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Mathematical Aspects of Feynman Path Integrals, Divergences, Quantum Fields and Diagrams, and Some More General Reflections

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Abstract

Feynman path integrals are first presented for the case of non-relativistic quantum mechanics, both in physical and mathematical terms. Then the case of scalar relativistic and Euclidean quantum fields is discussed, with a particular consideration of the mathematical problems arising when discussing interactions as non-linear functionals of the fields. The methods of (constructive) perturbation theory and renormalization theory in relation to Feynman path integrals are briefly discussed, in particular mentioning the visual help provided by Feynman diagrams. The paper ends with mentioning some open problems and presenting some philosophical remarks and reflections on the description of natural phenomena, in particular those of fundamental physics, in mathematical terms.

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Acknowledgements

I would like to thank all organizers, and in particular Luciano Boi for having given me the opportunity and challenge to speak to a truly interdisciplinary and philosophically oriented audience. The financial and logistic support of the Collège International de Philosophie, Paris, is also gratefully acknowledged, as well as the technical support by Timo Weiß, University of Bonn.

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  1. Institute for Applied Mathematics and Hausdorff Center for Mathematics, University of Bonn, Bonn, Germany

    Sergio Albeverio

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  1. Sergio Albeverio
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Correspondence to Sergio Albeverio .

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  1. Centre de Mathématiques, Ecole des Hautes Etudes en Sciences Sociales, Paris, France

    Luciano Boi

  2. Centre Gilles-Gaston Granger, Université d’Aix-Marseille, Aix & Marseille, France

    Carlos Lobo

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Albeverio, S. (2022). Mathematical Aspects of Feynman Path Integrals, Divergences, Quantum Fields and Diagrams, and Some More General Reflections. In: Boi, L., Lobo, C. (eds) When Form Becomes Substance. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-83125-7_9

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