Re: On the wave function of the photon

In article <d70pci$jbp$1@xxxxxxxxxxxxxxxxxxxx>, Ben Rudiak-Gould
<br276deleteme@xxxxxxxxx> wrote:

> I just found an interesting paper which presents a wave equation for a free
> photon, analogous to Dirac's equation for a free electron.[1] It motivates
> the equation by an argument virtually identical to Dirac's, but with a
> different choice of coefficient matrices. The resulting equation is (writing
> d_x for the partial with respect to x):
>
> i hbar d_t F = c (hbar / i) (S_x d_x + S_y d_y + S_z d_z) F,
>
> with the side condition that div F = 0, and the matrices
>
> / 0 0 0 \ / 0 0 i \ / 0 -i 0 \
> S_x = | 0 0 -i | S_y = | 0 0 0 | S_z = | i 0 0 |
> \ 0 i 0 / \ -i 0 0 / \ 0 0 0 /
>
> This is just the source-free Maxwell equations, with F = E+iB! This is a
> very pretty result, and I share the author's surprise that this equation
> isn't found alongside Dirac's in textbooks.

There are a number of sporadic SR field equations. These are essentially
one or two-particle equations, whose solutions, from the physical point of
view, are problematic. For the Dirac equation, see for example the book by
Itzykson-Zuber. In SR, the covariance principle forces that spin (p, q)
and (q, p) appear in equal amounts. There is a proof in one of the Gelfand
"Generalized Function" volumes.

One way to construct such equations is as follows: On a Lorentz manifold,
electromagnetism is represented by a two form F, which can be integrated
locally to an electromagnetic potential A, a 1-form. The algebra of
differential forms is isomorphic to the Clifford algebra on the Lorentz
metric, so these forms EM can be mapped into it. Then the groups SU2
groups one is interested in are subgroups of this Clifford algebra. So
just take some such representations, and one gets various sporadic field
equations. The Clifford multiplication is, one the differential algebra,
just the sum of exterior and interior multiplication so when expanded into
coordinates, one gets these nice looking equations

> In any case, reading the paper inspired a few questions which I hope someone
> on this group can answer.
>
> 1. Armed with wave equations for photons and electrons, one could try to
> construct a "fieldless QED", using a wave function on Fock space with some
> sort of coupling between electron and electron+photon states. Could one in
> principle reproduce the predictions of QED in a theory like this? My
> impression is that the answer is no, because I don't see how to accommodate
> virtual particles in Fock space. Which leads to question

This would be an attempt to construct a generalized Schroedinger equation
to SR or GR, which might be called the classical approach to the
unification problem. Nobody has had success with that. Hilbert space and
Feynman path integral methods gives some means of practically carry out
computations which agree with observation, so unification attempts have
often focused on such approaches.

> 1a. How is Fock space actually related to QFT, if at all? It's described in
> textbooks as a way to make a relativistic quantum theory that allows for the
> creation and annihilation of particles, but all the successful relativistic
> quantum theories are field theories, which don't have particles in the first
> place.

One can cook up Fock spaces for QFT; see for example the book by Streater
& Wightman, "PCT, Spin & Statistics and all that". When combined with the
covariance principle, the authors are able to prove the PCT theorem.

> 2. Can one derive similar wave equations for generic massive/massless
> particles of arbitrary half-integral spin? For the other particles in the
> standard model?

As indicated above, the covariance principle forces spin (p, q) and (q, p)
to exist in equal amounts; coupling these spins leads to difficulties in
physical interpretation. The covariance principle is incompatible with GR,
as it (apparently) imposes conditions on the Levi-Civita connection which
communicates gravity, but it is not clear what it should be replaced with.
PCT breaking has been observed in nature. The method only works for SU2
spin (because of its relation to the Clifford algebra, which only contains
SU2); for SU3 I recall a suggestion for a two particle coupling.

--
Hans Aberg

.