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'''Common integrals in quantum field theory''' are all variations and generalizations of [[Gaussian integral|Gaussian integrals]] to the [[complex plane]] and to multiple dimensions.<ref name="Zee">{{cite book|author=A. Zee|title=Quantum Field Theory in a Nutshell|publisher=Princeton University|year=2003|isbn=0-691-01019-6}}</ref>{{rp|pp=13–15}} Other integrals can be approximated by versions of the Gaussian integral. Fourier integrals are also considered.
== Variations on a simple Gaussian integral ==
=== Gaussian integral ===
The first integral, with broad application outside of quantum field theory, is the Gaussian integral.
<math display="block"> G \equiv \int_{-\infty}^{\infty} e^{-{1 \over 2} x^2}\,dx</math>
In physics the factor of 1/2 in the argument of the exponential is common.
Note that, if we let <math>r=\sqrt{x^2+y^2}</math> be the radius, then we can use the usual polar coordinate change of variables (which in particular renders <math>dx\,dy=r\,dr\,d\theta</math>) to get
<math display="block"> G^2 = \left ( \int_{-\infty}^{\infty} e^{-{1 \over 2} x^2}\,dx \right ) \cdot \left ( \int_{-\infty}^{\infty} e^{-{1 \over 2} y^2}\,dy \right ) = 2\pi \int_{0}^{\infty} r e^{-{1 \over 2} r^2}\,dr = 2\pi \int_{0}^{\infty} e^{- w}\,dw = 2 \pi.</math>
Thus we obtain
<math display="block"> \int_{-\infty}^{\infty} e^{-{1 \over 2} x^2}\,dx = \sqrt{2\pi}. </math>
=== Slight generalization of the Gaussian integral ===
<math display="block"> \int_{-\infty}^{\infty} e^{-{1 \over 2} a x^2}\,dx = \sqrt{2\pi \over a} </math>
where we have scaled
<math display="block"> x \to {x \over \sqrt{a}}. </math>
=== Integrals of exponents and even powers of ''x'' ===
<math display="block"> \int_{-\infty}^{\infty} x^2 e^{-{1 \over 2} a x^2}\,dx = -2{d\over da} \int_{-\infty}^{\infty} e^{-{1 \over 2} a x^2}\,dx = -2{d\over da} \left ( {2\pi \over a } \right ) ^{1\over 2} = \left ( {2\pi \over a } \right ) ^{1\over 2} {1\over a}</math>
and
<math display="block"> \int_{-\infty}^{\infty} x^4 e^{-{1 \over 2} a x^2}\,dx = \left ( -2{d\over da} \right) \left ( -2{d\over da} \right) \int_{-\infty}^{\infty} e^{-{1 \over 2} a x^2}\,dx = \left ( -2{d\over da} \right) \left ( -2{d\over da} \right) \left ( {2\pi \over a } \right ) ^{1\over 2} = \left ( {2\pi \over a } \right ) ^{1\over 2} {3\over a^2}</math>
In general
<math display="block"> \int_{-\infty}^{\infty} x^{2n} e^{-{1 \over 2} a x^2}\,dx = \left ( {2\pi \over a } \right ) ^{1\over {2}} {1\over a^{n}} \left ( 2n -1 \right ) \left ( 2n -3 \right ) \cdots 5 \cdot 3 \cdot 1 = \left ( {2\pi \over a } \right ) ^{1\over {2}} {1\over a^{n}} \left ( 2n -1 \right )!! </math>
Note that the integrals of exponents and odd powers of x are 0, due to [[odd function|odd]] symmetry.
=== Integrals with a linear term in the argument of the exponent ===
This integral can be performed by [[completing the square]]:
<math display="block"> \left( -{1 \over 2} a x^2 + Jx\right ) = -{1 \over 2} a \left ( x^2 - { 2 Jx \over a } + { J^2 \over a^2 } - { J^2 \over a^2 } \right ) = -{1 \over 2} a \left ( x - { J \over a } \right )^2 + { J^2 \over 2a } </math>
Therefore:
<math display="block">\begin{align}
\int_{-\infty}^\infty \exp\left( -{1 \over 2} a x^2 + Jx\right) \, dx
&= \exp\left( { J^2 \over 2a } \right )\int_{-\infty}^\infty \exp\left( -{1 \over 2} a w^2 \right) \, dw \\[8pt]
&=
\end{align}</math>
=== Integrals with an imaginary linear term in the argument of the exponent ===
The integral
<math display="block"> \int_{-\infty}^{\infty} \exp\left( -{1 \over 2} a x^2 +iJx\right ) dx = \left ( {2\pi \over a } \right ) ^{1\over 2} \exp\left( -{ J^2 \over 2a }\right ) </math>
is proportional to the [[Fourier transform]] of the Gaussian where {{mvar|J}} is the [[conjugate variables|conjugate variable]] of {{mvar|x}}.
