From: sarfatti@well.sf.ca.us (Jack Sarfatti)
Subject: Feynman42 -gravity explains electron mass?
Organization: Whole Earth 'Lectronic Link
Date: Mon, 28 Dec 1992 05:22:32 GMT


Feynman 42 QED & renormalization Part IV  gravitational cut-off in QED?

4. self-energy in momentum and energy space
".. the Fourier transform of D+(s21^2) is

-D+(s21^2) = (pi)^-1 S[e^-ikx21 (k/^-2) d^4k]   (10)

... noting that I+(2,1) for m^2 = 0 is D+(s21^2) ... the k/^-2 means ...
the limit as d->0 of (k.k +id)^-1 ... imagine that quanta (i.e., photons)
are particles of zero mass.. we make the general rule that all poles are to
be resolved by considering the masses of the particles (i.e. electrons) and
quanta to have infinitesimal negative imaginary parts. ... we see that the
self-energy is the matrix elementa between (spinors) uadjoint and u of the
matrix

(e^2/ipi) S[G(p/-k/-m)^-1Gk/^-2 d^4k]  (11)

                        / 2
                      / p
                    /
                  /
                 |\ 4
                 | \
                 |   \
         p-k     |     \ k^-2
                 |    /
                 |  /
                 |/
                / 3
              /
            / p
          /
        / 1

Simplest (and numerically largest) electron self-energy Feynman diagram in
energy-momentum space with one virtual "zero point vacuum fluctuation
"photon.

"the equation can be understood by imagining that the electron of (four)
momentum p emits (gamma G) a quantum of (four)momentum k and makes its way
now with momentum p - k to the next event (propagtor factor (p/ - k/ -m)^-
1) which is to absorb the quantum (another 4x4 gamma matrix factor G, note
4-vector subscript understood JS). The amplitude of propagation of quanta
is k^-2. (There is a factor e^2/ipi for each virtual quantum). One
integrates over all (virtual) quanta.

The reason an electron of 4-momentum propagates as 1/(p/ - m) is that this
operator is the reciprocal of the Dirac equation operator, and we are
simply solving this equation. Likewise light goes as 1/k^2, for this is the
reciprocal D'Alembertion operator of the wave equation of light. The first
gamma G reprsents the current which generates the vector potential, while
the second is the velocity operator by which this potential is multiplied
in the Dirac equation when an external field acts on an electron."

Sarfatti homework problems

1. Can we make a consistent, but causality-violating QED with superluminal
(i.e.,tachyonic) electrons with propagator 1/(p/-im) in which 3-vector
magnitude part varies only from m to infinity?  Do we have to use boson
quantum statistics for the tachyonic electrons? How do they interact with
ordinary subluminal electrons?

2. Can we replace simple Fourier transforms using plane waves by a "wavelet
transform" to get better "nonperturbative" convergence?

Back to Feynman: he then discuses radiative corrections (i.e. virtual
photon effects) to scattering and distinguishes three cases.  "We have now
achieved our simplification of the form of writing matrix elements arising
from virtual processes." He then describes Compton electron-real photon
scattering (Klein-Nishina formula)as an example.  "Pair annihilation with
emission of two quanta are given by the same (Compton scattering) matrix,
positron states being those with negative time components of p. Whether
(real) quanta are absorbed or emitted depends on whether the time component
of q is positive or negative."

5. The convergence of processes with virtual quanta.
"... the self-energy expression (9)

dE1 = (e^2)S[(uGK0+(4,3)Gu)e^ipx43D+(s43^2)d4]  (9)

gives an infinite result when evaluated. The infinity arises, apparently,
from the coincidence of the Dirac delta function singularities in K+(4,3)
and D+(s43^2). Only at this point is it necessary to make a real departure
from conventional electrodynamics ...."

Sarfatti homework problem: What happens if we just put in a gravitational
cut-off at the Planck frequency 10^43 Hz where Lorentz symmetry breaks down
anyway?

Don't forget that the Lorentz invariant D+(s^2) requires both retarded
waves that propagate from now to future and advanced waves that propagate
from now to past - future causation is intrinsic to Lorentz invariance
contrary to conventional wisdom! Forbidding future causes really violates
special relativity - though Einstein of course forbade future cause which I
believe narrows the range of application of relatvity unnecessarily. Causal
special relativity is to causality-violating special relativity as
Euclidean geometry is to non-Euclidean geometry. This feature is
independent of Feynman's boundary condition.

