From: sarfatti@well.sf.ca.us (Jack Sarfatti)
Subject: Feynman 40 QED part II
Organization: Whole Earth 'Lectronic Link
Date: Mon, 21 Dec 1992 06:31:11 GMT


Feynman 40 QED & renormalization Part II
1. Comparison with the Hamiltonian method.
"Electrodynamics can be looked upon in two equivalent ways. One is as the
description of the behavior of a field (Maxwell's equations). The other is
as a description of a direct interaction at a distance (albeit delayed in
time) between charges (the solutions of Lienard and Wiechert). From the
latter point of view light is considered as an interaction of the charges
in the source with those in the absorber. This is an impractical point
ofview because many kinds of sources produce the same kind of effects. The
field point of view separates these aspects into two simpler problems,
production of light, and absorption of light. On the other hand, the field
point of view is less practical when dealing with close collisions of
particles (or their action on themselves) ...

Roughly, the field point of view is most practical for problems involving
real quanta, while the interaction view is best for the discussion of the
virtual quanta involved... the important aspect in problems with which we
shall deal is the effect of virtual quanta.

The Hamiltonian method is not well adapted to represent the direct action
at a distance between charges becuse that action is delayed. The
Hamiltonian method represents the future as developing out of the present.
If the values of a complete set of quantities are known now, their values
can be computed at the next instant in time. If particles interact through
a delayed interaction, however, one cannot predict the future by simply
knowing the present motion of the particles.  One would also have to know
what the motions of the particles were in the past in view of the
interaction this may have on future motions. This is done in the
Hamiltonian electrodynamics, of course, by requiring that one specify
beside the present motion of the particles, the values of a host of new
variables (the coordinates of the field oscillators) to keep track of that
aspect of the past motions of the particles which determine their future
behavior. The use of the Hamiltonian forces one to choose the field
viewpoint rather than the interaction viewpoint.

....

We shall be discussing the solutions of equations rather than the time
differential equations from which they come. We shall discover that the
solutions, because of the over-all space-time view that they permit, are as
easy to understand when interactions are delayed as when they are
instantaneous. ... for different observers in relative motion the
instantaneous present is different, and corresponds to a different 3-
dimensional cut of space-time. Thus, the temporal analyses of different
observers is different and their Hamiltonian equations are developing the
process in different ways. These differences are irrelevant, however, for
the solution is the same in any space-time frame. By forsaking the
Hamiltonian method, the wedding of relativity and quantum mechanics can be
accomplished most naturally.

2. The interaction between charges.
...We start by considering the non-relativistic case described by the
Schrodinger equation ... The wave function at a given time t is a function
psi(xa,xb,t) of the coordinates xa and xb of each particle. Thus call
K(xa,xb,t;xa',xb',t') the amplitude that particle a at xa' at time t' will
get to xa at t while particle b at xb' at t' gets to xb at t."

-------xa------------------------xb---t






---xa'---------------xb'--------------t'

"If the particles are free and do not interact this is

K(xa,xb,t;xa',xb',t') = K0a(xa,t;xa',t')K0b(xbt;xb,t')

... we can obviously define a quantity like K, but for which the time t
need not be the same for particles a and b (likewise for t'); e.g.

K0(3,4;1,2) = K0a(3,1)K0b(4,2)           (1)

can be thought of as the amplitude that particle a goes from x1 at t1 to x3
at t3 and that particle b goes from x2 at t2 to x4 at t4.

When the particles do interact, one can only define the quantity K(3,4;1,2)
precisely if the interaction vanishes between t1 and t2 and also between t3
and t4. In a real physical system such is not the case. There is such an
enormous advantage, however, to the concept that we shall continue to use
it   this means choosing such long time intervals t3 - t1 and t4 - t2 that
the extra interactions near the end points have small relative effects. ...
imagine first that the interaction is simply that given by a Coulomb
potential e^2/r where r is the distance between the particles. If this is
turned on only for a very short time dto at time to, the first order
correction to K(3,3;1,2) ..

