Re: Mossbauer effect and retarded interactions

Eugene Stefanovich wrote:
> Igor Khavkine wrote:
>
> >> Here is the logical chain:
> >>1) In the classical limit, QFT must yield classical mechanics
> >>with trajectories.
> >>2) STR demands that classical trajectories transform by Lorentz
> >>formulas.
> >>3) CJS theorem says that in this case there can be no interactions.
> >>4) This disagrees with observations.
> >>
> >>Therefore, there must be an error somewhere in this logical chain.
> >>Do you agree? I think that the error is in point 2). I checked all
> >>other points and couldn't find an error there.
> >
> > In that case I suggest you check again. There is indeed an error in this
> > chain, but it is in point 3). And point 1) should be expanded to be
> > really correct.
> >
> > QFT in the classical limit reduces to the relativistic mechanics of
> > massive particles interacting with fields. Both the positions of the
> > particles and the field amplitudes, and the corresponding momenta, are
> > dynamical variables. In Hamiltonian form, this classical theory is
> > formulated on an infinite dimensional phase space, and it is Poincare
> > invariant. Lets call this limit (A).
> > From here, a preferred frame may be
> > chosen and the non-relativistic limit may be taken. In this limit, the
> > fields can be eliminated in the favor of interaction potentials between
> > the particles. Thus only the particle positions and corresponding
> > momenta are the dynamical variables. In Hamiltonian form, this classical
> > theory is formulated on a finite dimensional phase space, and it is
> > Galilei invariant. Lets call this limit (B).
>
> So far so good. Just a couple remarks.
>
> 1. I presume that a good example of your limit (A) is QED -> Maxwell
> theory. I haven't seen this limit derived rigorously. Do you have
> references? Can Maxwell theory be formulated in the Hamiltonian form?
> What is the interacting Hamiltonian in this case? What are expressions
> of other generators of the Poincare group? That's not the way Maxwell
> theory is normally formulated. I am very interested if this really
> can be done?

Derivation of the limit: Any introductory book on quantum field theory.
For the canonical formalism, lookup the WKB approximation and how it is
related to classical motions. For the path integral formalism, look up
the saddle point approximation and how it is related to the classical
action. This classical limit of QED has been known since the time of
Dirac and Pauli.

Maxwell theory in Hamiltonian form: Goldstein, Jackson, Landau &
Lifshitz (vol 2), some books on QFT. I think Weinberg deals with the
Hamiltonian formulation of Maxwell equations in the chapter on the
canonical formalism.

Poincare generators: The canonical variables are the particle positions
and momenta, as well as the electric field and the 3D vector potential.
It is known how they transform under Poincare transformations. Explicit
form for the transformation generators: exercise.

> 2. The limit (B) cannot be Galilei invariant. When taking the classical
> limit we take h -> 0, we do not take c -> infinity. Therefore the
> symmetry group of inertial observers remains the Poincare group, and the
> theory of particles is relativistically invariant. The rigorous way to
> obtain such a limit is presented in my book. Let us call this limit (C).

The limit (B) *is* Galilei invariant. I explicitly stated:

> > From here, a preferred frame may be chosen and the non-relativistic
> > limit may be taken. In this limit, [...]

You were not clear which limit you meant originally, so I accounted for
both possibilities that came to mind. I'm still not sure what you mean
by the limit (C). If you mean a Poincare invariant Hamiltonian theory
particles interacting through potentials, formulated on a finite
dimensional phase space (only the particle positions and momenta), then
you are wrong. This limit cannot be obtained from QFT. The correct
classical limit (hbar -> 0, only) is (A), where particles interact with
dynamical fields. All, formulated on Minkowski space-time. The CJS
theorem prevents what you call "limit (C)" from existing at all. Fields
can only be replaced by potentials in the non-relativistic limit (c ->
infinity).

[...]

The rest of the deleted speculations rest on the erroneous presumption
of the existence of "limit (C)".

> I agree with your argument that there is a (slight) chance of a
> loophole. My proof of STR inconsistency will be invalidated if
>
> 1. a relativistic Hamiltonian classical theory of particles + fields can
> be obtained as a classical (h -> 0) limit of QFT, and

It is. QED -> Maxwell + electrons, as hbar -> 0. Has been known for 70+
years.

> 2. some (unlikely) cancellations make boost transformation laws
> of particle
> observables interaction-independent, even though the generators
> of boosts are interaction-dependent.

Solutions to the equations of Maxwell + matter are trajectories
embedded in Minkowski space-time together with field amplitudes defined
on the same space-time. Poincare transformations are automatically
obeyed when the solutions are expressed in any choice of inertial
coordinate frame.

The chance of a loophole is not slight. It is 100%. Here is a quote
from the CJS paper I previously referenced:

In Sec. IV we show that if one assumes these equations and assumes
that the generators H, P, J, and K satisfy the Poisson bracket
equations characteristic of the Lorentz group, then at least for the
case of two particles (*not* two particles and a field) one can
conclude that both of the particles must have a constant velocity so
that the theory is unable to describe any nontrivial interactions
between the particles.

-- Last paragraph of the introduction, emphasis added.

In other words, not only does the proof of the CJS theorem not apply to
fields (phase space with inifinite number of dimensions), but there is
at least one counterexample: Maxwell + matter.

> These are interesting questions, and I don't know if anybody has
> the answers.

The answers are already there. You need only recognize them.

Igor

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