From: sarfatti@well.sf.ca.us (Jack Sarfatti)
Subject: Feynman 36: more on vacuum polarization
Organization: Whole Earth 'Lectronic Link
Date: Wed, 16 Dec 1992 22:33:10 GMT


Feynman 36 more on positron paper
Feynman then rewrites Dirac electron transition amplitudes in terms of
integrals over all space time rather than integrals over a spacelike slice
at a fixed time.  For example, the transition amplitude f - > g to second
order

-S^2[g*(2)A(2)K+(2,1)A(1)f(1)d1d2]     (23)

where g* is adjoint (i.e. complex conjugate followed by beta matrix). "If
g(2) is a negative energy state we are solving a problem of annihilation of
electron in f(1), positron in g(2), etc."

He then briefly discusses how to do bound states in addition to scattering
states arriving at an integral equation.

psi(2) = phi0(2) - iS[K=(2,1)V(1)psi(1)]    (25)

where phi0(2) is incident free electron state without any scatterings off
classical potential V.

4. Several charges
"... there are two (or more ) distince charges (in addition to pairs they
may produce in virtual state) ... we assume they do not interact ... if we
have two particles a and b, the amplitude that particle a goes from x1 at
t1 to x3 at t3 while particle b goes from x2 at t2 to x4 at t4 is the
product

K(3,4;1,2) = K+a(3,1)K+b(4,2)

"The particles are identical however and satisfy the exclusion principle.
The principle requires only that one calculate K(3,4;1,2) - K(4,3;1,2)"

note
              3    4
              |    |
K(3,4;1,2) =  |    |
              |    |
              1    2
              a    b

             3     4
               \  /
K(4,3;1,2) =    /\
              /    \
              1     2
              a     b

".. it is normalized assuming that when the integral is performed over
points 3 aqnd 4 ... one divides by 2..This expression is correct for
positrons also ...   For example the amplitude that an electron and a
psitron found initially at x1 and x4 (say t1 = t4) are later found at x3
and x2 (with t2 = t3 > t1 is given by the same expression (i.e.,no
intermediate states)

K+a(3,1)K+b(4,2) - K+a(4,1)K+b(3,2)   (27)

The first term represents the amplitude that the electron (a) proceeds from
1 to 3 and the positron (b) from 4 to 2

   3           2
   |           |
   |           |
 a |         b |    =   K+a(3,1)K+b(4,2)
   |           |
   |           |
   1           4

while the second term represents the interfering amplitude that the pair at
1, 4 annihilate and what is found at 3,2 is a pair newly created in the
potential.

  \            /
    \        /
      \    /
        \/
         V        =    K+a(4,1)K+b(3,2)
        /\
      /    \
    /        \
  /            \
/                \


*No account need be taken of the exclusion principle in intermediate
states"
           3              / 2
          /\            /
        /    \        /
      /        \    /
    /            \/
  /               4
/ 1

virtual pair when t4 < t3 (and t2 > t1) production at 4 annihilation at 3

"It may be objected that if it happens that the electron created at 4 is in
the same state as the one coming from 1, then the process cannot occur
because of the exclusion principle and we should not have included it in
our term ... We shall see, however, that considering the exclusion
principle also requires another change which reinstates the quantity.

For we are computing the amplitudes relative to the amplitude that a vacuum
at t1 will still be a vacuum at t2.  we are interested in the alteration in
this amplitude due to the presence of an electron at 1.  Now one process
that can be visualized as occurring in the vacuum is the creation of a pair
at 4 followed by the re-annihilation of the same pair at 3 ( a process
which we shall call a closed loop path).


    2                 3
    |                / \
    |              /     \
    |            /         \    = lowest order  closed loop path
    |            \        /
    |              \    /
    |                \/
    1                 4

  But if a real electron is present in a certain state at 1, those pairs
for which the electron was created in in state 1 (at 4) in the vacuum must
now be excluded.  We must therefore subtract from our relative amplitude
the term corresponding to this process.  But this just reinstates the
quantity which it was argued should not have been included, the necessary
minus sign coming automatically from the definition of K+.  It is obviously
simpler to disregard the exclusion principle completely in the intermediate
states.

All the amplitudes are relative and their squares give the relative
probabilities... Absolute probabilities result if one multiplies each ...
by Pv, the true probability that if one has no particles present initially
there will be none finally.  This quantity Pv can be calculate by
normalizing the relative probabilities such that the sum of the
probabilities of all mutually exclusive alternatives is unity. ...

When we come to deal with interactions between charges ... There is the
possibility that virtual electrons in the vacuum may interact
electromagnetically with the real electrons ...

to be continued.