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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Functional integration</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="Functional_integration_(neurobiology)" title="Functional integration (neurobiology)">functional integration (neurobiology)</a>.</div>

<p><b>Functional integration</b> is a collection of results in <a href="Mathematics" title="Mathematics">mathematics</a> and <a href="Physics" title="Physics">physics</a> where the domain of an <a href="Integral" title="Integral">integral</a> is no longer a region of space, but a <a href="Function_space" title="Function space">space of functions</a>. Functional integrals arise in <a href="Probability" title="Probability">probability</a>, in the study of <a href="Partial_differential_equations" class="mw-redirect" title="Partial differential equations">partial differential equations</a>, and in the <a href="Path_integral_formulation" title="Path integral formulation">path integral approach</a> to the <a href="Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> of particles and fields.

</p><p>In an ordinary integral (in the sense of <a href="Lebesgue_integration" class="mw-redirect" title="Lebesgue integration">Lebesgue integration</a>) 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.

</p><p>Functional integration was developed by <a href="Percy_John_Daniell" title="Percy John Daniell">Percy John Daniell</a> in an article of 1919<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> and <a href="Norbert_Wiener" title="Norbert Wiener">Norbert Wiener</a> in a series of studies culminating in his articles of 1921 on <a href="Brownian_motion" title="Brownian motion">Brownian motion</a>. They developed a rigorous method (now known as the <a href="Wiener_measure" class="mw-redirect" title="Wiener measure">Wiener measure</a>) for assigning a probability to a particle's random path. <a href="Richard_Feynman" title="Richard Feynman">Richard Feynman</a> developed another functional integral, the <a href="Path_integral_formulation" title="Path integral formulation">path integral</a>, 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.

</p><p>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 <a href="Quantum_electrodynamics" title="Quantum electrodynamics">quantum electrodynamics</a> and the <a href="Standard_model" class="mw-redirect" title="Standard model">standard model</a> of particle physics.

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<div class="mw-heading mw-heading2"><h2 id="Functional_integration">Functional integration</h2></div>

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<p>Whereas standard <a href="Riemann_integral" title="Riemann integral">Riemann integration</a> sums a function <i>f</i>(<i>x</i>) over a continuous range of values of <i>x</i>, functional integration sums a <a href="Functional_(mathematics)" title="Functional (mathematics)">functional</a> <i>G</i>[<i>f</i>], which can be thought of as a "function of a function" over a continuous range (or space) of functions <i>f</i>. Most functional integrals cannot be evaluated exactly but must be evaluated using <a href="Perturbation_methods" class="mw-redirect" title="Perturbation methods">perturbation methods</a>. The formal definition of a functional integral is

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<annotation encoding="application/x-tex">{\displaystyle \int G[f]\;{\mathcal {D}}[f]\equiv \int _{\mathbb {R} }\cdots \int _{\mathbb {R} }G[f]\prod _{x}df(x)\;.}</annotation>

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</p><p>However, in most cases the functions <i>f</i>(<i>x</i>) can be written in terms of an infinite series of <a href="Orthogonal_functions" title="Orthogonal functions">orthogonal functions</a> such as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=f_{n}H_{n}(x)}">

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<annotation encoding="application/x-tex">{\displaystyle f(x)=f_{n}H_{n}(x)}</annotation>

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</math></span><img src="./2f3b8f1c2aab0348d2bc4b8ad40e71793d9e6640.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.162ex; height:2.843ex;" alt="{\displaystyle f(x)=f_{n}H_{n}(x)}" loading="lazy"></span>, and then the definition becomes

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<annotation encoding="application/x-tex">{\displaystyle \int G[f]\;{\mathcal {D}}[f]\equiv \int _{\mathbb {R} }\cdots \int _{\mathbb {R} }G(f_{1};f_{2};\ldots )\prod _{n}df_{n}\;,}</annotation>

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</p><p>which is slightly more understandable. The integral is shown to be a functional integral with a capital <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {D}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}}</annotation>

