# Spontaneous symmetry breaking and conservation of Noether's charge, classically

This is an old question to which I previously posted an answer. I later found that answer to be unsatisfactory and is now deleted. It is newly edited on $$01.05.2020$$.

Classical conservation laws are outcomes of the symmetries of the action while spontaneously symmetry breaking (SSB) is all about the symmetry of the action not being shared by the vacuum. Since in a SSB scenario, the action never breaks the symmetry but only the vacuum does, one would expect (at least, classically) that the charge conservation continues to hold.

Here I recall the proof of the conservation of Noether's charge. It requires (i) the action to retain the symmetry and (ii) the vanishing of the 3-current $${\bf J}$$ at spatial infinity. Both of these requirements are satisfied when SSB takes place. For example, in a spontaneously broken scalar field theory, although the field $$\phi(x)$$ has a constant value everywhere (and hence, does not vanish at spatial infinity), the 3-current does vanish (because it contains spatial derivatives of $$\phi$$)!

Is this a fair judgement to conclude that conservation laws continue to hold classically, for a spontaneously broken symmetry?

Here are some related posts regarding the quantum situation:

$$1.$$ Physically, what is the Fabri-Picasso theorem really trying to say?

$$2.$$ Does Fabri-Picasso theorem imply non-conservation of charge?

$$3.$$ Why do the conserved charges in the case of SSB of a global symmetry not exist?

• The charge operator becomes a ill-defined operator (the one you define as space integral of the time-component of the conserved current). However, its commutator is still well defined and that's the only ingredient you need to generate the variation of the fields, namely $[Q,\phi] = \delta \phi$ is still a good definition of variation. Dec 29, 2017 at 17:53
• Look at this post physics.stackexchange.com/a/228616/47373 Dec 29, 2017 at 18:32
• Sure, why not, but what would you do with said charge? The classical transformations would keep the action and trajectories invariant. Can you be more explicit in the concrete classical quantities that bother you? May 1, 2020 at 17:43
• physics.stackexchange.com/questions/436398/… In this post, I asked if the charge is conserved quantumly. But before that, I wanted to be sure that the charge is conserved classically. I am not yet convinced why the charge is conserved or not conserved quantumly if SSB happens. Here is a related post: physics.stackexchange.com/questions/548317/… I did not get any concrete answer to whether charge is conserved in SSB case. @CosmasZachos
– SRS
May 1, 2020 at 17:59
• Both links deal with the quantum domain, but here you insist you wish to stick to classical field theory, without states or infrared problems, no? May 1, 2020 at 18:25