# Feynman path integral

A generalisation of multi-dimensional integral, written

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

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

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

and

$$F[\varphi ]=-\pi {\int}_{X}{\varphi}^{2}(x)\mathit{d}\mu (x)$$ |

in which case it can be argued that the result is $1$. The argument^{} is by analogy^{} to the Gaussian integral
${\int}_{{\mathbb{R}}^{n}}\mathit{d}{x}_{1}\mathrm{\cdots}\mathit{d}{x}_{n}{e}^{-\pi {\scriptscriptstyle \sum {x}_{j}^{2}}}\equiv 1$. 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 $\varphi $. This leads to some integral

$$\int \mathrm{\cdots}\mathit{d}\varphi ({x}_{1})\mathrm{\cdots}\mathit{d}\varphi ({x}_{n}){e}^{f(\varphi ({x}_{1}),\mathrm{\dots},\varphi ({x}_{n}))}.$$ |

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 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 |
---|---|

Canonical name | FeynmanPathIntegral |

Date of creation | 2013-03-22 12:41:45 |

Last modified on | 2013-03-22 12:41:45 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 19 |

Author | PrimeFan (13766) |

Entry type | Definition |

Classification | msc 81S40 |

Related topic | LpSpace |

Related topic | RichardFeynman |