How to (path-) integrate by differentiating

7th International Workshop DICE2014 Spacetime – Matter – Quantum Mechanics

物理学;力学

Kempf, Achim^1 ; Jackson, David M.^2 ; Morales, Alejandro H.^3

Departments of Applied Mathematics and Physics, University of Waterloo, ON

N2L 3G1, Canada^1

Department of Combinatorics and Optimization, University of Waterloo, ON

N2L 3G1, Canada^2

Department of Mathematics, UCLA, Los Angeles

CA

90095, United States^3

关键词: Fourier;    Functional integration;    Integral transform;    New results;    Path integral;    Quantum field theory;    Variational problems;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/626/1/012015/pdf
DOI  :  10.1088/1742-6596/626/1/012015

学科分类：力学，机械学

来源: IOP

【 摘 要 】

Path integrals are at the heart of quantum field theory. In spite of their covariance and seeming simplicity, they are hard to define and evaluate. In contrast, functional differentiation, as it is used, for example, in variational problems, is relatively straightforward. This has motivated the development of new techniques that allow one to express functional integration in terms of functional differentiation. In fact, the new techniques allow one to express integrals in general through differentiation. These techniques therefore add to the general toolbox for integration and for integral transforms such as the Fourier and Laplace transforms. Here, we review some of these results, we give simpler proofs and we add new results, for example, on expressing the Laplace transform and its inverse in terms of derivatives, results that may be of use in quantum field theory, e.g., in the context of heat traces.