By again completing the square we see that the Fourier transform of a Gaussian is also a Gaussian, but in the conjugate variable. The larger {{mvar|a}} is, the narrower the Gaussian in {{mvar|x}} and the wider the Gaussian in {{mvar|J}}. This is a demonstration of the [[uncertainty principle]].
This integral is also known as the [[
=== Integrals with a complex argument of the exponent ===
The integral of interest is (for an example of an application see [[Relation between Schrödinger's equation and the path integral formulation of quantum mechanics]])
<math display="block"> \int_{-\infty}^{\infty} \exp\left( {1 \over 2} i a x^2 + iJx\right ) dx. </math>
We now assume that {{mvar|a}} and {{mvar|J}} may be complex.
Completing the square
<math display="block"> \left( {1 \over 2} i a x^2 + iJx\right ) = {1\over 2} ia \left ( x^2 + {2Jx \over a} + \left ( { J \over a} \right )^2 - \left ( { J \over a} \right )^2 \right ) = -{1\over 2} {a \over i} \left ( x + {J\over a} \right )^2 - { iJ^2 \over 2a}. </math>
By analogy with the previous integrals
<math display="block"> \int_{-\infty}^{\infty} \exp\left( {1 \over 2} i a x^2 + iJx\right ) dx = \left ( {2\pi i \over a } \right ) ^{1\over 2} \exp\left( { -iJ^2 \over 2a }\right ). </math>
This result is valid as an integration in the complex plane as long as {{mvar|a}} is non-zero and has a semi-positive imaginary part. See [[Fresnel integral]].
== Gaussian integrals in higher dimensions ==
The one-dimensional integrals can be generalized to multiple dimensions.<ref>{{cite book | author=Frederick W. Byron and Robert W. Fuller| title=Mathematics of Classical and Quantum Physics | publisher= Addison-Wesley| year=1969 | isbn=0-201-00746-0}}</ref>
<math display="block">\int \exp\left( - \frac 1 2 x \cdot A \cdot x +J \cdot x \right) d^nx = \sqrt{\frac{(2\pi)^n}{\det A}} \exp \left( {1\over 2} J \cdot A^{-1} \cdot J \right)</math>
Here {{mvar|A}} is a real positive definite [[symmetric matrix]].
This integral is performed by [[Diagonalizable matrix|diagonalization]] of {{mvar|A}} with an [[orthogonal matrix|orthogonal transformation]]
<math display="block">D= O^{-1} A O = O^\text{T} A O</math>
where {{mvar|D}} is a [[diagonal matrix]] and {{mvar|O}} is an [[orthogonal matrix]]. This decouples the variables and allows the integration to be performed as {{mvar|n}} one-dimensional integrations.
This is best illustrated with a two-dimensional example.
=== Example: Simple Gaussian integration in two dimensions ===
The Gaussian integral in two dimensions is
<math display="block">\int \exp\left( - \frac 1 2 A_{ij} x^i x^j \right) d^2x = \sqrt{\frac{(2\pi)^2}{\det A}}</math>
where {{mvar|A}} is a two-dimensional symmetric matrix with components specified as
<math display="block"> A = \begin{bmatrix} a&c\\ c&b\end{bmatrix}</math>
and we have used the [[Einstein summation convention]].
==== Diagonalize the matrix ====
The first step is to [[Diagonalizable matrix|diagonalize]] the matrix.<ref>{{cite book| author=Herbert S. Wilf| title=Mathematics for the Physical Sciences| publisher=Dover| year=1978| isbn=0-486-63635-6| url-access=registration| url=https://archive.org/details/mathematicsforph0000wilf_w9m6}}</ref> Note that
<math display="block">A_{ij} x^i x^j \equiv x^\text{T}Ax = x^\text{T} \left(OO^\text{T}\right) A \left(OO^\text{T}\right) x = \left(x^\text{T}O \right) \left(O^\text{T}AO \right) \left(O^\text{T}x \right) </math>
where, since {{mvar|A}} is a real [[symmetric matrix]], we can choose {{mvar|O}} to be [[orthogonal matrix|orthogonal]], and hence also a [[unitary matrix]]. {{mvar|O}} can be obtained from the [[eigenvectors]] of {{mvar|A}}. We choose {{mvar|O}} such that: {{math|''D'' ≡ ''O<sup>T</sup>AO''}} is diagonal.
