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# 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 quantum field theory 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

## Mathematics Subject Classification

81S40*no label found*

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## Corrections

broken by yark ✓

Typo by bbukh ✓

capitalization by Mathprof ✓

what is the integral over? by Mathprof ✓