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- Classical
- Thread starter MidgetDwarf
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It makes sense to define

Furthermore, often (but not always) when an operator is not self-adjoint, it is possible to make a self-adjoint

But this is not as crazy as it may look. For instance, even in ordinary QM, there is no self-adjoint extension of the momentum operator on a half-line. In other words, if a particle in one dimension lives at ##x>0##, then it's not possible to choose boundary conditions at ##x=0## such that the momentum is a self-adjoint operator.

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- #74

samalkhaiat

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True.One additional insight. If ##P^{\mu}## is not gauge invariant as @samalkhaiat said, then [itex]\delta P^{\mu}=[iQ_{\Lambda},P^{\mu}] \neq 0[/itex]. But I think we also have [itex]\langle\psi |\delta P^{\mu}|\psi'\rangle=0[/itex] for any physical states [itex]|\psi\rangle, \ |\psi'\rangle[/itex]

Correct. And this means that the state [itex]| \chi \rangle \equiv \delta P^{\mu}|\Psi \rangle [/itex] is awhich implies that [itex]\delta P^{\mu}|\psi'\rangle[/itex] is orthogonal to any physical state [itex]|\psi\rangle[/itex].

- #75

atyy

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Is this true even for classical EM?But as you know, the Aharonov-Bohm gauge-invariant observable is expressed in terms of the potential, not in terms of the magnetic field, provided that you insist on alocaldescription. It all boils down to the fact that the integral ##\int dx^{\mu}A_{\mu}## is gauge invariant, so it's not really necessary to deal with ##F_{\mu\nu}## in order to have a gauge-invariant quantity.

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- #77

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Wald points out that AB effect is classical, in the sense that it exists even for a classical charged field coupled to EM field.Is this true even for classical EM?

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I've no clue, what he means by this. Which "charged field" has a classical meaning?

- #79

martinbn

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I think he by classical he means not quantum, not that it actually exists in what we observe.I've no clue, what he means by this. Which "charged field" has a classical meaning?

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If there is no such field, then there's also no classical AB effect.

- #81

martinbn

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Mathematically there are such fields. I might be wrong, but I think that the point is that it is not an effect of the quantum theory.If there is no such field, then there's also no classical AB effect.

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- #83

martinbn

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You may be talking about physics, but what is Wald talking about in this specific instance?

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I don't know, as the book is not out yet in conventiently readable form.

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Wald perhaps has a different definition of the difference between physics and math. We don't see classical charged fields in actual experiments, but in theory, before performing quantization of a complex scalar field, one can study its classical properties. One of those classical properties is classical interference of classical waves, which includes interference around solenoids. The latter is the theoretical classical AB effect.

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Actually it has, in the macroscopic Ginzburg-Landau theory of superconductivity.The "Schrödinger field" has no classical interpretation

- #87

martinbn

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All this makes me want to get the book. Is any of you Wald in disguise? (Or may be the publisher.)

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I'm Wald's ex wife, but he does not longer speak to me since I told him that Carroll's book on GR is better than his.All this makes me want to get the book. Is any of you Wald in disguise? (Or may be the publisher.)

- #89

Frabjous

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This showed up today in the today’s mail.

- #90

ergospherical

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Cool! What are your first impressions of the text?This showed up today in the today’s mail.

- #91

Frabjous

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It is clearly a physics text, not a mathematical methods book. My one word description is “Clean”. He knows the path he wants to tread and he does not wander from it. It’s short with 225 pages of text, but it is more than just a set of lecture notes. If you do not know the math he uses, you will need supplementary material. That being said, the math doesn’t look scary. I think I will need to refresh my knowledge of Green’s functions. After thumbing through it, I am still looking forward to reading it.Cool! What are your first impressions of the text?

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hutchphd

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dude662

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It is literally lecture notes though. I was one of his students when he was making the bookIt is clearly a physics text, not a mathematical methods book. My one word description is “Clean”. He knows the path he wants to tread and he does not wander from it. It’s short with 225 pages of text, but it is more than just a set of lecture notes. If you do not know the math he uses, you will need supplementary material. That being said, the math doesn’t look scary. I think I will need to refresh my knowledge of Green’s functions. After thumbing through it, I am still looking forward to reading it.

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Interestingly, the book has no references.

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