In talking about Higgs mechanism, the first part is always some introduction to the concept of spontaneously symmetry breaking (SSB), some people saying that Higgs mechanism is the results of SSB of local gauge symmetry, some people says that we can formulate Higgs mechanism in a gauge invariant way, some people also says that we need only a non-zero vaccum expectation value... I am confused about this different or maybe same point of views.
In this post: How does the Higgs mechanism work? , the most highly voted answer, I still can't feel how SSB worked in Higgs mechanism. It seem that the validity of last part, the appearance of a mass term for $A$, is guaranteed if we have a non-zero equilibrium value $\phi_0$ to expand around. I do not see that the requirement that the phase of the field $\phi$ need to be fixed at some particular value to generate mass term. Thus it seems to me it is not true that SSB is really indispensable for Higgs mechanism.
To put it simply:
The spontanously breaking of what is attributed to Higgs mechanism?
local gauge symmetry
global symmetry, since breaking of a "gauge symmetry" should not have any effect on physics. In higgs mechanism, the really broken symmetry is a global one. Mathematically, it is similar in looking as fixing a gauge, but one should not think it as a spontanously breakdown of local gauge symmetry.
other.
Is SSB really indispensable for Higgs mechanism?
yes, Higgs mechanism is relied on the SSB of some symmetry (above question), the other approches of description eventually has spontanously broke some symmetry.
No, the SSB is just one way to describe Higgs mechanism (or even not a complete way), what is really need is the non-zero vaccum expectation value, for example in the linked post the requirement for the mass term to occur is to have some non-zero expectation value of $\phi$ to expand around, we do not need the phase of the field to be fixed, thus the symmetry is not broken.
Other.
Is Elitzur's theorem valid only in lattice field theory? States that SSB of local gauge symmetry is impossible.
Gauge invariant accounts of the Higgs mechanism in the abstract states that:
gauge symmetries merely reflect a redundancy in the state description and therefore the spontaneous breaking can not be an essential ingredient. Indeed, as already shown by Higgs and Kibble, the mechanism can be explained in terms of gauge invariant variables, without invoking spontaneous symmetry breaking
- Is electromagnetic gauge invariance spontaneously violated in superconductors? In the introduction it says:
In particular, we emphasize that global U(1) phase rotation symmetry, and not gauge symmetry, is spontaneously violated, and show that the BCS wave function is, contrary to claims in the literature, fully gauge invariant
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