===== Eigenvalues of ''A'' =====
To find the eigenvectors of {{mvar|A}} one first finds the [[eigenvalues]] {{mvar|λ}} of {{mvar|A}} given by
<math display="block"> \begin{bmatrix}a&c\\ c&b\end{bmatrix} \begin{bmatrix} u\\ v \end{bmatrix}=\lambda \begin{bmatrix}u\\ v\end{bmatrix}.</math>
The eigenvalues are solutions of the [[characteristic polynomial]]
<math display="block">( a - \lambda)( b-\lambda) -c^2 = 0 </math>
which are found using the [[quadratic equation]]:
<math display="block">\begin{align}
&= \tfrac{1}{2} ( a+b) \pm \tfrac{1}{2} \sqrt{(a-b)^2+4c^2}.
\end{align}</math>
===== Eigenvectors of ''A'' =====
Substitution of the eigenvalues back into the eigenvector equation yields
<math display="block"> v = -{ \left( a - \lambda_{\pm} \right)u \over c }, \qquad v = -{cu \over \left( b - \lambda_{\pm} \right)}.</math>
From the characteristic equation we know
<math display="block"> {a - \lambda_{\pm} \over c } = {c \over b - \lambda_{\pm}}.</math>
Also note
<math display="block"> { a - \lambda_{\pm} \over c } = -{ b - \lambda_{\mp} \over c}. </math>
The eigenvectors can be written as:
<math display="block">\begin{bmatrix} \frac{1}{\eta} \\[1ex] -\frac{a - \lambda_-}{c\eta} \end{bmatrix},
\qquad
\begin{bmatrix} -\frac{b - \lambda_+}{c\eta} \\[1ex] \frac{1}{\eta} \end{bmatrix} </math>
for the two eigenvectors. Here {{mvar|η}} is a normalizing factor given by,
<math display="block">\eta = \sqrt{1 + \left(\frac{a -\lambda_{-}}{c} \right)^2 } = \sqrt{1 + \left(\frac{b - \lambda_{+}}{c} \right)^2}.</math>
It is easily verified that the two eigenvectors are orthogonal to each other.
===== Construction of the orthogonal matrix =====
The orthogonal matrix is constructed by assigning the normalized eigenvectors as columns in the orthogonal matrix
Note that {{math|det(''O'') {{=}} 1}}.
If we define
<math display="block"> \sin(\theta) = -\frac{a - \lambda_{-}}{c \eta }</math>
then the orthogonal matrix can be written
<math display="block">O = \begin{bmatrix}\cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta)\end{bmatrix} </math>
which is simply a rotation of the eigenvectors with the inverse:
<math display="block">O^{-1} = O^\text{T} = \begin{bmatrix} \cos(\theta) & \sin(\theta) \\ -\sin(\theta) & \cos(\theta) \end{bmatrix}.</math>
===== Diagonal matrix =====
The diagonal matrix becomes
<math display="block"> D = O^\text{T} A O = \begin{bmatrix}\lambda_{-}& 0 \\[1ex] 0 & \lambda_{+}\end{bmatrix}</math>
with eigenvectors
<math display="block">\begin{bmatrix} 1\\ 0\end{bmatrix}, \qquad \begin{bmatrix} 0\\ 1 \end{bmatrix}</math>
===== Numerical example =====
<math display="block">A = \begin{bmatrix} 2&1\\ 1 & 1\end{bmatrix}</math>
The eigenvalues are
<math display="block">\lambda_{\pm} = {3\over 2} \pm {\sqrt{ 5} \over 2}.</math>
The eigenvectors are
<math display="block">{1\over \eta}\begin{bmatrix} 1 \\[1ex] -{1\over 2} - {\sqrt{5} \over 2} \end{bmatrix}, \qquad
where
<math display="block">\eta = \sqrt{{5\over 2} + {\sqrt{5}\over 2}}.</math>
Then
<math display="block">\begin{align}
O &= \begin{bmatrix} \frac{1}{\eta} & \frac{1}{\eta} \left({1\over 2} + {\sqrt{5} \over 2}\right) \\ \frac{1}{\eta} \left(-{1\over 2} - {\sqrt{ 5} \over 2}\right) & {1\over \eta}\end{bmatrix} \\
O^{-1} &= \begin{bmatrix} \frac{1}{\eta} & \frac{1}{\eta} \left(-{1\over 2} - {\sqrt{5} \over 2}\right) \\ \frac{1}{\eta} \left({1\over 2} + {\sqrt{5} \over 2}\right) & \frac{1}{\eta} \end{bmatrix}
