The following takes up the December/January discussion from the thread
''EM field of photon''. Based on the details given in the entry
S2g. Particle positions and the position operator
of my theoretical physics FAQ at
http://www.mat.univie.ac.at/~neum/physics-faq.txt
(cf. also
S2f. What is a photon?
S2h. Localization and position operators),
I had concluded on January 21 that Hawton's approach does not provide
a probability density with the physically necessary invariance
properties.

In the mean time, Margaret Hawton wrote a paper in
http://lanl.arxiv.org/pdf/0804.3773v2
supposedly constructing a Lorentz invariant photon number density,
as I had challenged her in that discussion.

Unfortunately, the construction given in this paper is inconsistent,
thus reinforcing my earlier conclusions.

More specifically, Hawton constructs, starting from the free photon
Fock space, in equation (6) the standard momentum space wave functions
(or probability amplitudes) c_sigma(k), where sigma=+-1 and k is the
3D wave vector. Transversality implies that any such wave function
satisfies
k dot c_sigma(k) = 0 for all k. (*)
The invariant inner product of two such wave functions c and d is
d|c = sum_sigma integral dk/|k| d_sigma(k)^* c_sigma(k),
in agreement with standard practice.

Later, Hawton constructs in (18) supposedly physical states localized
at position r at fixed time t, with circular polarization sigma.
However, these states do not in general respect the transversality
relation (*), hence are usually not in the physical Hilbert space
of transversal states. Thus most of these states have no physical
meaning.

The mistake made seems to be that k and a phase chi that initially
(in the discussion following (1)) depends on k are later - starting
in (15), where a chi dependent position operator is defined -
taken to be independent. Already after (4), it is stated that ''the
chi=0 definite helicity basis will be used'' - but there is no such
basis respecting the condition chi=chi(k) required by transversality.

This invalidates her main conclusions. It also invalidates the
consistency check after (19), where it is supposedly shown that
the probability amplitudes corresponding to different pairs
(r,sigma) and (r',sigma') of position and helicity are orthogonal
at equal times. But this would follow only if chi were a parameter
independent of k.

I disagree with Neumaier’s conclusion that my Lorentz invariant
construction is inconsistent. Chi is never taken to be independent.
All the states considered are transverse, and (18) just expresses a
rotated transverse state in terms of the chi=0 basis.

MargH schrieb:
I disagree with Neumaier’s conclusion that my Lorentz invariant
construction is inconsistent. Chi is never taken to be independent.
All the states considered are transverse, and (18) just expresses a
rotated transverse state in terms of the chi=0 basis.

That you disagree doesn't change my assessment of the situation.

In any transversal state, the divergence of c_sigma(r,t)
must vanish. But the delta-function for c_sigma(r,t) which you
get in the second paragraph after (19) [which you derive from (18)]
is manifestly non-transverse.

So, contrary to what you say above, not all the states considered
are transverse.

Your sloppiness in ignoring the k-dependence of chi allows you
to ''derive'' the non-transverse delta state as integral over
transverse states, which is clearly impossible.

In the paper, you call this a consistency check.
Others call this a proof of faulty reasoning.


Arnold Neumaier