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Jul 18, 2008, 7:07:15 AM7/18/08

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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.

''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.

Jul 24, 2008, 3:25:14 PM7/24/08

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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.

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.

Aug 1, 2008, 10:21:10 AM8/1/08

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MargH schrieb:

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

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