Geodesic
Quantum Anomalies via Differential Properties of Lebesgue--Feynman Generalized Measures
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Selected issues of mathematics and mechanics, Tome 310 (2020), pp. 107-118.

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We address the problem concerning the origin of quantum anomalies, which has been the source of disagreement in the literature. Our approach is novel as it is based on the differentiability properties of families of generalized measures. To this end, we introduce a space of test functions over a locally convex topological vector space, and define the concept of logarithmic derivatives of the corresponding generalized measures. In particular, we show that quantum anomalies are readily understood in terms of the differential properties of the Lebesgue–Feynman generalized measures (equivalently, of the Feynman path integrals). We formulate a precise definition for quantum anomalies in these terms. 
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     author = {John E. Gough and Tudor S. Ratiu and Oleg G. Smolyanov},
     title = {Quantum {Anomalies} via {Differential} {Properties} of {Lebesgue--Feynman} {Generalized} {Measures}},
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John E. Gough; Tudor S. Ratiu; Oleg G. Smolyanov. Quantum Anomalies via Differential Properties of Lebesgue--Feynman Generalized Measures. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Selected issues of mathematics and mechanics, Tome 310 (2020), pp. 107-118. http://geodesic.mathdoc.fr/item/TM_2020_310_a6/

[1] L. Accardi, O. G. Smolyanov, and M. O. Smolyanova, “Change of variable formulas for infinite-dimensional distributions”, Math. Notes, 60:2 (1996), 212–215 | DOI | MR | Zbl

[2] Bogachev V.I., Smolyanov O.G., Topological vector spaces and their applications, Springer, Cham, 2017 | MR | Zbl

[3] Cartier P., DeWitt-Morette C., Functional integration: Action and symmetries, Cambridge Univ. Press, Cambridge, 2006 | MR | Zbl

[4] Chernoff P.R., “Note on product formulas for operator semigroups”, J. Funct. Anal., 2 (1968), 238–242 | DOI | MR | Zbl

[5] Dolan B.P., “A tale of two derivatives: Phase space symmetries and Noether charges in diffeomorphism invariant theories”, Phys. Rev. D, 98:4 (2018), 044009 ; arXiv: 1804.07689 [gr-qc] | DOI | MR

[6] Feynman R.P., “Space-time approach to non-relativistic quantum mechanics”, Rev. Mod. Phys., 20:2 (1948), 367–387 | DOI | MR | Zbl

[7] Feynman R.P., “An operator calculus having applications in quantum electrodynamics”, Phys. Rev., 84:1 (1951), 108–128 | DOI | MR | Zbl

[8] Fujikawa K., “Evaluation of the chiral anomaly in gauge theories with $\gamma _5$ couplings”, Phys. Rev. D, 29:2 (1984), 285–292 | DOI | MR

[9] Fujikawa K., Suzuki H., Path integrals and quantum anomalies, Oxford Univ. Press, Oxford, 2013 | MR | Zbl

[10] J. Gough, T. S. Ratiu, and O. G. Smolyanov, “Quantum anomalies and logarithmic derivatives of Feynman pseudomeasures”, Dokl. Math., 92:3 (2015), 764–768 | DOI | MR | Zbl

[11] J. Gough, T. S. Ratiu, and O. G. Smolyanov, “Noether theorems and quantum anomalies”, Dokl. Math., 95:1 (2017), 26–30 | DOI | MR | Zbl

[12] Kupsch J., Smolyanov O.G., “Functional representations for Fock superalgebras”, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 1:2 (1998), 285–324 | DOI | MR | Zbl

[13] Nelson E., “Feynman integrals and the Schrödinger equation”, J. Math. Phys., 5 (1964), 332–343 | DOI | MR | Zbl

[14] Pokorski S., Gauge Field Theories, Cambridge Univ. Press, Cambridge, 2000 | MR | Zbl

[15] Smolyanov O.G., Tokarev A.G., Truman A., “Hamiltonian Feynman path integrals via the Chernoff formula”, J. Math. Phys., 43:10 (2002), 5161–5171 | DOI | MR | Zbl

[16] Smolyanov O.G., von Weizsäcker H., “Differentiable families of measures”, J. Funct. Anal., 118:2 (1993), 454–476 | DOI | MR | Zbl