Clarifications of semantics in gauge theories

Question

I wrote a short note clarifying some semantics used in gauge theories. I collect some statements that were once confusing to me and clarify them. Since I'm distributing it among friends and acquaintances, I don't want to be embarrassed with mistakes, so I put it up here for suggestions and criticisms. The note is for pedagogy and maybe not exactly graduate level, but this is chat session so I hope it is ok.

Update： A updated version of the note has been uploaded after the discussion with Arnold.

Answer 1

To 1.: If, as in a gauge theory, all observables of a theory are invariant under some continuous symmetry, there are unobservable degrees of freedom - there is no way to observe the variables that change under the symmetry.

For a simpler example, suppose you can observe only length $|x_i-x_k|$, not position, then position is unobservable since $x_i\to x_i+s$ is a continuous translation symmetry preserving length. This is in fact the case in real physics - we observe position only since the state of the universe has broken translation symmetry so that we know a few rigid points relative to which we can deduce position by means of computing from measured lengths. 

Thus the Poincare symmetry of a QFT has the same status as a gauge symmetry - it is due to unobservable degrees of freedom. Only since we pretend that position has an absolute meaning (in a laboratory frame where experiments testing a QFT are performed), translation and rotation symmetry seem to have a different status.  

In general relativity, this is more pronounced, as there not only translations but all volume-preserving diffeomorphisms are continuous symmetries - which in some formalisms is considered a kind of gauge symmetry.

To 2.: A large gauge transformation is local only in a restricted sense. It is nonlocal in that it affects more than any compact region. This makes it large. It has nothing to do with changing boundary conditions. Indeed, boundary conditions equivalent under large gauge transformations are physically indistinguishable. Precisely this causes problems that must be addressed when discussing infrared effects.

To 3.: your Lagrangians as written are the same. To turn a field theory into a diffeomorphism invariant theory (which, according to 1. above, makes it gauged), you need to be more careful. See the entry ''Diffeomorphism invariant classical mechanics'' in Chapter B8: Quantum gravity of my Theoretical Physics FAQ.

To 4.: The Maxwell equations are described in terms of a closed 2-form, hence carry with them a connection (and hence a gauge field), even if this is not explicit in the formulation. That the gauge field is real can be seen from the Aharonov–Bohm effect, which cannot be explained by Maxwell alone since the topological information in the gauge field is lost. And the other interactions need nonabelian gauge fields, where it seems impossible at all to eliminate the gauge fields. (In you final sentence, an ''is'' is missing.)

Comments

1 

If, as in a gauge theory, all observables of a theory are invariant under some continuous symmetry, there are unobservable degrees of freedom - there is no way to observe the variables that change under the symmetry.

This seems to be a matter of taste, one can start with a theory with local symmetry, then determine the set of quantities that are invariant under the local symmetry transformation, and call these observables; or one can start with a set of observables and construct a theory with local symmetries leaving observable invariant. But either way, the presence of local symmetry is essential, that's why I take it as a definition(or more pictorially a phase space trajectory always multi-furcates if we don't fix a gauge).

 we observe position only since the state of the universe has broken translation symmetry so that we know a few rigid points relative to which we can deduce position by means of computing from measured lengths. 

I can't quite comprehend this, isn't the ability of measuring position equivalent to the ability of choosing a coordinate system? And the latter seems completely innocent.

2. Thanks, here I made a genuine mistake, after checking out the definitions again, I realized I confused large gauge transformation with the gauge transformations that diverges at infinity(divergence lured me into thinking this is the "large" gauge transformation). There is a physicsforums post discussing this, some call the one in my head "asymptotically nontrivial" gauge transformation. (see next comment)

3. This is a boring way of gauging it, but it does possess local symmetry and hence redundant degrees of freedoms, which according to the definitions I gathered, is a gauge theory.

4

That the gauge field is real can be seen from the Aharonov–Bohm effect, which cannot be explained by Maxwell alone since the topological information in the gauge field is lost

I have no dispute with "gauge theory is real", since my philosophy(or preferred semantics) is that since ontology is model-dependent, then whatever description that is more natural or more powerful deserves to be called "real".  But I'm not sure about the "cannot be explained" part, since if I rememember correctly, the observable phase shift in AB effect is proportional to magnetic flux, which isn't necessarily expressed in gauge potentials.

