# Spontaneous symmetry breaking of gauge symmetries

Gold Member

## Main Question or Discussion Point

hello all

gauge symmetries are redundencies of the description of a situation. Therefore they are not real symmetries. So in what sense does it mean to spontaneously break a gauge symmetry?

ian

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malawi_glenn
Homework Helper
How do you justify that it is not a "real" symmetry?

atyy
I don't really understand this, but I've seen discussions about it.

"...the 'gauge symmetry' is not a symmetry. ..........the 'gauge symmetry' can never be broken." Wen, Quantum Orders and Symmetric Spin Liquids, http://arxiv.org/abs/cond-mat/0107071

"...the true gauge is ... G* ...... there is no true breakdown of this gauge symmetry here. What is broken is the global part of the symmetry, corresponding to G/G*" Nair, Quantum Field Theory p268, http://books.google.com/books?id=J4BmTXo_RkEC&printsec=frontcover#PPA268,M1

Gold Member
hi

It's not a real symmetry in the sense that, for example electromagnetsim can be done in terms of electric and magnetic fields with no reference to gauge transformations at all. Gauge transformations only come into it if you formulate the theory in terms of gauge potentials - there's no reason why you have to do electromagnetism this way, although it is sometimes convienient. This is what I mean by gauge symmetries being a redundency of the description - it's a fictious symmetry put in by hand.

ian

blechman
You're right that you can formulate classical E&M without ever talking about potentials, but in QM it is not so obvious how to do this: topological effects such as Aharanov-Bohm Effect and Dirac monopoles (for what they're worth) are not easy to see without reference to vector potentials. It really is these potentials that get quantized, so in a sense, they ARE the physical thing...

[I want to avoid a philosophical outcry about what it means to be "physical" - I just mean that this is the object that contains the electodynamic degrees of freedom. that's all I meant by that last statement.]

The gauge "symmetry" would be better called the gauge "redundancy" perhaps, but now you're just arguing semantics.

I'm not quite sure why you say that they are not "real symmetries" though - any time you have a symmetry it is because some degree of freedom is redundant. That's what it means to be a symmetry!

malawi_glenn
Homework Helper
I am on the same track as bleckman...

I think it was Dirac who emphasized it is not a real symmetry but rather, a convenient way to deal with contrained systems. It's just two different point of views, ways of thinking about a problem. People like to think in terms of symmetry. If you have rotational symmetry with a mass at a constant radius from a center, it is not very intelligent to use cartesian coordinates, and gauge out the radius. But yes, you can do it. :tongue2:

samalkhaiat
hi

there's no reason why you have to do electromagnetism this way, although it is sometimes convienient. This is what I mean by gauge symmetries being a redundency of the description - it's a fictious symmetry put in by hand.
How about; necessary law of nature! Without the gauge principle, I can not understand interactions. Can you?

regards

sam

Gold Member
What suggested the question was the comment from thiemann's paper

"We stress, however, that the gauge symmetries of General Relativity have been
exactly taken care of in the reduced phase space approach. We are talking here about a symmetry group and not a gauge group. To break a local gauge group is usually physically inacceptable especially in renormalisable theories where the corresponding Ward identities find their way into the renormalisation theorems. However, it may or may not be acceptable that a physical symmetry is (spontaneouly, explicitly ...) broken. For instance, the explicit breaking of the axial vector current
Ward identity in QED, also called the ABJ anomaly, is experimentally verified."

I'm thinking there is a differenece between spontaneous and actual symmetry breaking: in spontaneous symetry breaking it is the vacuum state that breaks the symmetry whereas in actual breaking of symmetry one has an addition term to the action which breaks the symmetry - e.g an external magnetic field

ian