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</math></span><img src="./3277962e1959c3241fb1b70c7f0ac6dcefebd966.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.792ex; height:2.176ex;" alt="{\displaystyle {\mathcal {D}}}" loading="lazy"></span>. Sometimes the argument is written in square brackets <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {D}}[f]}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}[f]}</annotation>

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<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

<p>Most functional integrals are actually infinite, but often the limit of the <a href="Quotient" title="Quotient">quotient</a> of two related functional integrals can still be finite. The functional integrals that can be evaluated exactly usually start with the following <a href="Gaussian_integral" title="Gaussian integral">Gaussian integral</a>:

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\displaystyle \int \exp \left\lbrace -{\frac {1}{2}}\int _{\mathbb {R} }\left[\int _{\mathbb {R} }f(x)K(x;y)f(y)\,dy+J(x)f(x)\right]dx\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]}}=\exp \left\lbrace {\frac {1}{2}}\int _{\mathbb {R} ^{2}}J(x)\cdot K^{-1}(x;y)\cdot J(y)\,dx\,dy\right\rbrace \,,}</annotation>

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</math></span><img src="./8e84b5e590181838f5dc77809e9c36c8a9bb98a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:106.618ex; height:12.843ex;" alt="{\displaystyle {\frac {\displaystyle \int \exp \left\lbrace -{\frac {1}{2}}\int _{\mathbb {R} }\left[\int _{\mathbb {R} }f(x)K(x;y)f(y)\,dy+J(x)f(x)\right]dx\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]}}=\exp \left\lbrace {\frac {1}{2}}\int _{\mathbb {R} ^{2}}J(x)\cdot K^{-1}(x;y)\cdot J(y)\,dx\,dy\right\rbrace \,,}" loading="lazy"></span></dd></dl>

<p>in which <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(x;y)=K(y;x)}">

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<annotation encoding="application/x-tex">{\displaystyle K(x;y)=K(y;x)}</annotation>

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</math></span><img src="./bc10da26d6738aedefcb2a66d3573eb3df3bc83f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.887ex; height:2.843ex;" alt="{\displaystyle K(x;y)=K(y;x)}" loading="lazy"></span>. By functionally differentiating this with respect to <i>J</i>(<i>x</i>) and then setting to 0 this becomes an exponential multiplied by a monomial in <i>f</i>. To see this, let's use the following notation:

</p><p><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G[f,J]=-{\frac {1}{2}}\int _{\mathbb {R} }\left[\int _{\mathbb {R} }f(x)K(x;y)f(y)\,dy+J(x)f(x)\right]dx\,\quad ,\quad W[J]=\int \exp \lbrace G[f,J]\rbrace {\mathcal {D}}[f]\;.}">

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<annotation encoding="application/x-tex">{\displaystyle G[f,J]=-{\frac {1}{2}}\int _{\mathbb {R} }\left[\int _{\mathbb {R} }f(x)K(x;y)f(y)\,dy+J(x)f(x)\right]dx\,\quad ,\quad W[J]=\int \exp \lbrace G[f,J]\rbrace {\mathcal {D}}[f]\;.}</annotation>

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</math></span><img src="./b3250c38a2dbf325177e93e266da0b4903c8ef3a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:91.847ex; height:6.176ex;" alt="{\displaystyle G[f,J]=-{\frac {1}{2}}\int _{\mathbb {R} }\left[\int _{\mathbb {R} }f(x)K(x;y)f(y)\,dy+J(x)f(x)\right]dx\,\quad ,\quad W[J]=\int \exp \lbrace G[f,J]\rbrace {\mathcal {D}}[f]\;.}" loading="lazy"></span>

</p><p>With this notation the first equation can be written as:

</p><p><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dfrac {W[J]}{W[0]}}=\exp \left\lbrace {\frac {1}{2}}\int _{\mathbb {R} ^{2}}J(x)K^{-1}(x;y)J(y)\,dx\,dy\right\rbrace .}">

