 the upper semicircle must be
used and p4 = -E at the pole.  ... Explicitly the function is

I+(x,t) = -(4pi)^-1 Dirac (s^2) + (m/8pis)H1(2)(ms)  (34)

where s = +(t^2 - x^2)^1/2 for t^2 > x^2 (inside light cone) and s = -i(x^2
- t^2)^1/2 for t^2 < x^2 (outside light cone), H1(2) is the Hankel function
and Dirac (s^2) is the Dirac delta function of (invariant) s^2.  It behaves
asymptotically as e^-ims (inside light cone), decaying exponentially in
space-like directions (outside light cone)"

This choice of contour or boundary condition corresponds to positive
energies propagating as retarded waves forward in time (+FIT) and negative
energies propagating as advanced waves backward in time (-BIT) as in
Feynman's eq. (17).  The retarded (p.x - Et) and advanced waves (p.x + ET)
are pointed in the same direction of space.
                               / 2
                             /
                           /  Feynman's +|E| electron FIT wave to +x
                         /    1 -> 2
                       /
                       1
                       \
                         \
                           \  Feynman's -|E| electron BIT wave also to +
                             \ 1 -> 3    = +|E| positron FIT wave to -x,
                               \ 3   (i.e.,3 -> 1)

This is not the same as the transactional interpretation of Born's quantum
probability density for a history as the modulation of a retarded FIT
amplitude by a complex-conjugate advanced BIT wave of same sign in energy.
That is, in contrast to Feynman's subluminal matter-antimatter boundary
condition, the Born probability interpretation comes from:

                                 2
                               / /
                             / /
      +|E| FIT 1 -> 2      / /     +|E| BIT "Star Wave" 2 -> 1
                         / /
                       / /
                       1

                  loop in time


Thought Problem by Sarfatti #1: Creation From The Future?

I note that if we had a faster-than-light "tachyon" spinor field then

I'+ = (2pi)^-2S[(p/^2 + m^2)^-1 e^-ipx21 d^4p]

and the p4 part of the "propagating" integral is only from
p1^2 + p2^2 +p3^2 = m^2 to infinity in order that energy be real (c = 1).
The frame invariant mass m is pure imaginary for the tachyon so that the
tachyonic mass shell p4 plane poles for real particles are
E = +(-)(p1^2 + p2^2 +p3^2 - m^2)^1/2. Note that virtual "zero point"
particles of the vacuum structure are off this mass shell.

What is the appropriate contour in the energy part of the integral?  Do we
have to replace the Feynman rule that mass has infinitesimal negative
imaginary part?  No, it appears we can use the same rule, except now E has
a small positive imaginary part because of the -m^2 (rather than +m^2 in
subluminal causal case).  However, the distinction between FIT and BIT is
not frame invariant in the absolute elsewhere outside the light cone where
we want propagation.  The p4 contour must be chosen so that we have damping
in both forward and backward light cones.

I emphasize, the whole idea of this exercise is that we want to have
causality-violating space-like wave propagation in the absolute elsewhere
outside the field of parallel light cones at every point in flat space time
with damped time-like propagation both forward and backward in time inside
the light cones - just the opposite of Feynman's result above for
subluminal fields that obey causality.

All the talk on sci physics that tachyons do not violate causality is due
to picking the wrong contour for the physical boundary conditions on the
superluminal field propagating outside the light cone in real time.  Such
propagation does not violate the invariant light-cone 4D geometry of
special relativity but it does violate retarded causality. Such causality
violation may be essential for the existence of life and mind.

We must be prepared to violate the usual spin-statistics connection. Thus a
tachyonic causality-violating massive spinor (1/2) field would have boson
quanta just like the Faddeev-Popov ghosts (e.g., spin zero scalar fermions)
of non-abelian Yang-Mills gauge theory except these are not ghosts but real
particles!  Remember, I want gross causality-violation as the mechanism for
the intelligent creation of the big bang in a quantum fluctuation out of
nothing engineered from the inside of the far future of that amplified
fluctuation that is our universe!
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