In the Lagrangian path-integral formulation of QFT, an anomalous symmetry is defined to be a symmetry of the action which is not a symmetry of the measure of the path integral, and therefore not a symmetry of the partition function. How do we define an anomalous symmetry in the Hamiltonian formulation of QFT, where this is no path integral or partition function?
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$\begingroup$ Anomaly is an quantum effect. Even you start with Hamiltonian formulation, following equal time commutation relations, Feynman rules etc. Anomaly will manifest itself in triangular fermion loop (in 4-D). $\endgroup$– MassCommented Jun 11, 2016 at 6:52
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2$\begingroup$ Semantic comment to the post (v1): Note that there exists a Hamiltonian (=phase space) path integral formulation. $\endgroup$– Qmechanic ♦Commented Jun 11, 2016 at 6:55
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1$\begingroup$ As Qmechanic says, one could as well examine the path integral for the Hamiltonian. Are you trying to ask how one sees the anomaly in the canonically quantized formalism? There you'll just have to evaluate the tree-level diagrams associated to the conservation of the currents. $\endgroup$– ACuriousMind ♦Commented Jun 11, 2016 at 10:02
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$\begingroup$ @ACuriousMind I was actually wondering how to even define an anomaly without referring to the path integral, so that one could find it from Feynman diagrams. But I think I found the answer: a theory is defined to have an anomalous symmetry if any possible loop regulator necessarily breaks the symmetry. $\endgroup$– tparkerCommented Jun 12, 2016 at 7:53
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According to this paper, the Hamiltonian interpretation of anomalies is that one cannot formulate any Gauss-like law to constrain the physical states in the anomalous theories.
- Luis Alvarez-Gaumé and Philip Nelson, Hamiltonian Interpretation of Anomalies, Comm. Math. Phys. Volume 99, Number 1 (1985), 103-114.