Back to Feynman's boundary condition on propagators:
"The obvious corresponding modification in the quantum theory is to replace
the D+(s^2) appearing in the quantum mechanical interaction by a new
function f+(s^2). We can postulate that if the Fourier transform of the
classical f+(s^2) is the integral over all k of F(k^2)e^-ikx12 d^4k, then
the Fourier transform of f+(s^2) is the same integral taken only over
positive frequencies k4 for t2 > t1 and only over negative ones for t2 < t2
in analogy to the relation of D+(s^2) to D(s^2). The function f(s^2) can be
written as s^2 = x.x.

f(x.x) = (2pi)^-2 S(k4=0 to infinity)S(angular)[sin(k4|x4|)cos(K.X)

         dk4d^3K g(k.k)]

note that caps K and X are 3-vector parts of 4-vectorsk and x and

g(k.k) is k4^-1 times the density of oscillators and may be expressed for
positive k4 as

g(k^2) = S(0 to infinity)[(D(k^2) - D(k^2-@^2)G(@)d@]

where S(0 to infinity)G(@)d@ = 1 and G involves @ >> m"

For gravity cut-off 10^-5 gm compared to 10^-27 gm.

"This simply means that the amplitude for propagation of quanta of momentum
k is

-F+(k^2) = pi^-1S(0toinfinity)[(k^-2 - k^2-@^2)^-1)G(@)d@

rather than k^-2. That is, writing F+ = -pi^-1k^-2C(k^2)

-f+(s12^2) = pi^-1S[e^-ikx12k^-2C(k^2)d^4k  (16)

Every integral over an intermediate quantum which previously involved a
factor d^4k/k^2 is now supplied with a convergence factor where

C(k^2) = S(0toinfinity)[-@^2(K^2-@^2)^-1 G(@)d@]  (17)

The poles are defined by replacing k^2 by K^2 +id in the limit d->0. That
is @^2 may be assumed to have a infinitesimal negative imaginary part."

*Note, that although Feynman's models are visualizable there is still a lot
of mathematical detail that cannot be so easily visualized here-
Schwinger's method, for comparison, is much harder than Feynman's.

"The function f+(s12^2) may still have a discontinuity in value on the
light cone.... The condition that f is have no discontinuity ... on the
light cone implies k^2C(k^2) ->0 as k-> infinity or, equivalently,

S(0 to infinity)[@^2G(@)d@ = 0  (18)

The expression for the self-energy matrix is now (i.e.,to lowest order in
perturbation theory JS)

(e^2/ipi)S[G(p/-k/-m)^-1Gk^-2C(k^2)d^4k]    (19)

which ... converges.  For practical purposes we shall suppose hereafter
that C(k^2) is simply -@^2/(k2-@^2) implying that some average (with weight
G(@)d@) over values of @ may be taken afterwards. ....

Since in all processes the (virtual) quantum momentum (k) will be contained
in at least one extra factor of the form (p-k-m)^-1 representing
propagation of an electron while that (vacuum fluctuation) quantum is in
the field, we can expect all such integrals with their convergence factors
to converge... all such processes will now be finite and definite
(excepting the processes with closed loops.. in which the diverging
integrals are over the momenta of the electrons rather than the quanta.

The integral of (19) with C(k^2)= -@^2/(k2-@^2) noting that p^2 = m^2 and
@>>M to lowest order in m/@ is

(e^2/2pi)[4m(ln(@/m)+1/20 - p(ln(@/m)+5/4)]  (20)

deltam = m(e^2/2pi)(3ln(@/m)+3/4)    (21)"

Sarfatti,note for the Planck quantum gravity cut-off @/m is 10^-5/10^-27 =
10^22 so the order of magnitude of the dimensionless factor in (21) is
(1/137)3ln(1022) of order unity roughly - that is the gravitational cut off
suggests that the entire mass of the electron is a self-energy effect!
These simple QED - gravity considerations extended to Yang-Mills theory
with spontaneous broken symmetry of vacuum tell us something about Higgs
mass and inflation - probably.

Feynman does not discuss connection of QED to gravity in his paper but goes
on the discuss radiative corrections to scattering in his section 6 ,
vacuum polarization in his 7 longitudinal photon waves in 8 - the latter
may have some connection to Bearden type stuff- though I do not understand
anything Bearden says - but I have not tried hard to understand him. We
should not place the papers of Puthoff and Bearden in the same category.
Puthoff's papers are easy to follow and quite conventional realtive to
Bearden's. Puthoff's papers get published in Physical Review.

to be continued.


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