Quantum anomalies via differential properties of the Feynman measures

Venue: PovoZero, via Sommarive, 14 - Povo - Sala Seminari "-1"
At: 17:00

Speaker: 

• Oleg Smolyanov - Lomonosov Moscow State University

Abstract:

Generalized measures are linear functionals on some spaces of functions, like in the Sobolev-Schwartz theory of distributions; they need not be integrals with respect to some σ-additive measures. The typical example of the generalized measure is the Feynman measure. They are used to represent some objects related to differential or pseudodifferential operators whose symbols are Hamilton functions of classical Hamiltonian systems. The value that a Feynman measure takes on a function is called the Feynman path integral.

Quantum anomaly is a violation of the symmetry with respect to a transformation during a quantization procedure. This means that a quantum anomaly occurs if a quantization procedure of a classical system,  invariant  relative to  a transformation, yields a quantum system which is no longer invariant under this transformation.

On page 352 of the book by Cartier, P. and DeWitt-Morette, C., Functional Integration, Cambridge University Press, 2006, it is claimed that the explanation of of quantum anomalies given in the book by Fujikawa, K. and Suzuki, H., Path Integrals and Quantum Anomalies, Oxford University Press, 2004, second printing 2013, is not correct. This criticism refers to the first, 2004 edition, of the last book. But the second, 2013 edition of the same book does not address this criticism, the authors maintain their original claims. Using a mathematically rigorous alternative approach, one arrive to the conclusion by Fujikawa and Suzuki.

The discussed in the talk approach, which uses differential properties of the generalized measures, is conceptually much simpler than the global (and non-rigorous) approach presented in the cited monographs.

Contact person: Sonia Mazzucchi

The speaker is guest as part of the "research in pairs" program