Line 204 ⟶ 165:
The diagonal matrix becomes
<math display="block">D = O^\text{T}AO = \begin{bmatrix} \lambda_- &0\\ 0 & \lambda_+ \end{bmatrix} = \begin{bmatrix} {3\over 2} - {\sqrt{5} \over 2}& 0\\ 0 & {3\over 2} + {\sqrt{5} \over 2}\end{bmatrix} </math>
with eigenvectors
<math display="block">\begin{bmatrix} 1\\ 0\end{bmatrix}, \qquad \begin{bmatrix} 0\\ 1 \end{bmatrix}</math>
==== Rescale the variables and integrate ====
With the diagonalization the integral can be written
<math display="block">\int \exp\left( - \frac 1 2 x^\text{T} A x \right) d^2x = \int \exp\left( - \frac 1 2 \sum_{j=1}^2 \lambda_{j} y_j^2 \right) \, d^2y</math>
where
<math display="block">y = O^\text{T} x.</math>
Since the coordinate transformation is simply a rotation of coordinates the [[Jacobian matrix and determinant|Jacobian]] determinant of the transformation is one yielding
<math display="block"> d^2y = d^2x </math>
The integrations can now be performed:
<math display="block">\begin{align}
\int \exp\left( - \frac{1}{2} x^\mathsf{T} A x \right) d^2x
={}& \int \exp\left( - \frac 1 2 \sum_{j=1}^2 \lambda_{j} y_j^2 \right) d^2y \\[1ex]
={}& \prod_{j=1}^2 \left( { 2\pi \over \lambda_j } \right)^{1/2} \\
\end{align}</math>
which is the advertised solution.
=== Integrals with complex and linear terms in multiple dimensions ===
With the two-dimensional example it is now easy to see the generalization to the complex plane and to multiple dimensions.
==== Integrals with a linear term in the argument ====
==== Integrals with an imaginary linear term ====
==== Integrals with a complex quadratic term ====
=== Integrals with differential operators in the argument ===
As an example consider the integral<ref name="Zee"/>{{rp|pp=21‒22}}
<math display="block">\int \exp\left[ \int d^4x \left (-\frac{1}{2} \varphi \hat A \varphi + J \varphi \right) \right ] D\varphi</math>
where <math> \hat A </math> is a differential operator with <math> \varphi </math> and {{mvar|J}} functions of [[spacetime]], and <math> D\varphi </math> indicates integration over all possible paths. In analogy with the matrix version of this integral the solution is
<math display="block">\int \exp\left[ \int d^4x \left (-\frac 1 2 \varphi \hat A \varphi +J\varphi \right) \right ] D\varphi \; \propto \; \exp \left( {1\over 2} \int d^4x \; d^4y J(x) D( x - y) J(y) \right)</math>
where
<math display="block">\hat A D( x - y) = \delta^4 ( x - y)</math>
and {{math|''D''(''x'' − ''y'')}}, called the [[propagator]], is the inverse of <math> \hat A</math>, and <math> \delta^4( x - y)</math> is the [[Dirac delta function]].
Similar arguments yield
<math display="block">\int \exp\left[\int d^4x \left (-\frac 1 2 \varphi \hat A \varphi + i J \varphi \right) \right ] D\varphi \; \propto \; \exp \left( - { 1\over 2} \int d^4x \; d^4y J(x) D( x - y) J(y) \right),</math>
and
<math display="block">\int \exp\left[ i \int d^4x \left ( \frac 1 2 \varphi \hat A \varphi + J\varphi \right) \right ] D\varphi \; \propto \; \exp \left( -{ i\over 2} \int d^4x \; d^4y J(x) D( x - y) J(y) \right).</math>
See [[Static forces and virtual-particle exchange#Path-integral formulation of virtual-particle exchange|Path-integral formulation of virtual-particle exchange]] for an application of this integral.