K1(3,4;1,2) = -ie^2S^2[K0a(3,5)K0b(4,6)r56^-1K0a(5,1)K0b(6,2)d^3x5d^3x6dto]

where t5 = t6 = to...  .. if we include a delta-function D(t5-t6), t5-t6 =
t56

K1(3,4;1,2) = -ie^2S^2[K0a(3,5)K0b(4,6)(r56^-1)D(t56)K0a(5,1)K0b(6,2)d5d6]
(2)
where d5 = d^3x5dt5 etc. ... We know, however, in classical
electrodynamics, that the Coulomb potential does not act instantaneously,
but is delayed by a time r56, taking the speed of light as unity. This
suggests simply replacing (r56^-1)D(t56) in (2) by something like
(r56^-1)D(t56-r56) to represent the delay in the effect of b on a.

This turns out to be not quite right, for when this interaction is
represented by photons they must be of only positive energy, while the
Fourier transform of D(t56-r56) contains frequencies of both signs. It
should instead be replaced by D+(t56-r56) (remember D here stands for Dirac
delta function) where (t5 > t6)

D+(x) = S(0 to infinity)[e^-ifxdf/pi]

= Lim (z->0)[(ipi)^-1/(x - iz)

= D(x) + (ipix)^-1    (3)

This is to be averaged with (r56)^-1D+(-t56-r56) which arises when t5 < t6
and corresponds to a emitting the quantum which b receives. Since

(2r)^-1[D+(t-r) + D+(-t-r)] = D+(t^2 - r^2)

this means (r56^-1)D(t56) is replaced by D+(s56^2) where

s56^2 = t56^2 - r56^2

is the square of the relativistically invariant interval between points 5
and 6."

Feynman's footnote 7 is important because it says of (r56^-1)D(t56-r56)
"It, and a like term for the effect of a on b, leads to a theory which, in
the classical limit, exhibits interaction through half-advanced and half-
retarded potentials. Classically, this is equivalent to purely retarded
effects within a closed box from which no light escapes... Analogous
theorems exist in quantum mechanics.  Note that D+(t-r) is retarded
propagating FIT and D+(-t-r) is advanced propagating BIT in the same
direction of space; i.e., in space-time diagram

        /
      /
    /  D+(t-r)
  /
/_______________________ space
\
  \
    \  D+(-t-r)
      \
        \

Therefore, relativistic invariance in Feynman's quantum electrodynamics
still requires the classical Wheeler-Feynman feature that signals propagate
both forwards and backwards in time.  In classical general relativity
solutions of the expanding universe, the Wheeler-Feynman absorber boundary
condition "equivalent to purely retarded effects within a closed box".

"Since in classical electrodynamics there is also an interaction through
the vector potential, the complete interaction ... should be
(1-v5.v6)D+(s56^2), or in the relativistic case

(1-A5.A6)D+(s56^2) = BaBbGauGbuD+(s56^2)"

(where A means Dirac eq alpha matrix, B means Dirac eq beta matrix, G means
Dirac eq gamma matrix, u is space-time index 1 to 4.)

"Hence we have for electrons obeying the Dirac equation,

K1(3,4;1,2)=-ie^2S^2[K+a(3,5)K+b(4,6)GauGbuD+(s56^2)K+a(5,1)K+b(6,2)d5d6]
(4)

... the factor BaBb being absorbed in the definition... of K+

This is our fundamental equation for electrodynamics. It describes the
effect of exchange of one quantum (therefore first order in e^2) between
two electrons.... Relativistic invariance is clear..."

3                                  4
  \                              /
    \                          /
      \ K+a(3,5)             / K+b(4,6)
        \                  /
         \               /
       5  ---------------  6
        /    D+(s56^2)    \
      /                     \
    / K+a(5,1)                \ K+b(6,2)
  /                             \
/ 1                               \2


This is the Feynman space-time diagram corresponding to eq (4).
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