1
$\begingroup$

I'm just wondering what's the difference between symmetry breaking and anomaly.

From my understanding, symmetry breaking means: there is a symmetry in the action, but in the ground state of the theory(minimum of the action), the symmetry becomes smaller, which is a subgroup of the original symmetry.

While anomaly means that there is a symmetry in the classical theory ,but it's no longer a symmetry after quantization.

I think there should be some connections between them, but I don't know yet.

$\endgroup$
1
4
$\begingroup$

The answer to the linked question largely answers your question, but it assumes a bit. Contrast the following cases. I italicize your special interest parts, but they should be set in context.

  • Unbroken, not anomalous symmetry. Good symmetry, both classically and quantum, with visible and pleasing effects. Global or local (e.g. color). Symmetry currents conserved.

  • Explicitly broken symmetry. Both classically and quantum, normally considered when the breaking (the non-vanishing divergence of currents) is small, e.g. the flavor SU(3) symmetry of light quarks.

  • Spontaneously broken symmetry. The current is conserved, but quantum collective effects, induced by the hamiltonian, alter the vacuum to a non-symmetric type, so the realization of the symmetry is peculiar, in the Nambu-Goldstone mode, with less facile effects, even though visible. The chiral symmetries of massless quarks in QCD are such. The local version underlies EW interactions.

  • Anomalous symmetry The current is conserved classically, but due to a property of the Dirac sea, develops a non-vanishing divergence proportional to $\hbar$ computable via fermion loop corrections to it. It is a property of the fermion sea/measure, not the vacuum altered by the hamiltonian. Globally, as in flavor chiral ones, they control important physics processes, like neutral pion decay, and are generally summarized by WZWN actions. Locally, they invalidate gauge invariance and render gauge theories inconsistent, so are avoided/cancelled (by the addition of suitable fermions) with extreme prejudice.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.