== Integrals that can be approximated by the method of steepest descent ==
In quantum field theory n-dimensional integrals of the form
<math display="block">\int_{-\infty}^{\infty} \exp\left( -{1 \over \hbar} f(q) \right ) d^nq</math>
appear often. Here <math>\hbar</math> is the [[reduced Planck constant]] and ''f'' is a function with a positive minimum at <math> q=q_0</math>. These integrals can be approximated by the [[method of steepest descent]].
For small values of the Planck constant, ''f'' can be expanded about its minimum
<math display="block">\int_{-\infty}^{\infty} \exp\left[ -{1 \over \hbar} \left( f\left( q_0 \right) + {1\over 2} \left( q-q_0\right)^2 f^{\prime \prime} \left( q-q_0\right) + \cdots \right ) \right] d^nq.</math>Here <math> f^{\prime \prime} </math> is the n by n matrix of second derivatives evaluated at the minimum of the function.
If we neglect higher order terms this integral can be integrated explicitly.
<math display="block">\int_{-\infty}^{\infty} \exp\left[ -{1 \over \hbar} (f(q)) \right] d^nq \approx \exp\left[ -{1 \over \hbar} \left( f\left( q_0 \right) \right ) \right] \sqrt{ (2 \pi \hbar )^n \over \det f^{\prime \prime} }.</math>
== Integrals that can be approximated by the method of stationary phase ==
A common integral is a path integral of the form
<math display="block"> \int \exp\left( {i \over \hbar} S\left( q, \dot q \right) \right ) Dq </math>
where <math> S\left( q, \dot q \right) </math> is the classical [[Action (physics)|action]] and the integral is over all possible paths that a particle may take. In the limit of small <math> \hbar </math> the integral can be evaluated in the [[stationary phase approximation]]. In this approximation the integral is over the path in which the action is a minimum. Therefore, this approximation recovers the [[classical limit]] of [[classical mechanics|mechanics]].
== Fourier integrals ==
=== Dirac delta distribution ===
The [[Dirac delta distribution]] in [[spacetime]] can be written as a [[Fourier transform]]<ref name="Zee"/>{{rp|p=23}}
<math display="block"> \int \frac{d^4 k}{(2\pi)^4} \exp(ik ( x-y)) = \delta^4 ( x-y).</math>
In general, for any dimension <math> N </math>
<math display="block"> \int \frac{d^N k}{(2\pi)^N} \exp(ik ( x-y)) = \delta^N ( x-y).</math>
=== Fourier integrals of forms of the Coulomb potential ===
===
While not an integral, the identity in three-dimensional [[Euclidean space]]
<math display="block">-{1 \over 4\pi} \nabla^2 \left( {1 \over r} \right) = \delta \left( \mathbf r \right) </math>where<math display="block">r^2 = \mathbf r \cdot \mathbf r</math>is a consequence of [[Gauss's theorem]] and can be used to derive integral identities. For an example see [[Longitudinal and transverse vector fields]].
This identity implies that the [[Fourier integral]] representation of 1/''r'' is
<math display="block">\int \frac{d^3 k}{(2\pi)^3} { \exp \left ( i\mathbf k \cdot \mathbf r \right) \over k^2 } = {1 \over 4 \pi r }.</math>
==== Yukawa potential: the Coulomb potential with mass ====
The [[Yukawa potential]] in three dimensions can be represented as an integral over a [[Fourier transform]]<ref name="Zee"/>{{rp|p=26, 29}}
<math display="block">\int \frac{d^3 k}{(2\pi)^3} { \exp \left ( i\mathbf k \cdot \mathbf r \right) \over k^2 +m^2 } = {e^{-mr} \over 4 \pi r } </math>
where
<math display="block">r^2 = \mathbf{r} \cdot \mathbf r, \qquad k^2 = \mathbf k \cdot \mathbf k.</math>
See [[Static forces and virtual-particle exchange#Selected examples|Static forces and virtual-particle exchange]] for an application of this integral.
