Re: EM field of photon

CarlB schrieb:
Arnold, thank you for your detailed examination of this. Please
forgive my likely poor understanding and typing errors, and thanks
in advance for continuing to look at this.

On Jan 3, 9:36 pm, Arnold Neumaier <Arnold.Neuma...@xxxxxxxxxxxx>
wrote:
..
This is strange terminology which she uses repeatedly. Her position
operator is a 3x3 matrix, not a vector as one would reasonably demand?
On closer inspection it turns out that it is a 3-vector with 3x3 matrix
components.

An analogy to this would be the 3-vector of 2x2 matrices known as the
Pauli spin matrices. One doesn't normally think of Pauli spin matrices
as a position operator, but I'll get back to that later in the post.

Try to compute the probability of the photon being in some nonspherical
region, or the expectation of some spherically nonsymmetric function
of position. I predict that you'll find that these depend on the
gauge and hence are unphysical.

If one can calculate probabilities consistently in spherical regions,
then it seems to me that real analysis will show that all compact
regions follow.

Only if the centers are arbitrary.
So calculate the probabilities for spherical regions with arbitrary centers, and I predict you'll get problems.


She discusses the effect of the phase too superficially (on p.28)
to see what happens. You seem to have read the paper more carefully;
I'd appreciate if you could provide the missing details to check
what is going on.

I hope that what I will be showing applies to what they did. However,
life is short, and instead of carefully reviewing their paper to see
if what they meant is what I'm guessing, I'm just going to quickly
type the calculations to show that matrix wave functions are natural.
This turns out to be a subject that is near and dear to my heart
and one that I can type up very quickly. I'm guessing that it applies
to what they did, but I dread looking through the several dozen pages
to see if this is the case.

Instead, the method gives a position wave function that uses
a 3x3 matrix. If you rotate coordinates, you end up with a
geometric phase, sometimes called Pancharatnam phase,
or Berry phase or Berry-Pancharatnam phase.
Is this phase a number or also a 3x3 matrix? If it is the latter,
which is most likely (I doidn't wade through all details of the
42 page paper) the gauge transform will change probabilities.

Geometric phase is a psuedoscalar, at least in the Pauli algebra.
Geometric phase shows up as exp( i k \sigma_x\sigma_y\sigma_z),
where \sigma_n are the Pauli spin matrices. So you can call it
a "number" or a 2x2 matrix, depending on what you wish to call
the unit matrix multiplied by a complex number.

I have nothing against matrix-valued components, but she should say
so, and not call a vector of matrices a matrix.


I don't play
much with spin-1,

Spin 1/2 or spin 1 makes the difference between existence and
nonexistence of a position operator. Thus simple extrapolation
from spin 1/2 (which is the basis for the remainder of your mail)
is inadequate.


but I know that photons carry geometric
phase identical in formula to that of electrons and so I suspect
it is similar. From a Clifford algebra point of view, these are
all just multivectors.


Arnold Neumaier

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