An anomaly is a symmetry of the classical action that fails to be a symmetry of the path integral, due to non-invariance of the path integral measure. Does it ever occur that the opposite thing happens, i.e. that the classical action does not possess a symmetry, but the combined transformation of the action and the measure leaves the path integral invariant? Is there a name for such a symmetry of the quantized theory that does not exist in the classical theory?

  • $\begingroup$ What do you precisely mean by quantum symmetry? There are two notions. Wigner/Kadison symmetry, that is a map from states to states preserving transition probabilities/convex structure of mixed states. Thy are completely described by unitary and anti unitary operators. Moreover there is the notion of dynamical symmetry, i.e. a symmetry (in the above sense) which preserves the dynamics of the system. In the simplest case the unitary/anti unitary operator commutes with the temporal evolutor. $\endgroup$ Aug 2, 2014 at 8:17
  • $\begingroup$ @ValterMoretti Probably referring to a dynamical symmetry, since the path integral is defining dynamics for the system. $\endgroup$
    – asperanz
    Aug 2, 2014 at 21:46
  • $\begingroup$ I found this paper... $\endgroup$
    – Trimok
    Aug 3, 2014 at 13:04

1 Answer 1


The following situation is not uncommon: classically a symmetry may be (spontaneously) broken, but, quantum mechanically, the symmetry is restored. Put differently, quantum fluctuations can, under certain, well understood conditions, destroy the classical asymmetry ("order"). The simplest example is probably the one-dimensional double well potential, centered symmetrically around the origin, with minima at $\pm a$. These minima represent two - classically degenerate - ground states: a particle sliding from the central peak will end up at either the left or the right bottom of the well and rest there in the end once its kinetic energy is gone. So, the discrete reflection symmetry, $x \to -x$, of the classical action (or potential) is violated ("spontaneously broken) by the ground state of minimal energy.

Quantum mechanically, however, the symmetry is restored via tunneling: quantum fluctuations (around the instantons representing the classical solutions that connect the minima) produce two low lying states. Their level splitting can be calculated using a quantum mechanical path integral and is given by the fluctuation determinant resulting from the stationary phase (or WKLB) approximation. A nice exposition can be found in Coleman's book, Aspects of Symmetry, Ch. 7.

The whole story has ramifications in higher dimensions. A continuous (rather than a discrete) symmetry cannot be broken spontaneously in two dimensions as quantum fluctuations again dominate (often referred to as the Mermin-Wagner-Coleman theorem). All this may be rephrased in the language of statistical field theory (presence or absence of phase transitions). But the general idea is really that quantum fluctuations, if strong enough, can "wipe out" classical asymmetry, hence restore symmetry.

Please note that the classical asymmetry is one of the ground state only - the action itself remains symmetric. That's the the main feature of spontaneous symmetry breaking.

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    $\begingroup$ You are discussing a symmetry of a solution/vacuum, though? The OP is asking about symmetries of the path integral. $\endgroup$
    – innisfree
    Apr 21, 2015 at 18:55

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