Line 342 ⟶ 265:
To derive this result note:
<math display="block">\begin{align}
\int \frac{d^3 k}{(2\pi)^3} \frac{\exp \left (i \mathbf k \cdot \mathbf r\right)}{k^2 +m^2}
\end{align}</math>
==== Modified Coulomb potential with mass ====
where the hat indicates a [[unit vector]] in three dimensional space. The derivation of this result is as follows:
<math display="block">\begin{align}
&\int \frac{d^3 k}{(2\pi)^3} \left(\mathbf{\hat k}\cdot \mathbf{\hat r}\right)^2 \frac{\exp \left (i\mathbf{k}\cdot \mathbf{r}\right )}{k^2 +m^2} \\[1ex]
&= \int_0^{\infty} \frac{k^2 dk}{(2\pi)^2} \int_{-1}^{1} du \ u^2 \frac{e^{ikru}}{k^2 + m^2} \\[1ex]
&= 2 \
&=
\end{align} </math>
Note that in the small {{mvar|m}} limit the integral goes to the result for the Coulomb potential since the term in the brackets goes to {{math|1}}.
==== Longitudinal potential with mass ====
where the hat indicates a unit vector in three dimensional space. The derivation for this result is as follows:
<math display="block">\begin{align}
& \int \frac{d^3 k}{(2\pi)^3} \mathbf{\hat k} \mathbf{\hat k} \frac{\exp \left (i\mathbf k \cdot \mathbf r \right)}{k^2 +m^2} \\[1ex]
&= \frac{e^{-mr}}{4 \pi r}\left\{ 1+ \frac{2}{mr}- {2\over (mr)^2 } \left( e^{mr} -1 \right) \right \} \left\{\mathbf 1 - {1\over 2} \left[\mathbf 1 - \mathbf{\hat r} \mathbf{\hat r}\right] \right\} + \int_0^{\infty} \frac{k^2 dk}{(2\pi)^2 } \int_{-1}^{1} du \frac{e^{ikru}}{k^2 + m^2} {1\over 2} \left[ \mathbf 1 - \mathbf{\hat r} \mathbf{\hat r} \right] \\[1ex]
&= {1\over 2} \frac{e^{-mr}}{4 \pi r} \left[ \mathbf 1 - \mathbf{\hat r} \mathbf{\hat r} \right]+ {e^{-mr} \over 4 \pi r } \left\{ 1+\frac{2}{mr} - {2\over (mr)^2} \left( e^{mr} -1 \right) \right \} \left\{ {1\over 2} \left[\mathbf 1 + \mathbf{\hat r} \mathbf{\hat r}\right] \right\} \\[1ex]
&= {1\over 2} \frac{e^{-mr}}{4\pi r} \left (\left[ \mathbf{1}- \mathbf{\hat{r}} \mathbf{\hat{r}} \right] + \left\{1 + \frac{2}{mr} - {2 \over (mr)^2} \left(e^{mr} -1 \right) \right \} \left[\mathbf{1}+ \mathbf{\hat{r}} \mathbf{\hat{r}}\right] \right )
\end{align}</math>
Note that in the small {{mvar|m}} limit the integral reduces to
<math display="block">{1\over 2} {1 \over 4 \pi r } \left[ \mathbf 1 - \mathbf{\hat r} \mathbf{\hat r} \right]. </math>
==== Transverse potential with mass ====
<math display="block">\int \frac{d^3 k}{(2\pi)^3} \left[\mathbf{1} - \mathbf{\hat{k}} \mathbf{\hat{k}} \right] { \exp \left ( i \mathbf{k} \cdot \mathbf{r}\right ) \over k^2 +m^2 } = {1\over 2} {e^{-mr} \over 4 \pi r} \left\{ {2 \over (mr)^2 } \left( e^{mr} -1 \right) - {2\over mr} \right \} \left[\mathbf{1} + \mathbf{\hat{r}} \mathbf{\hat{r}}\right]</math>
In the small ''mr'' limit the integral goes to
For large distance, the integral falls off as the inverse cube of ''r''
<math display="block">\frac{1}{4 \pi m^2r^3 }\left[\mathbf 1 + \mathbf{\hat r} \mathbf{\hat r}\right].</math>
For applications of this integral see [[Darwin Lagrangian]] and [[Static forces and virtual-particle exchange#Darwin interaction in a vacuum|Darwin interaction in a vacuum]].
