How to (path) integrate by differentiating
Abstract
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.
 Publication:

Journal of Physics Conference Series
 Pub Date:
 July 2015
 DOI:
 10.1088/17426596/626/1/012015
 arXiv:
 arXiv:1507.04348
 Bibcode:
 2015JPhCS.626a2015K
 Keywords:

 Mathematical Physics;
 High Energy Physics  Theory;
 Quantum Physics
 EPrint:
 Based on a presentation given by AK at the 7th International Workshop DICE2014 in Castiglioncello, Italy, September 1519, 2014