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DrDu
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If I where a QFT guy I would certainly be able to mock up a confining interaction where the field carrying the interaction depends on c and then would discuss the limit c to infinity.
If you are not a “QFT guy”, shouldn’t you listen to someone who makes his living from QFT?DrDu said:If I where a QFT guy I would certainly be able to mock up a confining interaction where the field carrying the interaction depends on c and then would discuss the limit c to infinity.
I am certainly eager to do so! I did some reading. Apparently, in the Coulomb gauge, the Gauss constraint holds at operator level. Doesn't this clearly show that the extensions to the Maxwell equations for A which appear in other gauges, are artifacts of working with A instead of F?samalkhaiat said:If you are not a “QFT guy”, shouldn’t you listen to someone who makes his living from QFT?
Is it still local if you need an infinite number of auxiliary fields?DrDu said:Isn't the "law of local interactions" in field theory rather a tautology as you can transform any non-local interaction into a local one introducing sufficient auxiliary fields?
All Maxwell's equations hold as operator equations in the Coulomb gauge.DrDu said:Apparently, in the Coulomb gauge, the Gauss constraint holds at operator level.
I don’t understand what you are saying here. In this thread I stressed the points which represent our current understanding of QFT:Doesn't this clearly show that the extensions to the Maxwell equations for A which appear in other gauges, are artifacts of working with A instead of F?
bhobba said:This is a very interesting thread I am greatly enjoying.
My concern is the OP said he had not done much QM and very likely no QFT. So I thought the following paper explaining how gauge symmetry and the vector potential naturally come out in QM will help his understanding:
http://www.niser.ac.in/~sbasak/p303_2010/23.11.pdf
I want to also assure the OP that this gauge view is absolutely essential to the standard model which of course QED is part of. Its part of what's called Yang-Mills theory:
https://en.wikipedia.org/wiki/Yang–Mills_theory
Thanks
Bill
Phylosopher said:Thank you sir. I am enjoying the discussion too! This would be my future reference after studying QFT
It seems that QFT theories from what I am seeing from the discussion were not a direct implications of the previously famous theory of ElectroMagnetism/Maxwell's equations, or at least the view of the last was lost in the process. Otherwise, there won't be this much disagreement between the members of the forum. I actually had the same discussion with one of my professors and the results were the same as here.
samalkhaiat said:3) In the usual formulation of QED, i.e., the one based on [itex]A_{\mu}(x) , J^{\mu}(x)[/itex] and [itex]\psi (x)[/itex], you have two options:
i) Give up locality and have the Maxwell equations as operator equations. This happens in the Coulomb gauge quantization.
ii) Keep locality and covariance but modify the Maxwell equations. This is the widely use Gupta-Bleuler covariant quantization.
This is just a repeat of what I have been saying.DrDu said:This is quite interesting.
A gauge choice defines a slice in the gauge group orbits. Did you mean this?Apparently, different gauges live in different spaces,
Why is that so? In many cases, a unitary transformation does exist between two gauge choices.i.e. they are unitarily inequivalent.
No. That is precisely the definition of broken understanding of QED.Isn't this precisely the definition of a broken symmetry?
A. Neumaier said:Both classically and in quantum mechanics, the free Maxwell equations (for electromagnetic fields without sources) are linear, hence free electromagnetic waves and free photons don't interact.
A. Neumaier said:No. The free Maxwell equations are not about the vector potential but about the electromagnetic field, and these hold also on the operator level!
A. Neumaier said:QED is not about the Maxwell equations but about a bigger system of equations involving a fermionic field not known before 1925. Such a field does not figure in Maxwell's equations. Neither do Maxwell's equations demand a derivation from an action principle; they stand for themselves.
This just means that one has to avoid the usual covariant quantization of the em-field that one can find in almost all usual textbooks.samalkhaiat said:The free Maxwell equation ##\partial^{\mu}F_{\mu\nu}=0## does not hold as operator equation in the usual covariant quantization of the em-field that one can find in almost all usual textbooks. And that is a complete answer to the remarks raised in #1.
Which you confirm:A. Neumaier said:What is constructed is a redundant reducible representation of the free massless spin 1 field on a vector space of unphysical wave functions that not even forms a Hilbert space, thus violating one of the basic principles of quantum mechanics. This is the reason why spurious terms are present in your equations. The physically relevant representation is the restriction to the wave functions defined by transverse momenta, and on this (irreducible) physical representation space, the free Maxwell equations hold in their classical form.
Maxwell's equations as stated everywhere do not contain physical charged fields but only the electromagnetic field strength and a charge current. These are gauge invariant fields, hence operate in the quantum case on a physical Hilbert space with a positive definite metric.samalkhaiat said:The space [itex]\mathcal{H}_{ph}[/itex] carries one good news for you. Indeed, the Maxwell equations do hold as operator equations in [itex]\mathcal{H}_{ph}[However, this comes with heavy price, because all elements of [itex]\mathcal{H}_{ph}[/itex] have zero electric charge, and [itex][J^{0}(f)][/itex] generates trivial automorphism, i.e., [itex]\mathcal{H}_{ph}[/itex] has no room for electrons.
But nothing in the theorem you cite prevents one from formulating QED in terms of a Lagrangian with nonlocal fields, without introducing an indefinite metric. For scalar electrons this was done in Section 3 of the 1962 paper Quantum electrodynamics without potentials by S Mandelstam. The spinor case (i.e., full perturbative QED) was treated in a subsequent 1964 paper by Levi. The interacting Maxwell equations hold on the operator level.samalkhaiat said:I
The theorem say that “one cannot hope to formulate QED in term of [itex]F_{\mu\nu}[/itex], [itex]j^{\mu}[/itex] and local charged fields without essentially going to the Gupta-Bleuler formulation.
We don't disagree on the fact that the standard treatment of QED is based on a gauge theory and usually represented by field operators on an indefinite metric space. We only disagree on the claim that the gauged representation and the indefinite metric are unavoidable. The fact is that they are not; the theorem cited by samalkhaiat only holds under assumptions made for convenience, not for physical necessity.bhobba said:we have a few people whose knowledge of such things is extremely advanced - Samalkhaiat and Dr Neumaier are among them. It's unusual for any of those to disagree, but it happens occasionally and we all learn something.