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A Time–Frequency Analysis Perspective on Feynman Path Integrals

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Landscapes of Time-Frequency Analysis

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

Abstract

The purpose of this expository paper is to highlight the starring role of techniques from time–frequency analysis in some recent contributions concerning the mathematical theory of Feynman path integrals. We hope to draw the interest of mathematicians working in time–frequency analysis on this topic, as well as to illustrate the benefits of this fruitful interplay for people working on path integrals.

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Notes

  1. 1.

    We remark that in physics literature the term “propagator” is usually reserved to the integral kernel u t of U(t), see below. This may possibly lead to confusion since it is in conflict with the traditional nomenclature adopted in the analysis of PDEs.

  2. 2.

    We set \(\mathcal {Q}_k\) for the Fourier multiplier whose symbol is a suitably smoothed version of the characteristic function of the dyadic annulus \(R_k = \{\xi \in \mathbb {R}^d \, : \, 2^{k-1} \le |\xi | < 2^k \}\), \(k \in \mathbb {N}\); we also define R 0 to be the unit ball.

  3. 3.

    We denote by \(C^0_b(\mathbb {R},X)\) the space of continuous and bounded functions \(\mathbb {R}\to X\).

  4. 4.

    To be precise, the result provided here concerns conditions under which the embedding \(M_{s}^{p,q}\cdot M_{s}^{p,q}\hookrightarrow M_{s}^{p,q}\) is continuous; this means that the algebra property eventually holds up to a constant. It is well known that one may provide an equivalent norm for which the boundedness estimate holds with C = 1 (cf. [61, Thm. 10.2]). This condition will be tacitly assumed whenever concerned with Banach algebras from now on.

  5. 5.

    The factor 2π derives from the normalization of the Fourier transform adopted in this section.

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Authors and Affiliations

  1. Dipartimento di Scienze Matematiche (DISMA) “G. L. Lagrange”, Politecnico di Torino, Torino, Italy

    S. Ivan Trapasso

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  1. S. Ivan Trapasso
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Correspondence to S. Ivan Trapasso .

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Editors and Affiliations

  1. Department of Mathematics, University of Turin, Torino, Italy

    Paolo Boggiatto

  2. Department of Mathematics, Ghent University, Ghent, Belgium

    Tommaso Bruno

  3. Department of Mathematics, University of Turin, Torino, Italy

    Elena Cordero

  4. Institute of Mathematics, University of Vienna, Wien, Austria

    Hans G. Feichtinger

  5. Department of Mathematical Sciences, Polytechnic University of Turin, Torino, Italy

    Fabio Nicola

  6. Department of Mathematics, University of Turin, Torino, Italy

    Alessandro Oliaro

  7. Department of Mathematical Sciences, Polytechnic University of Turin, Torino, Italy

    Anita Tabacco

  8. Department of Mathematical Sciences, Polytechnic University of Turin, Torino, Italy

    Maria Vallarino

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Trapasso, S.I. (2020). A Time–Frequency Analysis Perspective on Feynman Path Integrals. In: Boggiatto, P., et al. Landscapes of Time-Frequency Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-56005-8_10

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