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| {{hatnote|You may also be looking for [[functional integration (neurobiology)]].}}
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| '''Functional integration''' is a collection of results in [[mathematics]] and [[physics]] where the [[domain (mathematics)|domain]] of an [[integral]] is no longer a [[manifold|region of space]], but a [[Function space|space of functions]]. Functional integrals arise in [[probability]], in the study of [[partial differential equations]], and in the [[path integral formulation|Feynman approach]] to the [[quantum mechanics]] of particles and fields.
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| In an [[Lebesgue integration|ordinary integral]] there is a function to be integrated—the integrand—and a region of space over which to integrate the function—the domain of integration. The process of integration consists of adding up the values of the integrand for each point of the domain of integration. Making this procedure rigorous requires a limiting procedure, where the domain of integration is divided into smaller and smaller regions. For each small region the value of the integrand cannot vary much so it may be replaced by a single value. In a functional integral the domain of integration is a space of functions. For each function the integrand returns a value to add up. Making this procedure rigorous poses challenges that continue to be topics of current research.
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| Functional integration was developed by [[Percy John Daniell|P. J. Daniell]] in a paper of 1919<ref>{{Cite journal
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| | volume = 20
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| | issue = 4
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| | pages = 281–288
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| | last = Daniell
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| | first = P. J.
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| | title = Integrals in An Infinite Number of Dimensions
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| | journal = The Annals of Mathematics
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| | series = Second Series| date = July 1919
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| | jstor = 1967122
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| | doi = 10.2307/1967122
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| }}</ref> and [[Norbert Wiener|N. Wiener]] in a series of studies culminating in his papers 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|R. 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.
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| 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 [[standard model]] of particle physics.
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| ==Functional Integration==
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| {{Confusing|section|date=January 2014}}
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| Whereas normal integration sums a function, f(x), over a continuous range of values of x, functional integration sums a [[functional (mathematics)|functional]], G[f], over a continuous range of functions, f. Most functional integrals cannot be evaluated exactly but must be evaluated using [[perturbation methods]]. The formal definition of a functional integral is:
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| :<math>
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| \int{ G[f] [Df] } \equiv \int\limits_{-\infty}^\infty{ ... \int\limits_{-\infty}^\infty{ G[f] } }\prod_x df(x)
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| </math>
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| However in most cases the functions f(x) can be written in terms of an infinite series of orthogonal functions such as <math>f(x) = f_n H_n(x)</math> and then the definition becomes:
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| :<math>
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| \int{ G[f] [Df] } \equiv \int\limits_{-\infty}^\infty{ ... \int\limits_{-\infty}^\infty{ G(f_1,f_2,..) } }\prod_n df_n
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| </math>
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| which is slightly more understandable. The integral is shown to be a functional integral with a capital D. Sometimes it is written in square brackets [Df] or D[f] to indicate f is a function.
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| ==Examples==
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| Most functional integrals are actually infinite but the [[quotient]] of two functional integrals can be finite. The functional integrals that can be solved exactly usually start with the following [[Gaussian integral]]:
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| :<math>
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| \frac{
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| \int{ e^{i \int{ -\frac{1}{2}f(x) \cdot K(x,y) \cdot f(y) dxdy} + \int{ J(x) \cdot f(x) dx} }[Df] }
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| }
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| {
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| \int{ e^{i \int{ -\frac{1}{2}f(x) \cdot K(x,y) \cdot f(y) dxdy} } [Df] }
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| }
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| =
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| e^{i \frac{1}{2}\int{ J(x) \cdot K^{-1}(x,y) \cdot J(y) dxdy } }
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| </math>
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| By functionally differentiating this with respect to J(x) and then setting J to 0 this becomes an exponential multiplied by a polynomial in f. For example setting <math>K(x,y)=\Box\delta(x-y)</math> we find:
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| :<math> | |
| \frac{
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| \int{ f(a) f(b) e^{i \int{ f(x) \Box f(x) dx^4}} }[Df]
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| }{
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| \int{ e^{i \int{ f(x) \Box f(x) dx^4}} }[Df]
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| }
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| = K^{-1}(a,b) = \frac{1}{|a-b|^2}
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| </math>
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| where a,b and x are 4-dimensional vectors. This comes from the formula for the propagation of a photon in quantum electrodynamics. Another useful integral is the functional delta function:
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| :<math>
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| \int{ e^{i \int{ f(x) g(x) dx}} }[Df] = \delta[g] = \prod_x\delta( g(x) )
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| </math>
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| which is useful to specify constraints. Functional integrals can also be done over [[Grassmann number|Grassmann-valued]] functions <math>\psi(x)</math> where <math>\psi(x)\psi(y)=-\psi(y)\psi(x)</math> which is useful in quantum electrodynamics for calculations involving [[fermions]].
