The Heisenberg Relations for the Photon -- the Symplectic Approach

[Alfred Einstead]

On May 24, 4:52 pm, Alfred Einstead <federation2...@netzero.com>
wrote:
> I decided to give scribd out a try and (if it works out okay) to place
> all my notes [and other legacy material in my archive]...

This is the last in a series of articles (3 for this week).

I uploaded the following to Scribd.

"The Photon Sector"
http://www.scribd.com/doc/1449590952/Heisenberg-Relations-for-the-Photon-Symplectic-Approach.

The analysis covers *both* the relativistic and non-relativistic
versions of the photon (as further elaborated below). In addition are
articles covering the symplectic version of the Wigner classification
-- in a form suitable for both relativistic and non-relativistic
theory.

These other articles may be found
http://www.scribd.com/RockBrentwood
(in particular "A Unified Framework for Symplectic Wigner
Classification")

and will be subcategorized, when the numbers reach high enough levels.

Summary:
In the folklore is a well-known “non-existence” theorem for the photon
position operator. The nature of this “result” is that “non-existence”
is only in the same sense that spherical coordinates do “not exist”
for the sphere. In fact, the Darboux Theorem always guarantees the
local existence of such coordinates; and the existence of an atlas of
charts, each parametrized by such coordinates (and for photons, like
spheres, only 2 charts are needed, though we will not prove this
here.)

These are some notes I wrote up on the symplectic representation class
I’ve termed the “helical luxons”, or “helion” for short. This is the
sector that includes the photons, but not the other luxons – the
luxons with “continuous spin” (a.k.a.”cylindrical luxons”). Its
distinguishing feature is that the helicity (i.e. component of angular
momentum parallel to its momentum) is an invariant. Like the spin 0
case, there are 3 pairs of complimentary coordinates, as there are for
the corresponding Heisenberg relations

The complementarity relations for the photon are developed within a
unified framework that encompasses both classical and quantum theory,
and both non-relativistic and relativistic theory. The analysis is
extended to the nonrelativistic domain, deriving the complementarity
relations for the “helical synchron” – the non-relativistic analogue
of the photon. As has been noted elsewhere (LNP 188, section 5.6) the
symplectic representation of photons and the situation that arises
with its coordinates is, in fact, quite analogous to what happens with
the magnetic monopole.

A more comprehensive analysis that includes the symplectic classes
corresponding to all the inhomogeneous representations, in both the
relativistic and non-relativistic settings, will be added later.

– Mark Hopkins