Symmetries and Conversation Laws

[humanino]

Maybe more qualified people will contribute, but in the meantime, since you ask

let's try to promote this symmetry to a local one and see if the resulting gauge theory works. Is this an accurate picture of how one thinks of local gauge theories or is there some more rigorous way to motivate gauging symmetries?

It's not the way I think about it. Let's get back to U(1) again. You have wave functions, they are (sets of) complex numbers. The phase difference matters between them, but not the absolute phase. You can multiply them by a unitary complex number. This is a global gauge transformation. It amounts to a choice for the "origin" of the phase. Then you easily realize that you could choose this origin at every point in space-time. But by doing so, you monkey around with the derivative, which "connects" two neighboring space-time points. So you need in order to fix the derivative to introduce an additional field, and luck or magic happens, this is the photon (potential vector) field. It seems to me from this point of view rather well motivated that the absolute phase should be arbitrary at every point in space-time. This construction unravel something deeper, which can be geometrically described in terms of what mathematician call connections and fiber bundles. It produces interactions from the constraints that derivative should transform accordingly to the arbitrariness we want in our physics.

Do you have a good resource for this stuff?

There are many. Dirac Yeshiva's lecture in 1964, "The geometrical setting of gauge theories of the Yang-Mills type" by Daniel and Viallet (RMP 1980), or "the geometry of physics" by Frankel (Cambridge 2004) are some I like.

 

[jensa]

Thank you very much for your reply.

Then you easily realize that you could choose this origin at every point in space-time.

I think this is the step I have difficulty understanding... It seems to me that requiring that you can choose this origin differently at different space-time points is to impose an additional constraint, on top of having a global choice of phase-convention (i.e. origin of phase). The justification for imposing this stronger constraint seems to be that, if we do so, we end up with gauge fields which correspond to the EM field.

But by doing so, you monkey around with the derivative, which "connects" two neighboring space-time points. So you need in order to fix the derivative to introduce an additional field, and luck or magic happens, this is the photon (potential vector) field. It seems to me from this point of view rather well motivated that the absolute phase should be arbitrary at every point in space-time. This construction unravel something deeper, which can be geometrically described in terms of what mathematician call connections and fiber bundles. It produces interactions from the constraints that derivative should transform accordingly to the arbitrariness we want in our physics.

I think I understand these steps/arguments, given that we have the freedom to choose a different phase-convention at each space-time point. Question is just if there is some more fundamental motivation for imposing this freedom/arbitrariness other than "because it gives us the right physics".

There are many. Dirac Yeshiva's lecture in 1964, "The geometrical setting of gauge theories of the Yang-Mills type" by Daniel and Viallet (RMP 1980), or "the geometry of physics" by Frankel (Cambridge 2004) are some I like.

Thanks again! I will definitely try to get a hold of these books and check them out. Titles sound promising.

 

[humanino]

I think this is the step I have difficulty understanding... It seems to me that requiring that you can choose this origin differently at different space-time points is to impose an additional constraint, on top of having a global choice of phase-convention (i.e. origin of phase). The justification for imposing this stronger constraint seems to be that, if we do so, we end up with gauge fields which correspond to the EM field.

Well, I thought the difficult step was to understand how gauge invariance relates to the arbitrariness of the phase. Once you have imposed the global gauge transformation, will you accept that your choice in Washington today should automatically be valid on Saturn yesterday !?

 

[jensa]

Well, I thought the difficult step was to understand how gauge invariance relates to the arbitrariness of the phase. Once you have imposed the global gauge transformation, will you accept that your choice in Washington today should automatically be valid on Saturn yesterday !?

Thanks again humanino. I really appreciate you taking time to explain this.
So it is a natural consequence of relativity? That different observers cannot agree on a common choice for the origin of phase and therefore it must be arbitrary at every space-time point...or something like that? That sounds like reasonable justification.

Just a note to defend my stupidity: I deal only with non-relativistic QFT (in condensed matter) so such ideas usually pass me by.

 

[genneth]

jensa: don't feel stupid --- these are very deep questions you're trying to ask. As the end of Wen's book points out, the conventional picture of gauge theories are as systems where the configuration is defined by some connection on a fibre bundle. As for why: the historical path is Weyl's attempt to take the idea of general relativity and apply it further. In GR, we say that we cannot compare directions at different spacetime points without something to tell us how to; we call the object that informs us how neighbouring points' tangent spaces relate to each other the connection. Weyl thought that maybe the size of objects should also be local --- so there would need to be something that compares sizes at neighbouring points. He called this a gauge theory, because it's a theory that changes the size-measuring apparatus, i.e. a literal gauge. This produced some very neat maths, and actually causes something very much like EM to drop out. However, it also predicts things which are entirely contradictory to experimental evidence (the original paper was published with a footnote by Einstein explaining why it could not be correct). With the advent of quantum mechanics, the idea to replace size with phase was not too much of a stretch, with the advantage that it actually produces the right predictions.

This view that gauge fields are the result of needing to compare quantities at different spacetime points is fairly accepted, and used in both high-energy and condensed matter. However, it certainly raises the question of whether the phase is then supposed to be a real physical thing --- there are very good arguments from quantum information theory that the wavefunction (phase and amplitude) should only be thought of as book-keeping by some observer, and not actually some physical object. Nevertheless, even ignoring meta-physical pondering, it's certainly somewhat mysterious why phase should be incomparable, but size is.

Wen's angle is that the gauge theories we've looked at are only the tip of the iceberg of quantum systems with many-to-one labelling of states. For instance, he shows that you can get U(1) (and also non-abelian) gauge theories with massless fermions as the low-energy theory of some bosonic system. In this case, these things are all just book-keeping, and we choose to work with them because we know how too. In fact, the only "gauge structures" (Wen's term for theories with a many-to-one labelling) we can deal with are: identical particles of bosons or fermions (but not anyons), and certain gauge field theories (the word gauge now used in the traditional sense). So far, we can only solve problems that can be decomposed into combinations of these. As an example of a system we don't yet know how to perform the decomposition: non-linear sigma models.

Is Wen right? I don't know. But it's certainly an interesting point of view which deserves more wide-spread knowledge, especially outside of condensed matter.

 

[jensa]

Thanks genneth. Your reply has been very interesting. You have definitely raised my interest for Wen's book, which I have so far only read parts of.

 

[humanino]

There is no stupid question and it's very good to ask about fundamentals
I don't think it's strictly fair to say that's simply relativity, because absolute phase is not an observable. So we're playing around with the concepts, that's true, but enforcing self consistency.

 

[genneth]

Just to add to my entirely incoherent ramblings: there may be a very good reason for favouring gauge field theories (in the sense of having a configuration space that consists of a connection) over other multi-labelled systems. Mass terms are deadly for the naturalness of a field theory with small couplings; any mass term would grow very fast with renormalisation (in the Kadanoff-Wilson sense), and naturally all masses at low energy would be on the order of the theory's UV cutoff. Gauge field theories are protected by symmetry, so produced vector bosons which are guaranteed to be massless.

Open question: does this picture hold for theories which are not weakly coupled? The key question is what the scaling dimension of various local operators actually are, near whichever fixed point the theory actually flows to.