For a system of point charged particles interacting with one another, the Fokker-Tetrode-Schwartz action is in terms of the particles' relative coordinates from one another without any reference to fields, yet derives Maxwell's equations including the
Lorentz force law. Haag's theorem states that the interaction picture does not exist in an interacting, relativistic, quantum field theory so how does it affect a relativistic classical field theory like classical electrodynamics above?

Thanks in advance for any comments,

JMc

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On 21/09/09 7:16 AM, john mcandrew wrote:

For a system of point charged particles interacting with one another, the Fokker-Tetrode-Schwartz action is in terms of the particles' relative coordinates from one another

The use of relative coordinates isn't really what would count as the "interaction picture" in QM. As the definition states, <https://en.wikipedia.org/wiki/Interaction_picture#Definition>
that would involve splitting the computation into a part that is
"well understood and exactly solvable" and another part that is
(hopefully) easier to deal with than the original problem.

... without any reference to fields, yet derives Maxwell's equations including the Lorentz force law.

If there is no reference to fields then we cannot expect Haag's
theorem to say much about it. (Actually it should be not just
classical fields, but *quantum fields* to apply the theorem, but
it would be reasonable of course to check whether some classical
analogue exists..)

Haag's theorem states that the interaction picture does not exist in an interacting, relativistic, quantum field theory so how does it affect a relativistic classical field theory like classical electrodynamics above?

If - despite the applicability issues mentioned - there would be
something similar to the QM effect, then that would have to be a
"choice problem" in the representation of your system and/or an
ambiguity in commutation relations, as in the description: <https://en.wikipedia.org/wiki/Haag%27s_theorem#Formal_description>

In your (classical) example the problems with commutation relations
might then be visible in the Poisson bracket <https://en.wikipedia.org/wiki/Canonical_commutation_relation#Relation_to_classical_mechanics>
but as explained, there is not much reason to expect any true analogue. (Although with sufficiently far-stretched generalizations one can
probably always construct one, but not convincingly..)

--
Jos

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On Friday, September 10, 2021 at 10:02:03 AM UTC+1, Jos Bergervoet wrote:

On 21/09/09 7:16 AM, john mcandrew wrote:

For a system of point charged particles interacting with one another, the Fokker-Tetrode-Schwartz action is in terms of the particles' relative coordinates from one another

The use of relative coordinates isn't really what would count as the "interaction picture" in QM. As the definition states, <https://en.wikipedia.org/wiki/Interaction_picture#Definition>
that would involve splitting the computation into a part that is
"well understood and exactly solvable" and another part that is
(hopefully) easier to deal with than the original problem.

... without any reference to fields, yet derives Maxwell's equations including the Lorentz force law.

If there is no reference to fields then we cannot expect Haag's
theorem to say much about it. (Actually it should be not just
classical fields, but *quantum fields* to apply the theorem, but
it would be reasonable of course to check whether some classical
analogue exists..)

Haag's theorem states that the interaction picture does not exist in an interacting, relativistic, quantum field theory so how does it affect a relativistic classical field theory like classical electrodynamics above?

If - despite the applicability issues mentioned - there would be
something similar to the QM effect, then that would have to be a
"choice problem" in the representation of your system and/or an
ambiguity in commutation relations, as in the description: <https://en.wikipedia.org/wiki/Haag%27s_theorem#Formal_description>

In your (classical) example the problems with commutation relations
might then be visible in the Poisson bracket <https://en.wikipedia.org/wiki/Canonical_commutation_relation#Relation_to_classical_mechanics>
but as explained, there is not much reason to expect any true analogue. (Although with sufficiently far-stretched generalizations one can
probably always construct one, but not convincingly..)

--
Jos

Thanks. So it appears to me that the Fokker-Tetrode-Schwartz action is exactly correct for CED in modelling point charges interacting with one another, including radiation, making the whole theory consistent and elegant. The downside is having to know
where all the charges in the universe are making the field approach more practical, but then having to deal with 'radiation reaction' as an approximate add-on to the theory from what I know so far.

JMc

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