![]() In SU(2) gauge theory in 4 dimensional Minkowski space, a gauge transformation corresponds to a choice of an element of the special unitary group SU(2) at each point in spacetime. Witten anomaly and Wang–Wen–Witten anomaly In this case the large gauge transformations do not act on the system and do not cause the path integral to vanish. If the disconnected gauge symmetries map the system between disconnected configurations, then there is in general a consistent truncation of a theory in which one integrates only over those connected components that are not related by large gauge transformations. The sum of the phases in every other subgroup of U(1) is equal to zero, and so all path integrals are equal to zero when there is such an anomaly and a theory does not exist.Īn exception may occur when the space of configurations is itself disconnected, in which case one may have the freedom to choose to integrate over any As there is an anomaly, not all of these phases are the same, therefore it is not the identity subgroup. The sum of the gauge orbit of a state is a sum of phases which form a subgroup of U(1). Such symmetries and possible anomalies occur, for example, in theories with chiral fermions or self-dual differential forms coupled to gravity in 4 k + 2 dimensions, and also in the Witten anomaly in an ordinary 4-dimensional SU(2) gauge theory.Īs these symmetries vanish at infinity, they cannot be constrained by boundary conditions and so must be summed over in the path integral. In known examples such symmetries correspond to disconnected components of gauge symmetries. Global anomalies in symmetries that approach the identity sufficiently quickly at infinity do, however, pose problems. In particular the corresponding anomalous symmetries can be fixed by fixing the boundary conditions of the path integral. For example, the large strength of the strong nuclear force results from a theory that is weakly coupled at short distances flowing to a strongly coupled theory at long distances, due to this scale anomaly.Īnomalies in abelian global symmetries pose no problems in a quantum field theory, and are often encountered (see the example of the chiral anomaly). Since regulators generally introduce a distance scale, the classically scale-invariant theories are subject to renormalization group flow, i.e., changing behavior with energy scale. The most prevalent global anomaly in physics is associated with the violation of scale invariance by quantum corrections, quantified in renormalization. Technically, an anomalous symmetry in a quantum theory is a symmetry of the action, but not of the measure, and so not of the partition function as a whole.Ī global anomaly is the quantum violation of a global symmetry current conservation.Ī global anomaly can also mean that a non-perturbative global anomaly cannot be captured by one loop or any loop perturbative Feynman diagram calculations-examples include the Witten anomaly and Wang–Wen–Witten anomaly. The relationship of this anomaly to the Atiyah–Singer index theorem was one of the celebrated achievements of the theory. In quantum theory, the first anomaly discovered was the Adler–Bell–Jackiw anomaly, wherein the axial vector current is conserved as a classical symmetry of electrodynamics, but is broken by the quantized theory. Perhaps the first known anomaly was the dissipative anomaly in turbulence: time-reversibility remains broken (and energy dissipation rate finite) at the limit of vanishing viscosity. In classical physics, a classical anomaly is the failure of a symmetry to be restored in the limit in which the symmetry-breaking parameter goes to zero. In quantum physics an anomaly or quantum anomaly is the failure of a symmetry of a theory's classical action to be a symmetry of any regularization of the full quantum theory. Not to be confused with Anomaly (natural sciences).
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