Feynman path integralFeynmannPathIntegral2002-05-31 12:30:472008-10-17 23:05:39DefinitionFeynmanpath integralquantum field theoryFeynman diagrams\usepackage{amssymb}
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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
\[
\phi \in L^2[X,\RR]
\]
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
$\int_{\RR^n} dx_1\cdots dx_n e^{-\pi\sum x_j^2} \equiv 1$. Alas, one can absorb the $\pi$ into the measure on $X$.
Alternatively, following Pierre Cartier and others, one can use this analogy to {\em define} a measure on $L^2$
and proceed axiomatically.
One can bravely trudge onward and hope to come up with something, say \`a 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
\[
\dotsint d\phi(x_1) \cdots d\phi(x_n) e^{f(\phi(x_1),\ldots,\phi(x_n))}.
\]
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 \PMlinkescapetext{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.
{\bf Remark:}
Note however that in solving quantum field theory problems one attacks the problem in the Feynman approach by
`dividing' it \emph{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.
\begin{thebibliography}{2}
\bibitem{hk} Hui-Hsiung Kuo, {\it Introduction to Stochastic Integration}. New York: Springer (2006): 250 - 253
\bibitem{jk} J. B. Keller \& D. W. McLaughlin, ``The Feynman Integral'' {\it Amer. Math. Monthly} {\bf 82} 5 (1975): 451 - 465
\end{thebibliography}