Re: On the wave function of the photon

Arnold Neumaier wrote:
> Eugene Stefanovich wrote:
>
>>
>> Ben Rudiak-Gould wrote:
>>
>>> 1. Armed with wave equations for photons and electrons, one could try
>>> to construct a "fieldless QED", using a wave function on Fock space
>>> with some sort of coupling between electron and electron+photon
>>> states. Could one in principle reproduce the predictions of QED in a
>>> theory like this? My impression is that the answer is no, because I
>>> don't see how to accommodate virtual particles in Fock space. Which
>>> leads to question
>>>
>>> 1a. How is Fock space actually related to QFT, if at all? It's
>>> described in textbooks as a way to make a relativistic quantum theory
>>> that allows for the creation and annihilation of particles, but all
>>> the successful relativistic quantum theories are field theories,
>>> which don't have particles in the first place.
>>
>>
>> This is not true. Quantum field theories are actually about particles
>> and their interactions. Most QFT textbooks successfully hide this fact
>> by placing too much emphasis on fields.
>
>
> _All_ QFT textbooks (except your misleading draft book which should
> never be published)display the fact that your assertion that ''quantum
> field theories are actually about particles and their interactions''
> gives a very distorted view of QFT.
>
> Once one has a representation of the Poincare group (and no one
> is disputing that this is necessary) one a momentum operator p
> and from this many quantum fields
> f(x) = exp(-ix dot p) f exp(i x dot p),
> an the expectations and correlations of some of these are observable.
> These are used in so many real applications of quantum mechanics
> that it is a sign of utter ignorance to make the claim that
> quantum field theory is not about quantum fields.

What physically measurable quantity correspond to the Dirac's
electron-positron field

psi(x) = \inp dp \omega(p)^{-1}
[exp(-ipx) u(p) a^*(p) + exp(ipx) v(p) b(p)] ? (1)

I think that there is no such a quantity. The field psi(x) is
just a linear combination of electron creation a^*(p)
and positron annihilation b(p) operators. One can form
Poincare invariant interaction operators from such fields.
Once all 10 Poincare generators H_0+V, P, J, K_0+Z are
constructed as functions of creation and annihilation
operators, fields are not needed anymore. They can be
thrown away, and any desired physical result can be
calculated from H_0+V, P, J, K_0+Z.

When you calculate scattering amplitudes, like

<0| aa S a^*a^*|0>

they refer to certain configurations (e.g., a^*a^*|0>) of particles,
not fields. When experimentalists measure scattering cross-sections
they prepare and detect states of few particles with well-defined
momenta, polarizations, etc.

Observables are represented by Hermitian operators in the Fock space.
All such operators can be expressed as functions of creation and
annihilation operators. Creation and annihilation operators can be
written through fields (by using formulas inverse to (1)).
So, if one likes one can express observable quantities through fields,
but this is no more than mathematical manipulation, and doesn't
justify the claims that fields are "fundamental" quantities.

Eugene Stefanovich.

.

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