Feynman path integral

A generalisation of multi-dimensional integral, written

$$\int 𝒟\varphi \exp \left(ℱ[\varphi ]\right)$$

where $\varphi$ ranges over some restricted set of functions from a measure space $X$ to some space with reasonably nice algebraic structure. The simplest example is the case where

$$\varphi \in L^2[X,ℝ]$$

and

$$F[\varphi ]=-\pi {\int }_X{\varphi }^2(x)𝑑\mu (x)$$

in which case it can be argued that the result is $1$. The argument is by analogy to the Gaussian integral${\int }_{{ℝ}^n}𝑑x_1\mathrm{\cdots }𝑑x_n{e}^{-\pi \sum x_j^2}\equiv 1$. Alas, one can absorb the $\pi$ into the measure on $X$.Alternatively, following Pierre Cartier and others, one can use this analogy to define a measure on $L^2$and proceed axiomatically.

One can bravely trudge onward and hope to come up with something, say à la Riemann integral, by partitioning $X$,picking some representative of each partition, approximating the functional $F$ based on theseand calculating a multi-dimensional integral as usual over the sample values of $\varphi$. This leads to some integral

$$\int \mathrm{\cdots }𝑑\varphi (x_1)\mathrm{\cdots }𝑑\varphi (x_n)e^{f(\varphi (x_1),\mathrm{\dots },\varphi (x_n))}.$$

One hopes that taking successively finer partitions of $X$ will give a sequence of integrals which converge on some nice limit. I believe Pierre Cartier has shown that this doesn’t usually happen, except for the trivial kind of example given above.

The Feynman path integral was constructed as part of a re-formulation of by Richard Feynman, based on the sum-over-histories postulate of quantum mechanics, and can be thought of as an adaptation of Green’s function methods for solving initial/boundary value problems. No appropriate measure has been found for this integral and attempts at pseudomeasures have given mixed results.

Remark:Note however that in solving quantum field theory problems one attacks the problem in the Feynman approach by‘dividing’ it via Feynman diagrams that are directly related to specific quantum interactions;adding the contributions from such Feynman diagrams leads to high precision approximations to the final physicalsolution which is finite and physically meaningful, or observable.