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Feynman 35 "Theory of Positrons 1" From the future revisted!

*All of this may be important in the physics of our consciousness because
of virtual electron-positron "Lamb shift" frequency shifts of electron
transitions in atoms and molecules.  These shifts are in the megacycle
region in hydrogen and are probably smaller in bigger atoms and molecules.
They could affect neurotransmitter release and capture in synaptic clefts!

1. Let's take quick (not quite superficial) look at Feynman's classic
papers on QED.  This is mainly meant for non-experts, though pundits may
learn a thing or two.

"The main principle is to deal directly with the solutions to the
Hamiltonian differential equations rather than with these equations
themselves ...The same equation, Dirac's, which describes the deflection of
the world line of an electron in a field, can also describe the deflection
... when it is large enough to reverse the time-sense of the world line,
and thereby correspond to pair annihilation .... the direction of the world
line is replaced by the direction of propagation of the waves.

This view is quite different from that of the Hamiltonian method which
considers the future as developing continuously from out of the past.  Here
we imagine the entire space-time history laid out, and that we just become
aware of increasing portions of it successively. ...  The temporal order of
events during the scattering, which is analyzed in such detail by the
Hamiltonian differential equation is irrelevant. ....

... in non-relativistic quantum mechanics the amplitude for a given process
can be considered as a sum of an amplitude for each spacetime path
available .... remove, in the relativistic case, the (non-relativistic)
restriction that the paths must proceed always in one direction in time ...

2. Green's function (non-relativistic semi-classical limit in which
electromagnetic field is classical but the electron is quantum mechanical)

The Schrodinger equation

idpsi/dt = Hpsi  (1)

describes change in the wave function psi in an infinitesimal time dt as
due to the operation of an operator e^-iHdt .... if psi(x1,t1) is the wave
function at x1 at time t1, what is the wave function t time t2 > t1?

psi(x2,t2) = S[K(x2,t2;x1,t1)psi(x1,t1)d^3x1]    (2)

"S" = Integral or Sum depending on context, K is a Green's function

... if H (i.e., the Hamiltonian) is a constant operator having eigenvalues
En, eigenfunctions phin, so that psi(x,t) can be expanded as
S(n)[Cnphin(x)], then psi(x,t2) = S(n)[e^-iEn(t2 - t1)Cnphin(x1).  Since
Cn = S[phin(x1)*psi(x1,t1)d^3x1], one finds (where we write 1 for x1,t1 and
2 for x2,t2)

K(2,1) = S(n)[phin(x2)phin(x1)*e^-iEn(t2 - t1)]    (3)

fot t2 > t1.  We shall find it convenient for t2 < t1 to define K(2,1) = 0
(Eq. (2) is then not valid for t2 < t1) .... K can be defined by that
solution of

(id/dt2 - H2)K(2,1) = iDirac delta (2,1)     (4)

which is zero for t2 < t1....  H2 means that the operator acts on variables
2 of K(2,1).  When H is not constant ,,, K is less easy to evaluate than
(3).

Feynman  then gives a physical picture for the Green's function:

We can call K(2,1) the total amplitude for arrival at x2,t2 starting from
x1,t1. (It results from adding an amplitude e^iS, for each space time path
between these points, where S is the action along the path.)

I add that K(1,2) is the time reverse of K(2,1).

*Note S = 2pi Classical Action/Planck's constant

Example 1 for non-relativistic free particle phin = e^ipx, En = p^2/2m

Ko(2,1) = S{e^-i[p(x1 - x2) - p^2(t2 - t1)/2m]d^3p (2pi)^-3}

= (2pi im^-1(t2 - t1))^-1/2 e^im(x2-x1)^2(t2 - t1)^-1/2

The transition amplitude for finding a particle in state chi(t2) at time
t2, if at t1 it was in state psi(t1), is

S[chi*(2)K(2,1)psi(1)d^3x1 d^3x2]     (5)

I add note that the integral is over both space-like slices, so that the
retarded (forward in time) transition amplitude is not a local function of
space.  The advanced transition amplitude backward in time (t2 > t1) is

S[psi*(1)K(1,2)chi(2)d^3x1 d^3x2]     (5')

this is not in the original Feynman paper -but I add it because of John
Cramer's "transactional interpretation" from the old Wheeler-Feynman
classical action-at-a-distance electrodynamics.

