The perturbative approach to path integrals: a succinct mathematical treatment (Q2825549)

The paper attempts to give an elucidating approach to Feynman's path integral, which -- especially in Quantum Field Theory -- lacks a proper mathematical definition. The main core of this work is about proving several quantities, statements and theorems (e.g. Wick's theorem) that are frequently used in applied calculations within Quantum Field Theory. For that, we have to deal with the \textit{finite dimensional} (in contrast to the usually infinite dimensions of quantum mechanics) integral NEWLINE\[NEWLINEI=\int d^d x f(x) e^{-S(x)/\hbar},NEWLINE\]NEWLINE where \(S(x)\) is seen to be the action and \(f(x)\) some observable. The parameter \(\hbar\) is explicitly kept as expansion parameter, where \(\hbar\to 0\) reproduces the semi-classical limit. In the course of the paper, Wick's Theorem for this class of integrals is introduced and proven that the Wick expansion around a critical point is invariant under coordinate transformations. Later on it is proven that the Wick expansion is also gauge invariant, which is an important point for the application to actual problems in physics. The connection to these is finally approached in Section V with remarks on Quantum Field Theory. Here, a ``Formal Path Integration Principle'' is stated: ``Any formal manipulation of a perturbatively defined path integral yields provisional identities whose legitimacy depend upon an analysis of the regularization and renormalization scheme employed.'' These identities are known as Schwinger-Dyson equations, Ward identities and Slavnov-Taylor identities. To prove this identities, one has again to prove some invariance conditions of the path integral as generalization of the integral from above, NEWLINE\[NEWLINEI=\frac{1}{Z}\int D\phi\; O(\phi) e^{-S(\phi)/\hbar},NEWLINE\]NEWLINE that are explicitly worked out for the mentioned identities (by field transformation, gauge fixing, BRST transformations).