Functional integration: Difference between revisions

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}}</ref> and [[Norbert Wiener]] in a series of studies culminating in his articles of 1921 on [[Brownian motion]]. They developed a rigorous method (now known as the [[Wiener measure]]) for assigning a probability to a particle's random path. [[Richard Feynman]] developed another functional integral, the [[path integral formulation|path integral]], useful for computing the quantum properties of systems. In Feynman's path integral, the classical notion of a unique trajectory for a particle is replaced by an infinite sum of classical paths, each weighted differently according to its classical properties.

Functional integration is central to quantization techniques in theoretical physics. The algebraic properties of functional integrals are used to develop series used to calculate properties in [[quantum electrodynamics]] and the [[standardStandard modelModel]] of particle physics.

==Functional integration==

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<math>

\frac{\displaystyle\int f(a)f(b)\exp\left\lbrace-\frac{1}{2} \int_{\mathbb{R}^2} f(x) K(x,;y) f(y)\, dx\,dy\right\rbrace \mathcal{D}[f]}

{\displaystyle\int \exp\left\lbrace-\frac{1}{2} \int_{\mathbb{R}^2} f(x) K(x,;y) f(y) \,dx\,dy\right\rbrace \mathcal{D}[f]} =

K^{-1}(a;b)\,.

</math>

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*[[Victor Popov]], Functional Integrals in Quantum Field Theory and Statistical Physics, Springer 1983

*[[Sergio Albeverio]], Sonia Mazzucchi, A unified approach to infinite-dimensional integration, Reviews in Mathematical Physics, 28, 1650005 (2016)

*[[John R. Klauder|Klauder, John]]. "[https://www.phys.ufl.edu/functional-integration/ Lectures on Functional Integration]." ''University of Florida.'' [https://web.archive.org/web/20240708182058/http://www.phys.ufl.edu/functional-integration/ Archived] on July 8th, 2024.