From: sarfatti@well.sf.ca.us (Jack Sarfatti)
Subject: Feynman 44 Is it unitary? Sarfatti predicts new solid state release of vacuum energy based upon Feynman's musings in 1949?
Organization: Whole Earth 'Lectronic Link
Date: Thu, 31 Dec 1992 07:32:00 GMT


Feynman 44 QED & renormalization Part VI Feynman has problems with
unitarity and gauge invariance. Sarfatti introduces new idea to release
vacuum energy in solid state physics - it is right?
7. The problem of vacuum polarization
the virtual electron-positron pairs not included in above radiative
corrections to scattering which only dealt with virtual photons emitted and
reabsorbed on the electron worldlines.


                              / real electron p2 hit by virtual photon
                            /
           virtual photon  /
         G ______________/ G
          /\            |
  pa+q  /    \ pa       |
        \    /          |
          \/ A          | real electron p1
  virtual pair "closed loop"

This is the biggest vacuum polarization Feynman diagram in 4-momentum
space. I shall paraphrase a little - the quotes are loose - see original
paper for details.

"The potential Ae^-iqx creates a pair of electrons of momenta pa, -pb. This
pair then reannihilates, emitting a quantum q = pa-pb, which quantum
scatters the original electron from state 1 to state 2. The matrix element
for this process ... is (30)

-(e^2/ipi)(uad2Gu1)S{Tr[(pa/+q/-m)^-1 G(pa/-m)^-1G]d^4pa q^-2 C(q^2)A

... the potential produces the pair with amplitude proportional to AG, the
electrons of momenta pa and -(pa+q) proceed from there to annihilate
producing a quantum (factor G) which propagates (factor q^-2C(q^2)) over to
the other electron, by which it is absorbed (matrix element of G between
states 1 and 2 of the original electron, i.e., (uad2Gu1),. All momenta pa
and spin states of the virtual electron pair are admitted which means the
trace and the d^4pa integral are calculated.

One can imagine that the closed loop path of the positron-electron produces
a (vacuum polarization) current

4piju = JuvAv  (31)

(uv are here are not electron spinors but Lorentz indices ranging from 1 to
4 with summation convention on RHS of 31)

which is the source of the virtual photons which act on the second
electron. One sees .. that

Juv = S{Tr[(pa/+q/-m)^-1 G(pa/-m)^-1G]d^4pa  (32)

diverges very badly. The modification of D to f alters the amplitude with
which the vacuum polarization current will affect the scattered real
electron , but it can do nothing to prevent the divergence ....

***The closed loops are a consequence of the usual hole theory in
electrodynamics (i.e., vacuum structure is a filled Fermi sea of negative
energy electrons propagating forward in time. Lift a negative FIT energy
electron into a positive energy FIT state and the hole left behind is a
postive energy FIT positron - this is the Dirac picture of the QED vacuum
structure. It is similar to non-relavistic ground state of a simple metal
in solid state physics.   The Pauli exclusion principle makes the vacuum
stable. Changing the quantum statistics from Fermi to Bose by changing
dimension from 3+1 to 2+1 using slabs or "quantum wells" would appear to
make the metallic ground state unstable inducing a first order phase
transition releasing energy? -Sarfatti note of 12/30/92)***

Was that an unanticipated telepatype - a message from the future?

Back to Feynman:
"The closed loops are .... required to keep probability conserved. The
probability that no pair is produced by a potential is not unity and its
deviation from unity arises from the imaginary part of Juv. Again with
closed loops excluded, a pair of electrons once created cannot be
annihilated, the scattering of light by light would be excluded etc.....

To be sure, it is always possible that these matters of probability
conservation will work themselves out as simply in the case of interacting
charges as for those in a fixed potential...It would be interesting to
calculat the Lamb shift accurately enough to be sure that the 20 megacylce
expected from vacuum polarization are actually present" Remember this is
1949! Feynmen then talks of intruducing still another convergence factor
for the electrons but he cautions that "the result is unsatisfactory"
because it violates gauge invariance (i.e. local conservation of electric
charge).

to be continued





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