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<annotation encoding="application/x-tex">{\displaystyle {\dfrac {W[J]}{W[0]}}=\exp \left\lbrace {\frac {1}{2}}\int _{\mathbb {R} ^{2}}J(x)K^{-1}(x;y)J(y)\,dx\,dy\right\rbrace .}</annotation>

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</math></span><img src="./fd1e44f6afd739751867dcace98340f4a7d3bce2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:47.629ex; height:6.509ex;" alt="{\displaystyle {\dfrac {W[J]}{W[0]}}=\exp \left\lbrace {\frac {1}{2}}\int _{\mathbb {R} ^{2}}J(x)K^{-1}(x;y)J(y)\,dx\,dy\right\rbrace .}" loading="lazy"></span>

</p><p>Now, taking functional derivatives to the definition of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W[J]}">

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</math></span><img src="./29ebdc65f8e1fc5204380a81951ea53868aca978.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.2ex; height:2.843ex;" alt="{\displaystyle W[J]}" loading="lazy"></span> and then evaluating in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J=0}">

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</p><p><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dfrac {\delta }{\delta J(a)}}W[J]{\Bigg |}_{J=0}=\int f(a)\exp \lbrace G[f,0]\rbrace {\mathcal {D}}[f]\;,}">

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<annotation encoding="application/x-tex">{\displaystyle {\dfrac {\delta }{\delta J(a)}}W[J]{\Bigg |}_{J=0}=\int f(a)\exp \lbrace G[f,0]\rbrace {\mathcal {D}}[f]\;,}</annotation>

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</p><p><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dfrac {\delta ^{2}W[J]}{\delta J(a)\delta J(b)}}{\Bigg |}_{J=0}=\int f(a)f(b)\exp \lbrace G[f,0]\rbrace {\mathcal {D}}[f]\;,}">

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<annotation encoding="application/x-tex">{\displaystyle {\dfrac {\delta ^{2}W[J]}{\delta J(a)\delta J(b)}}{\Bigg |}_{J=0}=\int f(a)f(b)\exp \lbrace G[f,0]\rbrace {\mathcal {D}}[f]\;,}</annotation>

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</math></span><img src="./f266d3a38180e28f148c61150a391f56df677b15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:48.34ex; height:6.676ex;" alt="{\displaystyle {\dfrac {\delta ^{2}W[J]}{\delta J(a)\delta J(b)}}{\Bigg |}_{J=0}=\int f(a)f(b)\exp \lbrace G[f,0]\rbrace {\mathcal {D}}[f]\;,}" loading="lazy"></span>

</p><p><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \qquad \qquad \qquad \qquad \vdots }">

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<annotation encoding="application/x-tex">{\displaystyle \qquad \qquad \qquad \qquad \vdots }</annotation>

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</p><p>which is the result anticipated. More over, by using the first equation one arrives to the useful result:

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<annotation encoding="application/x-tex">{\displaystyle {\dfrac {\delta ^{2}}{\delta J(a)\delta J(b)}}\left({\dfrac {W[J]}{W[0]}}\right){\Bigg |}_{J=0}=K^{-1}(a;b)\;;}</annotation>

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</math></span><img src="./fc48127d61d29e93f3e4c1cd51992459dadbeced.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:39.862ex; height:6.509ex;" alt="{\displaystyle {\dfrac {\delta ^{2}}{\delta J(a)\delta J(b)}}\left({\dfrac {W[J]}{W[0]}}\right){\Bigg |}_{J=0}=K^{-1}(a;b)\;;}" loading="lazy"></span></dd></dl>

<p>Putting these results together and backing to the original notation we have:

</p><p><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\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)\,.}">

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<annotation encoding="application/x-tex">{\displaystyle {\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)\,.}</annotation>

</semantics>

</math></span><img src="./844d11a6d3dd609814e9635695d69f459c86db1d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:67.11ex; height:12.843ex;" alt="{\displaystyle {\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)\,.}" loading="lazy"></span>

</p><p>Another useful integral is the functional <a href="Delta_function" class="mw-redirect" title="Delta function">delta function</a>:

</p>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</mrow>

</munder>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int \exp \left\lbrace \int _{\mathbb {R} }f(x)g(x)dx\right\rbrace {\mathcal {D}}[f]=\delta [g]=\prod _{x}\delta {\big (}g(x){\big )},}</annotation>

</semantics>

</math></span><img src="./76d31aaa6c9f8d92eb119a39321eb6ac89ccb9f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:49.392ex; height:6.676ex;" alt="{\displaystyle \int \exp \left\lbrace \int _{\mathbb {R} }f(x)g(x)dx\right\rbrace {\mathcal {D}}[f]=\delta [g]=\prod _{x}\delta {\big (}g(x){\big )},}" loading="lazy"></span></dd></dl>

<p>which is useful to specify constraints. Functional integrals can also be done over <a href="Grassmann_number" title="Grassmann number">Grassmann-valued</a> functions <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi (x)}</annotation>

</semantics>

</math></span><img src="./a596a1fb4130a47f6b88c66150497338bd6cbccc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.652ex; height:2.843ex;" alt="{\displaystyle \psi (x)}" loading="lazy"></span>, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (x)\psi (y)=-\psi (y)\psi (x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi (x)\psi (y)=-\psi (y)\psi (x)}</annotation>

</semantics>

</math></span><img src="./c3aca22d6fb1d05db38c37b9e86fa4e456a094b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.167ex; height:2.843ex;" alt="{\displaystyle \psi (x)\psi (y)=-\psi (y)\psi (x)}" loading="lazy"></span>, which is useful in quantum electrodynamics for calculations involving <a href="Fermions" class="mw-redirect" title="Fermions">fermions</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Approaches_to_path_integrals">Approaches to path integrals</h2></div>

<p>Functional integrals where the space of integration consists of paths (<i>ν</i> = 1) can be defined in many different ways. The definitions fall in two different classes: the constructions derived from <a href="Wiener_process" title="Wiener process">Wiener's theory</a> yield an integral based on a <a href="Measure_(mathematics)" title="Measure (mathematics)">measure</a>, 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 differently for different classes of functions.

</p>

<div class="mw-heading mw-heading3"><h3 id="The_Wiener_integral">The Wiener integral</h3></div>

<p>In the <a href="Wiener_process" title="Wiener process">Wiener integral</a>, a probability is assigned to a class of <a href="Brownian_motion" title="Brownian motion">Brownian motion</a> paths. The class consists of the paths <i>w</i> 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 <a href="Normal_distribution" title="Normal distribution">Gaussian-distributed</a> with a <a href="Variance" title="Variance">variance</a> that depends on the time <i>t</i> and on a diffusion constant <i>D</i>:

</p>

</mrow>

</mrow>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<mrow>

<mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Pr {\big (}w(s+t),t\mid w(s),s{\big )}={\frac {1}{\sqrt {2\pi Dt}}}\exp \left(-{\frac {\|w(s+t)-w(s)\|^{2}}{2Dt}}\right).}</annotation>

</semantics>

</math></span><img src="./0342290dcf0b8567742461150a146b9065c434b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:64.822ex; height:7.509ex;" alt="{\displaystyle \Pr {\big (}w(s+t),t\mid w(s),s{\big )}={\frac {1}{\sqrt {2\pi Dt}}}\exp \left(-{\frac {\|w(s+t)-w(s)\|^{2}}{2Dt}}\right).}" loading="lazy"></span></dd></dl>

<p>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.