==== Angular integration in cylindrical coordinates ====
There are two important integrals. The angular integration of an exponential in cylindrical coordinates can be written in terms of Bessel functions of the first kind<ref name="Zwillinger_2014">{{cite book |author-first1=Izrail Solomonovich |author-last1=Gradshteyn |author-link1=Izrail Solomonovich Gradshteyn |author-first2=Iosif Moiseevich |author-last2=Ryzhik |author-link2=Iosif Moiseevich Ryzhik |author-first3=Yuri Veniaminovich |author-last3=Geronimus |author-link3=Yuri Veniaminovich Geronimus |author-first4=Michail Yulyevich |author-last4=Tseytlin |author-link4=Michail Yulyevich Tseytlin |author-first5=Alan |author-last5=Jeffrey |editor-first1=Daniel |editor-last1=Zwillinger |editor-first2=Victor Hugo |editor-last2=Moll |editor-link2=Victor Hugo Moll |translator=Scripta Technica, Inc. |title=Table of Integrals, Series, and Products |publisher=[[Academic Press, Inc.]] |date=2015 |orig-year=October 2014 |edition=8 |language=English |isbn=978-0-12-384933-5 |lccn=2014010276 <!-- |url=https://books.google.com/books?id=NjnLAwAAQBAJ |access-date=2016-02-21 -->|title-link=Gradshteyn and Ryzhik <!-- |pages= was on page 402 and 679 in 1965 edition, but page numbers probably changed meanwhile -->}}</ref><ref name="Jackson">{{cite book|
<math display="block">\int_0^{2 \pi} {d\varphi \over 2 \pi} \exp\left( i p \cos( \varphi) \right)=J_0 (p)</math>
and
<math display="block">\int_0^{2 \pi} {d\varphi \over 2 \pi} \cos( \varphi) \exp\left( i p \cos( \varphi) \right) = i J_1 (p). </math>
For applications of these integrals see [[Static forces and virtual-particle exchange#Magnetic interaction between current loops in a simple plasma or electron gas|Magnetic interaction between current loops in a simple plasma or electron gas]].
== Bessel functions ==
=== Integration of the cylindrical propagator with mass ===
==== First power of a Bessel function ====
See [[Abramowitz and Stegun]].<ref name="AbramowitzStegun">{{cite book|
For <math> mr \ll 1 </math>, we have<ref name="Jackson"/>
<math display="block">K_0 (mr) \to -\ln \left( {mr \over 2}\right) + 0.5772.</math>
For an application of this integral see [[Static forces and virtual-particle exchange#Two line charges embedded in a plasma or electron gas|Two line charges embedded in a plasma or electron gas]].
==== Squares of Bessel functions ====
The integration of the propagator in cylindrical coordinates is<ref name="Zwillinger_2014"/>
<math display="block">\int_0^{\infty} {k\; dk \over k^2 +m^2} J_1^2 (kr) =I_1 (mr)K_1 (mr).</math>
For small mr the integral becomes
<math display="block">\int_o^{\infty} {k\; dk \over k^2 +m^2} J_1^2 (kr) \to {1\over 2 }\left[ 1 - {1\over 8} (mr)^2 \right].</math>
For large mr the integral becomes
<math display="block">\int_o^{\infty} {k\; dk \over k^2 +m^2} J_1^2 (kr) \to {1\over 2} \left( {1\over mr}\right).</math>
For applications of this integral see [[Static forces and virtual-particle exchange#Magnetic interaction between current loops in a simple plasma or electron gas|Magnetic interaction between current loops in a simple plasma or electron gas]].
In general,
<math display="block">\int_0^{\infty} {k\; dk \over k^2 +m^2} J_{\nu}^2 (kr) = I_{\nu} (mr)K_{\nu} (mr) \qquad \Re (\nu) > -1. </math>
=== Integration over a magnetic wave function ===
The two-dimensional integral over a magnetic wave function is<ref name="AbramowitzStegun"/>{{rp|at=§11.4.28}}
<math display="block">{2 a^{2n+2}\over n!} \int_0^{\infty} { dr }\;r^{2n+1}\exp\left( -a^2 r^2\right) J_{0} (kr) = M\left( n+1, 1, -{k^2 \over 4a^2}\right).</math>
Here, ''M'' is a [[confluent hypergeometric function]]. For an application of this integral see [[Static forces and virtual-particle exchange#Charge density spread over a wave function|Charge density spread over a wave function]].
== See also ==
* [[Relation between Schrödinger's equation and the path integral formulation of quantum mechanics]]
== References ==
{{reflist}}
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