Feynman then develops perturbation theory as multiple scattering of a
particle in a weak classical potential U(x,t) that differs from zero only
for t between t1 and t2

K(2,1) = K0(2,1) + K1(2,1) + K2(2,1) + K3(2,1) + ...   (6)

and he derives

K1(2,1) = -iS[K0(2,3)U(3)K0(3,1)d3]                (9)

note this is an integral over all space and time (i.e., d3 = dx3dt3)
although in this non-relativistic limit it is only in the slab between
space-like slices from t1 to t2.  Similarly,

K2(2,1) = (-i)^2S^2[K0(2,4)U(4)K0(4,3)U(3)K0(3,1)d3d4]    (10)

where S^2 means double integral etc.

3. Feynman analyses the Dirac equation. K is now a 4x4 matrix operating on
a 4-component wave function.  phin* is replaced by the "adjoint" (i.e., it
is multiplied by beta matrix from right).  However, using the same boundary
conditions as in the non-relativistic case is the "one electron" theory in
which the electron scattering off the classical potential U can only
scatter forward in time with both positive and negative energies. Feynman
defines energy as "rate of change of phase". "On the other hand, according
to the positron theory negative energy states are not available to the
electron after the scattering    But there are other solutions .... We
shall choose the solution defining K+ so that K+(2,1) for t2 > t1 is the
sum of (3) over positive energy states only. Feynman then proves that since
K0 - K+ is the sum of (3) over all negative energy states for all times,
and since K0(2,1) = 0 for t2 < t1, therefore K+(2,1) for t2 < t1 is the
negative of the sum (3) over negative energy states.  Feynman interprets
the negative sign in terms of the Pauli exclusion principle.  "The fact
that the entire sum is taken as negative in computing K+(4,3) (e.g., see eq
(10)) is reflected in the fact that in hole theory the amplitude has its
sign reversed in accordance with the Pauli principle and the fact that the
electron arriving at 2 has been exchanged with one in the sea.

                          / 2
                        /
                      /
        3           /             here is a Feynman diagram for eq.(10)
        /\        /
      /    \    /
    /        \/ 4
   /     virtual pair if t4 < t3
1/


To this, and to higher orders, all processes involving virtual pairs are
correctly described in this way.

"The scatterings may, however, be toward both future and past times, an
electron propagating backwards in time being recognized as a positron.
This therefore suggests that negative energy components created by
scattering in a potential be considered as waves propagating from the
scattering point toward the past, and that such waves represent the
propagation of a positron annihilating the electron in the potential. ...
With this interpretation real pair production is also described correctly..
... All these amplitudes are relative to the amplitude that a vacuum will
remain a vacuum, which is taken as unity."

Feynman then discusses the relativistic generalization of eq (2) (i.e., his
eq. (19)) for the wave function which involves "the closed 3-dimensional
surfacr of a region of space time containing point 2   the wave function
psi(2) ... is determined at any point inside a four-dimensional region if
its values on the surface of that region are specified... That is, the
amplitude for finding a charge at 2 is determined both by the amplitude for
finding an electron previous to the measurement and by the amplitude for
finding a positron after the measurement.  This might be interpreted as
meaning that even in a problem involving but one charge the amplitude for
finding the charge at 2 is not determined when the only thing known is the
amplitude for finding an electron (or a positron) at an earlier time.
There may have been no electron present initially but a pair was created in
the measurement (or also by other external fields). The amplitude for this
contingency is specified by the amplitude for finding a positron in the
future."

*All of this may be important in the physics of our consciousness because
of virtual electron-positron "Lamb shift" frequency shifts of electron
transitions in atoms and molecules.  These shifts are in the megacycle
region in hydrogen and are probably smaller in bigger atoms and molecules.
They could affect neurotransmitter release and capture in synaptic clefts!

to be continued