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| ==In symbolic algebra software==
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| Most symbolic algebra packages such as Maple or Mathematica do not support functional (path) integration as standard although additional packages can be constructed for them.
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| ==Approaches to path integrals==
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| {{expand section|date=October 2009}}
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| Functional integrals where the space of integration consists of paths (''ν'' = 1) can be defined in many different ways. The definitions fall in two different classes: the constructions derived from [[Wiener process|Wiener's theory]] yield an integral based on a [[Measure (mathematics)|measure]]; whereas the constructions following Feynman's path integral do not. Even within these two broad divisions, the integrals are not identical, that is, they are defined for different classes of functions.
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| ===The Wiener integral===
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| In the [[Wiener process|Wiener integral]] a probability is assigned to a class of [[Brownian motion]] paths. The class consists of the paths ''w'' that are known to go through a small region of space at a given time. The passage through different regions of space is assumed independent of each other and the distance between any two points of the Brownian path is assumed to be [[Normal distribution|Gaussian distributed]] with a [[variance]] that depends on the time ''t'' and on a diffusion constant ''D'':
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| :<math> \mathrm{Pr}(w(s+t),t|w(s),s) = \frac{1}{\sqrt{2\pi D t}} \exp\left({-\frac{\|w(s+t) - w(s)\|^2}{2Dt}} \right)</math>
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| The probability for the class of paths can be found by multiplying the probabilities of starting in one region and then being at the next. The Wiener measure can be developed by considering the limit of many small regions.
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| * Itō and Stratonovich calculus
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| ===The Feynman integral===
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| * Trotter formula
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| * The Kac idea of Wick rotations.
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| * Using x-dot-dot-squared or i S[x] + x-dot-squared.
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| * The Cartier DeWitt-Morette relies on integrators rather than measures
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| ===The Lévy integral===
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| * [[Fractional quantum mechanics]]
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| * [[Fractional Schrödinger equation]]
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| * [[Lévy process]]
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| * [[Fractional statistical mechanics]]
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| ==See also==
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| *[[Feynman path integral]]
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| *[[Partition function (quantum field theory)]]
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| *[[Saddle point approximation]]
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| ==References==
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| {{Reflist}}
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| ==Further reading==
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| * [[Hagen Kleinert|Kleinert, Hagen]], ''Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets'', 4th edition, World Scientific (Singapore, 2004); Paperback ISBN 981-238-107-4 '' (also available online: [http://www.physik.fu-berlin.de/~kleinert/b5 PDF-files])''
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| *{{ cite journal|author-link=Nick Laskin|arxiv=0811.1769|doi=10.1103/PhysRevE.62.3135|title=Fractional quantum mechanics|year=2000|last1=Laskin|first1=Nick|journal=Physical Review E|volume=62|issue=3|pages=3135|bibcode = 2000PhRvE..62.3135L }}
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| *{{ cite journal|author-link=Nick Laskin|arxiv=quant-ph/0206098 |doi=10.1103/PhysRevE.66.056108|title=Fractional Schrödinger equation|year=2002|last1=Laskin|first1=Nick|journal=Physical Review E|volume=66|issue=5|bibcode = 2002PhRvE..66e6108L }}
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| * O.G. Smolyanov, E.T.Shavgulidze. Сontinual integrals. Moscow, Moscow State University Press, 1990. (in Russian). http://lib.mexmat.ru/books/5132
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| [[Category:Integral calculus]]
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| [[Category:Functional analysis]]
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| [[Category:Mathematical physics]]
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| [[Category:Quantum mechanics]]
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| [[Category:Quantum field theory]]
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| [[de:Pfadintegral]]
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| [[pt:Integração funcional]]
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Jerrie Swoboda is what the public can call me and as well , I totally dig where it name. Managing people is my time job now. As a girl what The way we wish like is to have croquet but I still can't make it my careers really. My hubby and I chose to live a life in Massachusetts. Go to my web site to find out more: http://prometeu.net
Take a look at my website; hack Clash Of clans ifunbox