In quantum field theory we say that gauge symmetry is a redundancy, and also, in Xiao-Gang Wen's book, it reads that gauge symmetry is not a symmetry, so it can never be broken. And the Higgs mechanism is about the breaking of global symmetry. Then how can we understand the symmetry breaking from electroweak theory $SU(2) \times U(1)$ to QED $U(1)$?


Gauge symmetry is rather like the distinction between passive and active transformations of space; the first is a change of coordinates and the second is an actual transformation of the space itself; however the end result can be the same.

For example, if you rotate a coordinate system by 15 degrees anti-clockwise, this is equivalent to rotating the underlying space 15 degrees clockwise. The end result is the same.

Yet, the first rotation we did above was merely a change of coordinates; nothing physically is changed by this - and it's for this reason it's called a passive transformation; in the second rotation the coordinate system does not move, but the underlying space does - and this is why this case is called an active transformation; and so the first rotation is unphysical, but the second is physical.

Now, the first rotation is not a real symmetry - nothing physical moved; in the second rotation something physical did move - and so it's called a real symmetry. Gauge symmetry is rather like the example of the first rotation - and this is why it's called 'a redundancy' or 'not a symmetry'.

I'd question the characterisation that gauge symmetries can't be broken; for example, we have here, in an article on Goldstone bosons:

These spinless bosons correspond to spontaneously broken internal symmetries.

Internal symmetries are equivalent to gauge symmetries and so they are saying such symmetries can be broken.


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