From: sarfatti@well.sf.ca.us (Jack Sarfatti)
Subject: Feynman 45 - gravity cut-off in QED has zero bare mass and charge
Organization: Whole Earth 'Lectronic Link
Date: Sun, 3 Jan 1993 07:02:46 GMT


Feynman 45,renormalization of charge by vacuum polarization and a
conjecture on the nature of cosmological dark matter and that quantum
gravity creates mass and charge of electron from zero rather than infinite
bare values!. Part VII

Recall from the last lecture that:
"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.

.......

A method of making (32) convergent without spoiling the gauge invariance
has been found by Bethe and by Pauli. The convergence factor for light can
be looked upon as the result of a superpostion of the effects of quanta of
various masses (some contributing negatively) ...  likewise (i.e. the
convergence factor for electrons) if we we take the factor C(p^2 - m^2) = -
@^2(p^2 - m^2 -@^2)^-1 so that (p^2 - M^2)^-1C(p^2 - m^2) = (p^2 - m^2)^-1
- (p^2 - m^2 - @^2)^-1 we are taking the difference of the result for
electrons of mass m and mass (@^2 + m^2)^1/2. But we have taken this
difference for each propagation between interactions with photons.... That
is, if ...(32) integrated over some finite range of p is called Juv(m^2),
but with m replaced by (@^2 + m^2)^1/2 is called Juv((@^2 + m^2)^1/2) we
should calculate (32')

Juv^P = S(0 to infinity)[Juv(m^2) - Juv((@^2 + m^2)^1/2)]G(@)d@ ... then
... the range of p integration can be extended to infinity as the integral
now converges.....(33) is an explicit gauge invariant solution for JuvP.
See original paper. The formal statement of U(1) em gauge invariance in 4-
momentum space is

qu(quqv - q^2duv) = 0. There is a term -(1/3)(ln(@^2/m^2)(quqv-q^2duv)
which "gives a current proportional to the current producing the potential.
This would have the same effect as a change in charge, so that we would
have a difference de^2 between e^2 and the experimentally observed
(squared) charge e^2 + de^2, analogous to the (self-energy) difference
between m and the observed mass. This charge depends logarithmically on the
cut-off, de^2/e^2 = -(2e^/3pi)ln(@/m). After this renormalization of charge
is made, no effects will be sensitive to the cut-off"  There is a second
more complicated cut-off independent term in the vacuum response function
Juv.

*Suppose we use the same gravity cut-off we did in the mass renormalization
of of the self-energy of the electron in which all the mass of the electron
is caused by gravity or curvature - as it should be.

@^2/m^2 is 10^-10/10^-54 = 10^44 so ln@^2/m^2 is order of 44/loge = 44/.43
= 102 and e^2 is 1/137 so de^2/e^2 is very roughly .16 and de/e is .4 - not
bad! It may be that a more accurate calculation with more Feynman diagrams
using the gravity cut-off gives both mass and charge starting from the idea
of zero bare mass and zero bare charge! In standard QED the bare quantities
are infinite.*

*What does Feynman mean, in the case of the convergence factor for light
only, by "some contributing negatively" does that mean spin 1 vector
particles with the wrong Fermi-Dirac statistics like the Faddeev-Popov
ghosts? The ghosts do not exist asymptotically as real particles - but
perhaps that is a mistake - perhaps they compose the cosmological dark
matter which is more than 90% of the mass of the universe. This dark matter
would be superluminal in real time and transluminal in imaginary time which
is orthogonal to real time but co-existent in a "shadow universe"?*




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