I often say that too, but people usually don't get it. Now when Wald says that, I hope people will finally start to pay attention.caz said:
May I ask what is novel about this textbook, or in other words, it's "raison d'être"? The contents list and the material in the Google books preview looks like the standard electromagnetism theory covered already quite extensively in many existing books. (But then again, it is Wald, so I would not like to judge it so soon!).caz said:
If we take the usual view that charges are the source of EM fields, then you have an analogous question: How did the charges get there in the first place?dextercioby said:
Well, let's wait what Wald writes. I'm very skeptical. What's physically significant are observables, and observables are gauge invariant. In quantum field theory (retarded) correlation functions of gauge-invariant local observables are the mathematical expressions which describe observable quantities. That also includes the observables related to the Aharonov-Bohm effect(s).Demystifier said:
You don't need to wait. This section can already be seen in the link in #4.vanhees71 said:
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 a local description. 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.vanhees71 said:
OK, I am buying this explanation. Thank you!Demystifier said:
Right, but ##A_{\mu}## modulo gauge transformations is not the same as ##F_{\mu\nu}##. So the point is that the fundamental quantity is ##A_{\mu}## modulo gauge transformations, not ##F_{\mu\nu}##.vanhees71 said:
Exactly! But still, the fundamental quantity is ##\psi##, not ##\bar{\psi} \Gamma \psi##.vanhees71 said:
Trivially I would point to preface where the author gives his reason.ergospherical said:
Keith_McClary said:
vanhees71 said:
Not to derail the thead, but I think that you and he have a different view on this although you both say it.Demystifier said:
What's the difference?martinbn said:
The way I understand him and you, you might disagree, is that he has a very clear view on the metaphysics. For him the electromagnetic field has ontology on equal footing with the other types of matter say electrons. This much he says. The ##E## and ##B## and the ##A## and ##\varphi## on the other hand are part of the mathematical discription. The second pair is more fundamental, but that is question of the mathematics not metaphysis. For you, the way I understand your view, the ##E## and ##B## and the ##A## and ##\varphi## can have ontology, which leads to a very confused notion of what exists and what is real.Demystifier said:
If you look at my arguments in this thread, you will see that I don't refer to ontology and reality.martinbn said:
Yes, but thisDemystifier said:
means other threads where you have expressed your view as well. But if I have misunderstood you then you can correct me.Demystifier said:
Well, sometimes I say that in the context of ontology, but that's not the only context where I say that.martinbn said:
So do you have the same view as him the way I wrote it hereDemystifier said:
I agree, at least in the context of classical physics.martinbn said:
May be this point is made in Chapter 9.vanhees71 said:
If you go to the Princeton URL given by @robphy ,vanhees71 said:
vanhees71 said:
The last sentence of the second paragraph of section 1.1 is "In other words, an electromagnetic field is an equivalence class of potentials φ, A under the transformation eq. (1.13).", and I am sure that Wald will treat this with more care and detail later in the book.martinbn said:
My definition of "readable" does not involve destroying woods.vanhees71 said:
One can easily prove the statement: “In a gauge-invariant theory (such as QED), the Poincare generators (P_{\mu} , J_{\mu\nu}) cannot be gauge invariant”. Aren’t these (P_{\mu} , J_{\mu\nu}) physical quantities?vanhees71 said:
1) The tensor F_{\mu\nu} is an observable physical field. However, as dynamical variables F_{\mu\nu} gives incomplete description in the quantum theory.Demystifier said:
For a comparison, would you say that Wald's book on general relativity provides better physical insight than Landau and Weinberg?samalkhaiat said:
It depends on what one means by "physical". I think @vanhees71 meant measurable.samalkhaiat said:
There are things in Wald that are not covered in Landau nor Weinberg.Demystifier said:
In classcial relativistic field theory and local relativistic QFT the "Poincare generators" are built from local fields (i.e., not the gauge-dependent four-potentials), i.e., via the Belinfante energy-momentum tensor (gauge-invariant) and not the canonical one (gauge-dependent).samalkhaiat said:
As a textbook, no. Weinberg and Landau give the student better physical insight to GR than Wald’s. However, if you require refined mathematic then go to Wald or (even better) Hawking & Ellis.Demystifier said:
Physical quantity or measurable quantity, they should both mean the same thing. In QM, it is represented by Hermitian operator and it is called observable. In quantum gauge field theory (such as QED), It can be shown that a necessary condition for an operator A to be observable is A |\Psi \rangle \in \mathcal{V}_{ph}, \ \ \ \forall |\Psi \rangle \in \mathcal{V}_{ph} , \ \ \ \ \ (1) where \mathcal{V}_{ph} is the physical vector space in which the scalar products are positive semi-definite. The positive definite Hilbert space is given by \mathcal{H}_{ph} = \mathcal{V}_{ph} / \mathcal{V}_{0}, where \mathcal{V}_{0} is the subspace of \mathcal{V}_{ph} consisting of zero-norm vectors. The essential point is that the condition Eq(1) does not necessarily require an observable operator to be gauge invariant, i.e., it does not need to commute with the generator of gauge transformation.Demystifier said:
The generators are Hermitian operators. They act on (gauge-dependent) states not “potentials”.Demystifier said:
Hehehe, Jackson is for students not for me. I spent big chunk of my academic life working with those two tensors. But any way, see my reply to vanhees71 bellow.Demystifier said:
Are you aware about the problems with the Belinfante’s expressions for the "generators"? Well, 1) the algebra closes only on-shell, or 2) they fail to generate the correct Poincare transformations on local fields. But, even if we ignore this difficulty and declare that (P_{\mu} , J_{\mu\nu})_{Bel} = (P_{\mu} , J_{\mu\nu})_{Can} , we can still prove (on general ground) that (P_{\mu}, J_{\mu\nu})_{Bel} cannot be gauge invariant. I can show you the easy proof if you want.vanhees71 said:
I would like to see the proof. In fact, I cannot imagine how generators based on Belinfante can fail to be gauge invariant.samalkhaiat said:
But physical states in QED are gauge-independent, aren't they?samalkhaiat said:
It's a question of physics, not grammar. An observable is never a self-adjoint operator. It is represented by a self-adjoint operator in the mathematical formalism of QT. Only things that are observable (and in physics it should even be quantifiable, i.e., measurable) are relevant for the physics. A gauge-dependent quantity cannot be observable by definition, because it is not unique, given a physical situation.Demystifier said:
The generators of the Poincare transformation are gauge invariant and the same for the Belinfante and the canonical expressions. It's about the local densities (energy, momentum, stress, angular momentum density) of the corresponding observables, which are not unique and not a priori observable. For that you need gauge invariant expressions.samalkhaiat said:
It is a matter of precise language, to avoid confusion. A gauge-dependent quantity cannot be measurable, but it can satisfy the formal definition of "observable" as an arbitrary self-adjoint operator.vanhees71 said:
Let me guess, the proof is axiomatic, not constructive. You assume that the generators satisfy some expected properties and then prove that they are in contradiction with gauge invariance. But you don't actually prove that the generators explicitly constructed from Belinfante really have those expected properties. Am I close?samalkhaiat said:
).