Spontaneously broken gauge symmetry

atyy

Struyve discusses that a global symmetry can be considered operationally distinguishable or not depending on whether the system is viewed as being the whole universe or a subsystem (p15):

"The other partial transformations do not correspond to a local transformation and can hence be viewed as changing the physical state of the subsystem. However, the observational difference only arises when comparing the fields of the subsystem with those of the environment and not from within the subsystem itself. This is analogous to the familiar view one can take on, for example, translation and rotation symmetry in classical mechanics. While translations or rotations of the whole universe are unobservable, and hence may be regarded as gauge transformations, translations or rotations of (approximately isolated) subsystems relative to their environment will yield (in principle) observable differences."

Presumably Haag must be taking the superconductor to be the whole universe so that the ground states are operationally indistinguishable? OTOH, since the global symmetry is not a gauge symmetry in the sense of producing nondeterministic EOM, if the superconductor is considered a subsytem, then the ground states are operationally distinguishable, say with respect to a second superconductor via the Josephson effect?

atyy

I believe Greiter's Eq 11-13 are correct. The are the same definition of gauge invariance as Scholarpedia's Eq 17 & 18. There isn't any conflict with Haag's analysis, because Greiter's gauge invariance is not the same as Haag's gauge invariance. Rather Haag's gauge invariance is Greiter's global invariance (Eq 98), which is not the same as Greiter's gauge invariance (Eq 11-13).

DrDu

You (and Greiter) are mixing up state space and position representation. A state [itex] |\phi>[/itex] transforms into [itex]U|\phi>[/itex]. To get the transformation in the position representation, you have to project onto [itex]<x|[/itex], ie [itex]\phi(x)=<x|\phi>[/itex] gets transformed into [itex]\phi'(x)=<x|U|\phi>[/itex]. Greiter claims that [itex] U|\phi>=|\phi>[/itex], so that [itex]\phi(x)=\phi'(x)[/itex] (note that the function [itex]\phi[/itex] in eq. 13 is, in contrast to the claim by Greiter, not the wavefunction in position representation) which is at variance with Scholarpedia.

atyy

If I understand Greiter correctly, since a change of gauge is a different description of the same physical situation, the two wave functions correspond to the same state in different gauges. This seems to match Wen's description of a gauge transformation as being by definition a "do nothing" transformation. Greiter's and Wen's definitions include "local" gauge transformations, so they're different from Haag's "global gauge" transformation.

So

[tex]U^{\Lambda}|\phi> = U^{\Lambda} \prod (u_{k}+v_{k}e^{i\phi}c^{\dagger}_{k \uparrow}c^{\dagger}_{-k \downarrow})|0> = \prod (u_{k}+v_{k}e^{i\phi^{\Lambda}}c^{\dagger\Lambda}_{k \uparrow}c^{\dagger\Lambda}_{-k \downarrow})|0> = |\phi^{\Lambda}>= |\phi> \text{ ,}[/tex]

where the equality between the gauge-transformed [itex]|\phi^{\Lambda}> \text{ and } |\phi>[/itex] is due to
[tex]c^{\dagger\Lambda}_{k \uparrow} = e^{i\frac{e}{\hbar c}} c^{\dagger}_{k \uparrow} \text{ ,}[/tex]
[tex]\phi^{\Lambda}=\phi-\frac{2e}{\hbar c} \Lambda \text{ .}[/tex]

DrDu

Ok, so U^Lambda would be strictly the identity transformation in that scheme.
I still don't see where this leads to.
Who is Wen?

atyy

I think Greiter is basically saying the same thing as Struyve. I think Greiter is pointing out that there is a definition of gauge transformations as basically identity transformations, which we use for gauge-equivalent vector potentials describing the same physical situation. This includes local gauge transformations, and can't be spontaneously broken (by definition).

In Greiter's terminology, the global phase symmetry is a transformation between different physical states. This symmetry is spontaneously broken in the superconducting state. Haag calls this "global gauge symmetry" since if we don't admit a second superconductor to judge relative phases, then the different states are observationally identical. Greiter prefers not to use the term "gauge" for the global symmetry, because he reserves the term for "do nothing" transformations, and also because the phase can be observed relative to a second superconductor.

Xiao-Gang Wen wrote a textbook about many-body physics that I've been slowly reading. Incidentally, he was my undergraduate thesis advisor. I was a biology major, and a biologist by profession, but did some undergrad physics for fun. The reason he gives in the book for why a gauge symmetry can't be spontaneously broken, and my question of how it's related to Elitzur's theorem, started out this thread.

Ok, hopefully now that we got some terminlogy straight, let's try to get back to the OP. Let's stick to "gauge transformations" in the sense of Greiter and Wen for the moment. As you pointed out, Elitzur's theorem seems to allow local symmetries to be spontaneously broken by non-local order parameters. When a continuous symmetry is spontaneously broken by a local order parameter, we get massless Goldstone bosons. In the Anderson-Higgs mechanism, a continuous global symmetry is spontaneously broken, but there is no massless Goldstone boson, presumably because the order parameter is non-local (Hansson et al, p5). Is there any example where a continuous local symmetry is not a gauge transformation (so that it can be broken), and is spontaneously broken by a non-local order parameter (since Elitzur's theorem says a local order parameter won't work)?