Re: Hawton Position Operator. (Was: EM field of photon)

sr schrieb:
This thread seems to have gone quiet, but without a satisfying
resolution. I'd like to request further discussion about MargH's
position operator, and Arnold Neumaier's objections.

Summarizing....

Arnold Neumaier wrote:

[...objections to MargH's position operator...]

[...] the position vector must satisfy the standard commutation rules (3) with
the angular momentum, which is violated, as she says explicitly on p.5. [...]

MargH said:

I don't agree that the standard commutation relations must be obeyed.

I don't understand your reasons behind this opinion. It seems natural to
me that the position vector should satisfy the standard commutation
rules with the angular momentum, so I was surprised that you think
differently. Could you elaborate on your reasons, please?

There is a very nice blog that discusses our position operator at
http://carlbrannen.wordpress.com/2008/01/14/consistent-histories-and-...

??? This seems to be about consistent histories, not about photon
positions...

There is discussion about the Hawton position operator further down on
that webpage.

Ah, I had given up up about halfway...

The statement there about ''Newton's and Wigner's insistence on a complete rotational manifold'' is misguided. Actuall, Newton and Wigner only treated the massive case, where one indeed has the full set of
spin states. The massless case is treated in passing at the end of
their paper, where they mention that the conclusion then are different
for spin > 1/2 (and, reading between the lines, just _because_ of
the transversality of the photon.)


(Note that the rather puzzling statement therein about
"our probabilities will end up not as real numbers, but instead as real
multiples of matrices" must be read carefully in the context of the
preceding consistent histories stuff to understand what is meant.)

I have no idea which meaning could be assigned to matrix-valued probabilities; neither is it clear to me why Hawton's approach
should yield such probabilities; the blog does not seem to give details.


The trace is discarded without good reasons in the paragraph before
the heading ''Hawton's Photon Position Operator'',

In the formula for Pr(H) a few paragraphs before, the trace
is already missing after each equal sign. But the trace is necessary
(and it is indeed mentioned in the English context).
With the trace, the probabilities discussed there become
ordinary numbers; without the trace, the formulas simply become wrong:
The third equality would not be correct since, as the context states,
it uses the fact that tr AB = tr BA.


Thus if the probabilities for being in some region of space
are indeed matrices rather than numbers then an interpretation
of the resulting probability matrix calculus - that would make the formalism meaningful - is missing.

On the other hand, if the probabilities for being in some region
of space are ordinary numbers (as I think they should be)
then their independence of the Cartesian coordinate system used
is in doubt. If these probabilities change under a rotation
of the coordinate system, they cannot be physical.

I fear the latter is the case...


Arnold

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