</p>

<ul><li>Itō and Stratonovich calculus</li></ul>

<div class="mw-heading mw-heading3"><h3 id="The_Feynman_integral">The Feynman integral</h3></div>

<ul><li>Trotter formula, or <a href="Lie_product_formula" title="Lie product formula">Lie product formula</a>.</li>

<li>The Kac idea of Wick rotations.</li>

<li>Using x-dot-dot-squared or i S[x] + x-dot-squared.</li>

<li>The Cartier DeWitt–Morette relies on integrators rather than measures</li></ul>

<div class="mw-heading mw-heading3"><h3 id="The_Lévy_integral">The Lévy integral</h3></div>

<ul><li>Fractional quantum mechanics</li>

<li>Fractional Schrödinger equation</li>

<li><a href="L%C3%A9vy_process" title="Lévy process">Lévy process</a></li>

<li>Fractional statistical mechanics</li></ul>

<ul><li><a href="Path_integral_formulation" title="Path integral formulation">Feynman path integral</a></li>

<li><a href="Partition_function_(quantum_field_theory)" title="Partition function (quantum field theory)">Partition function (quantum field theory)</a></li>

<li><a href="Saddle_point_approximation" class="mw-redirect" title="Saddle point approximation">Saddle point approximation</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFDaniell1919" class="citation journal cs1">Daniell, P. J. (July 1919). "Integrals in An Infinite Number of Dimensions". <i>The Annals of Mathematics</i>. Second Series. <b>20</b> (4): <span class="nowrap">281–</span>288. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1967122">10.2307/1967122</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1967122">1967122</a>.</cite></span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://www.scholarpedia.org/Path_integral">Jean Zinn-Justin (2009), <i>Scholarpedia</i> <b>4</b>(2):8674</a>.</li>

<li><a href="Hagen_Kleinert" title="Hagen Kleinert">Kleinert, Hagen</a>, <i>Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets</i>, 4th edition, World Scientific (Singapore, 2004); Paperback <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>981-238-107-4</bdi> <i> (also available online: <a rel="nofollow" class="external text" href="http://www.physik.fu-berlin.de/~kleinert/b5">PDF-files</a>)</i></li>

<li><cite id="CITEREFLaskin2000" class="citation journal cs1">Laskin, Nick (2000). "Fractional quantum mechanics". <i>Physical Review E</i>. <b>62</b> (3): <span class="nowrap">3135–</span>3145. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0811.1769">0811.1769</a></span>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2000PhRvE..62.3135L">2000PhRvE..62.3135L</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevE.62.3135">10.1103/PhysRevE.62.3135</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/11088808">11088808</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:15480739">15480739</a>.</cite></li>

<li><cite id="CITEREFLaskin2002" class="citation journal cs1">Laskin, Nick (2002). "Fractional Schrödinger equation". <i>Physical Review E</i>. <b>66</b> (5): 056108. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0206098">quant-ph/0206098</a></span>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2002PhRvE..66e6108L">2002PhRvE..66e6108L</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevE.66.056108">10.1103/PhysRevE.66.056108</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/12513557">12513557</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:7520956">7520956</a>.</cite></li>

<li><cite id="CITEREFMinlos2001" class="citation cs2">Minlos, R. A. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Integral_over_trajectories">"Integral over trajectories"</a>, <i><a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite></li>

<li>O. G. Smolyanov, E. T. Shavgulidze. <i>Continual integrals</i>. Moscow, Moscow State University Press, 1990. (in Russian). <a rel="nofollow" class="external free" href="http://lib.mexmat.ru/books/5132">http://lib.mexmat.ru/books/5132</a></li>

<li><a href="Victor_Popov" title="Victor Popov">Victor Popov</a>, Functional Integrals in Quantum Field Theory and Statistical Physics, Springer 1983</li>

<li><a href="Sergio_Albeverio" title="Sergio Albeverio">Sergio Albeverio</a>, Sonia Mazzucchi, A unified approach to infinite-dimensional integration, Reviews in Mathematical Physics, 28, 1650005 (2016)</li>

<li><a href="John_R._Klauder" title="John R. Klauder">Klauder, John</a>. "<a rel="nofollow" class="external text" href="https://www.phys.ufl.edu/functional-integration/">Lectures on Functional Integration</a>." <i>University of Florida.</i> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20240708182058/http://www.phys.ufl.edu/functional-integration/">Archived</a> on July 8th, 2024.</li></ul></div><div><div class="zim-footer">

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