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Quantum Anomalies via Differential Properties of Lebesgue–Feynman Generalized Measures

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Abstract

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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Acknowledgments

J. Gough thanks the Institut Henri Poincaré for their kind hospitality during the Measurement and Control of Quantum Systems trimester in 2018. O. G. Smolyanov thanks the School of Mathematical Sciences of the Shanghai Jiao Tong University for the excellent working conditions provided during his visit in May 2017, when this paper was begun.

Funding

T. S. Ratiu was partially supported by the National Natural Science Foundation of China (grant no. 11871334) and the NCCR SwissMAP grant of the Swiss National Science Foundation. O. G. Smolyanov was supported by the Lomonosov Moscow State University (grant “Fundamental problems of mathematics and mechanics”) and by the Moscow Institute of Physics and Technology within the Russian Academic Excellence Project “5-100.”

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Authors and Affiliations

  1. Institute of Mathematics, Physics and Computer Science, Aberystwyth University, Penglais, Aberystwyth, Ceredigion, SY23 3BZ, UK

    John E. Gough

  2. School of Mathematical Sciences, Shanghai Jiao Tong University, 800 Dongchuan Road, Minhang District, Shanghai, 200240, China

    Tudor S. Ratiu

  3. Section de Mathématiques, Université de Genève, 2-4 rue du Lièvre, Case postale 64, 1211, Genève 4, Switzerland

    Tudor S. Ratiu

  4. École Polytechnique Fédérale de Lausanne, 1015, Lausanne, Switzerland

    Tudor S. Ratiu

  5. Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, 119991, Russia

    Oleg G. Smolyanov

  6. Moscow Institute of Physics and Technology (National Research University), Institutskii per. 9, Dolgoprudnyi, Moscow oblast, 141701, Russia

    Oleg G. Smolyanov

Authors
  1. John E. Gough
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  2. Tudor S. Ratiu
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  3. Oleg G. Smolyanov
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Correspondence to John E. Gough.

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Gough, J.E., Ratiu, T.S. & Smolyanov, O.G. Quantum Anomalies via Differential Properties of Lebesgue–Feynman Generalized Measures. Proc. Steklov Inst. Math. 310, 98–107 (2020). https://doi.org/10.1134/S0081543820050077

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  • DOI: https://doi.org/10.1134/S0081543820050077

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