From: sarfatti@well.sf.ca.us (Jack Sarfatti)
Subject: Feynman 46 -The Master shows Coulomb force is faster than light!
Organization: Whole Earth 'Lectronic Link
Date: Wed, 6 Jan 1993 05:26:43 GMT


Feynman 46. The coulomb force really is faster-than-light in QED!
8. Longitudinal waves. A spacelike action for the Coulomb potential!
"In the usual form of quantum electrodynamics the longitudinal and
transverse waves are given separate treatment. Alternately the condition
4divpsi = 0 is carried along as a supplementary condition. In  the present
form no such special considerations are necessary for we are dealing with
the solutions of the equation -WAVEA = 4pij with a current j which is
conserved 4Divj = 0. That means at least WAVE4DivA = 0 and in fact our
solution also satisfies 4DivA = 0.

To show that this is the case we consider the amplitude for emission (real
or virtual) of a photon and show that the divergence of this amplitude
vanishes. The amplitude for emission for photons polarized in the u
direction involves matrix elements of the u gamma matrix. Therefore what we
have to show is that the corresponding matrix elements of quGu = q/
vanish...." see original paper  "Hence any wave emitted will satisfy 4DivA
= 0. Likewise longitudinal waves (that is waves for which A = 4grad@ or a/
= q/) cannot be absorbed and will have no effect, for the matrix elements
of emission and absorption are similar. (We have said little more than that
a vector potential A = 4grad@ has no effect on a Dirac electron since a
transformation psi' = e^-i@ psi removes it. ....

This has a useful practical consequence in that in computing probabilities
for transition for unpolarized light one can sum the squared matrix over
all four directions rather than just the two special polarization vectors.
...Thus suppose the matrix element for some process for light polarized in
direction eu is euMu. If the light has wave vector qu we know from the
argument above (i.e., 4DivA = 0) that quMu = 0. For unpolarized light
progressing in the z direction we would ordinarily calculate Mx^2 + My^2.
But we can as well sum Mx^2 +My^2 + Mz^2 - Mt^2 for quMu = 0 implies Mt =
Mz since qt = qz for free (i.e., real) quanta (i.e. photons). This shows
that unpolarized light is a relativistically invariant concept.....

Real processes correspond to poles in the formulas for virtual processes.
The pole occurs when k/^2 = 0 ... it looks at first as though ,,,, we would
have four kinds of polarization instead of two. Now it is clear that only
two perpendicular to k are effective.

The usual elimination of longitudinal and scalar virtual photons (leading
to an instantaneous Coulomb potential) can of course be performed here
too....K is the 3-vector part of k (4-vector)  k/ = k4Gt - KGK. Feynman
shows that a typical term in a virtual transition reduces to a product of
timelike gamma matrices multiplied by d^4k/K^2 where K is the 3-vector part
not the 4 vector. This is why the Coulonmb potential is really spacelike in
its influence. If in the rest frame of the charge the potential is
instantaneous, then in any other frame the interval between field point and
source point, while not simultaneous, is spacelike!

In more detail Feynman shows that a general virtual process involves the
matrix part

(Gt...Gt) - (GK...GK) = [(K^2 - k4^2)/K^2](Gt...Gt)

because "we have shown above that k/ replacing the G gives zero ... so that
multiplying by k/^-2-d^4k = d^4k (k4^2 -K^2)^-1 the net effect is -
(Gt...Gt)d^4k/K^2. The Gt means just scalar waves, that is, potentials
produced by charge density (not currents). The fact that 1/K^2 does not
contain k4 means that k4 can be integrated first, resulting in an
instantaneous interaction, and the d^3K/K^2 is just the momentum
representation (Fourier transform) of the Coulomb potential 1/r."

Thus, Feynman proves that the elimination of the longitudinal and scalar
virtual photons required by the Maxwell field equation with conserved
currents really requires a Coulomb force that is outside the light cone!
The virtual longitudinal and scalar photons exert a spacelike influence!