Feynman path integral

A generalisation of multi-dimensional integral, written

 $\int\mathcal{D}\phi\;\mathrm{exp}\left(\mathcal{F}[\phi]\right)$

where $\phi$ 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

 $\phi\in L^{2}[X,\mathbb{R}]$

and

 $F[\phi]=-\pi\int_{X}\phi^{2}(x)d\mu(x)$

in which case it can be argued that the result is $1$. The argument is by analogy to the Gaussian integral $\int_{\mathbb{R}^{n}}dx_{1}\cdots dx_{n}e^{-\pi\sum x_{j}^{2}}\equiv 1$. 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 these and calculating a multi-dimensional integral as usual over the sample values of $\phi$. This leads to some integral

 $\int\cdots d\phi(x_{1})\cdots d\phi(x_{n})e^{f(\phi(x_{1}),\ldots,\phi(x_{n}))}.$

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 physical solution which is finite and physically meaningful, or observable.

References

• 1 Hui-Hsiung Kuo, Introduction to Stochastic Integration. New York: Springer (2006): 250 - 253
• 2 J. B. Keller & D. W. McLaughlin, “The Feynman Integral” Amer. Math. Monthly 82 5 (1975): 451 - 465
Title Feynman path integral FeynmanPathIntegral 2013-03-22 12:41:45 2013-03-22 12:41:45 PrimeFan (13766) PrimeFan (13766) 19 PrimeFan (13766) Definition msc 81S40 LpSpace RichardFeynman