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After some further digging, I found that the usage of "large gauge transformation" is person-dependent,  some, e.g. Jeffrey Harvey in his lecture notes explicitly stated (page 21)

In gauge theory it is important to make a distinction between small gauge transformations g which are those approaching the identity at spatial infinity and large gauge transformations which do not approach the identity at spatial infinity.

and our users @Hunter and @drake (and I myself previously) adopted the same definition , see this SE post.

On the other hand, perhaps more frequently, some only consider gauge transformations that tend to identity, then all the maps $R^n\to \text{Gauge group}$ can be consistently compactified to maps $S^n\to \text{Gauge group}$, then they define the "small" gauge transformations to be the ones that are homotopically trivial, otherwise they call them "large".

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The second definition for a group G is equivalent to the first for the universal cover (acting on a cover of spacetime), if the system under consideration has with the original group no large gauge transformations. In general think the first definition is the more appropriate one.

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1. What is observable is determined by our ability to observe it in principle, i.e., to make it macroscopically visible in a measurement. This is impossible both for unbroken local gauge transformations (because of confinement) and for global Poincare (or diffeomorphism) transformations (because we can measure only Loretnz distances).

3. Your Lagrangians $L$ and $L'$ are fully identical since the extra terms cancel exactly . Of course any field theory becomes a gauge field theory  if it is interpreted as a field theory containing fields not present in the formulation - as obviously nothing about these is observable. if you really want to keep the example, delete the apparent dependence on $q_2(t)$; this doesn't change anything and simplifies your argument, while now it looks (at first) like there was a misprint somewhere. 

4. Maxwell fields $F_{\mu\nu}$ and hence anything in Maxwell theory are fully explained by $F=dA$. The point is that (unlike your in my eyes ridiculous example in 3.) gauge theory is needed to explain a macroscopic effect in electrodynamics, hence Maxwell theory is even macroscopically incomplete without the gauge formulation. The Aharonov–Bohm effect proves what you took as heading in 4.

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1. "because we can measure only Loretnz distances", it seems you are trying to say our physics must be modeled on affine space, but I just don't see what is wrong with adding an origin for an observer.

3. Thanks, I'll try to make that clearer.

4. I am trying to be very cautious here, based on my experience, one has very large freedom to choose the way to build up an equivalent theory with the conventional one, if he/she is willing to tweak and twist the theory hard enough, not that it's a worthwhile thing to do, though. For example the geocentric model wasn't really wrong in the sense that it's incapable of producing right numbers, but it was just so much more cumbersome compared to heliocentric model that we are completely justified to call it "wrong".

I think the same might be accomplished for QED(in fact, I'm thinking, if hypothetically the equations of motions (field operator equations) of QED can be well defined, shouldn't it contain all information you can possibly extract from a Lagrangian formulation? I forgot to consider fermion equation of motion, where gauge potential appears), and yes, for non-abelian theories the gauge fields appear even in (bosonic)equations of motions, but it could mean we just need to tweak even harder(again, not that I think it's a worthwhile task to do).  

That's why I narrowed the context to the extent that I think is safe, i.e. Lagrangian formulation with manifest Poincare symmetries and manifest locality.

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The second definition for a group G is equivalent to the first for the universal cover (acting on a cover of spacetime), if the system under consideration has with the original group no large gauge transformations. In general think the first definition is the more appropriate one.

Could you elaborate the mathematics? Universal cover has fundamental group being trivial, but it seems here the relevant object is the higher homotopy group $\pi_n$. And what do you mean by "a cover of space-time"?

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adding an origin for an observer makes your theory observer-dependent. The hallmark of physics is objectivity, i.e., observer independence.

Universal cover has fundamental group being trivial, but it seems here the relevant object is the higher homotopy group $π_n$.

You are right; maybe I got that wrong. I don't have enough experience here.

"a cover of space-time" = several copies, labelled by an index. Thus all fields depend on this extra index, like with spinor fields which correspond to a double cover.