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Fascinating. And what would you think of the fact that fields don't have well defined values at a point, i.e. ##\phi(x)## is undefined? That the fundamental things are the smeared fields, ##\phi(f) = \int_{\mathcal{M}}{\phi(x)f(x)d^{4}x}##?PeterDonis said:Yes.

These are not different sets of fields. They are differentdescriptionsin terms of fields. I look at it the same as a change of inertial frames in SR: you're describing the same underlying thing, just in different coordinates. Similarly, symmetry breaking doesn't change the underlying thing, but it does change which description of it is most useful.