Common integrals in quantum field theory: Difference between revisions

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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}} pp. 13-15</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==

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===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>

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The eigenvectors can be written as:

<math display="block">\begin{bmatrix} \dfracfrac{1}{\eta} \\[1ex] -\dfracfrac{a - \lambda_-}{c\eta} \end{bmatrix},

\qquad

\begin{bmatrix} -\dfracfrac{b - \lambda_+}{c\eta} \\[1ex] \dfracfrac{1}{\eta} \end{bmatrix} </math>

for the two eigenvectors. Here {{mvar|η}} is a normalizing factor given by,

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The diagonal matrix becomes

<math display="block"> D = O^T A O = \begin{bmatrix}\lambda_{-}& 0 \\[1ex] 0 & \lambda_{+}\end{bmatrix}</math>

with eigenvectors

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The eigenvectors are

<math display="block">{1\over \eta}\begin{bmatrix} 1 \\[1ex] -{1\over 2} - {\sqrt{5} \over 2} \end{bmatrix}, \qquad
{1\over \eta} \begin{bmatrix} {1\over 2} + {\sqrt{5} \over 2 } \\ [1ex] 1 \end{bmatrix}</math>

where

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<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\over /2} \\

={}& \left( { (2\pi)^2 \over \prod_{j=1}^2 \lambda_j } \right)^{1\over /2} \\[1ex]

={}& \left( { (2\pi)^2 \over \det{ \left( O^{-1}AO \right)} } \right)^{1\over /2} \\[1ex]

={}& \left( { (2\pi)^2 \over \det{ \left( A \right)} } \right)^{1\over /2}

\end{align}</math>

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===Integrals with differential operators in the argument===

As an example consider the integral<ref> name="Zee, pp. 21-22.<"/ref>{{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>

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===Dirac delta distribution===

The [[Dirac delta distribution]] in [[spacetime]] can be written as a [[Fourier transform]]<ref name="Zee"/>Zee, {{rp|p. =23.</ref>}}

<math display="block"> \int \frac{d^4 k}{(2\pi)^4} \exp(ik ( x-y)) = \delta^4 ( x-y).</math>

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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]].

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

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====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"/>Zee, {{rp|p. =26, 29.</ref>}}

<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>

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<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}

={}& \int_0^{\infty} \frac{k^2 dk}{(2\pi)^2} \int_{-1}^1 du {e^{ikru}\over k^2 + m^2} \\[2pt1ex]

={}& {2\over r} \int_0^{\infty} \frac{k dk}{(2\pi)^2} {\sin(kr) \over k^2 + m^2} \\[2pt1ex]

={}& {1\over ir} \int_{-\infty}^{\infty} \frac{k dk}{(2\pi)^2} {e^{ikr} \over k^2 + m^2} \\[1ex]

={}& {1\over ir} \int_{-\infty}^{\infty} \frac{k dk}{(2\pi)^2} {e^{ikr} \over (k + i m)(k - i m)} \\[1ex]

={}& {1\over ir} \frac{2\pi i}{(2\pi)^2} \frac{im}{2im} e^{-mr} \\[1ex]

={}& \frac{1}{4 \pi r} e^{-mr}

\end{align}</math>

====Modified Coulomb potential with mass====

<math display="block">\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} = \frac{e^{-mr}}{4 \pi r} \left\{[1 + \frac{2}{mr} - \frac{2}{(mr)^2} \left(e^{mr}-1 \right) \right \}]</math>

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 \int_0^{\infty} \frac{k^2 dk}{(2\pi)^2} \frac{1}{k^2 + m^2} \left\{[\frac{1}{kr} \sin(kr) + \frac{2}{(kr)^2} \cos(kr)- \frac{2}{(kr)^3} \sin(kr) \right \}] \\[1ex]

&={}& \frac{e^{-mr}}{4\pi r} \left\{[1 + \frac{2}{mr} - \frac{2}{(mr)^2} \left(e^{mr}-1 \right) \right \}]

\end{align} </math>

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<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]

&={}& \int \frac{d^3 k}{(2\pi)^3} \left[ \left( \mathbf{\hat k}\cdot \mathbf{\hat r}\right)^2\mathbf{\hat r} \mathbf{\hat r} + \left( \mathbf{\hat k}\cdot \mathbf{\hat \theta}\right)^2\mathbf{\hat \theta} \mathbf{\hat \theta} + \left( \mathbf{\hat k}\cdot \mathbf{\hat \phi}\right)^2\mathbf{\hat \phi} \mathbf{\hat \phi} \right] \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>

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====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|author last = Jackson, | first = John D.|title=Classical Electrodynamics (| edition = 3rd ed.)| publisher=Wiley| year=1998| isbn=0-471-30932-X}} p. 113</ref>{{rp|p=113}}

<math display="block">\int_0^{2 \pi} {d\varphi \over 2 \pi} \exp\left( i p \cos( \varphi) \right)=J_0 (p)</math>

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<math display="block">\int_0^{\infty} {k\; dk \over k^2 +m^2} J_0 \left( kr \right)=K_0 (mr). </math>

See Abramowitz and Stegun.<ref name="AbramowitzStegun">{{cite book| authorauthor1=M. Abramowitz and| author2 = I. Stegun| title=Handbook of Mathematical Functions| publisher=Dover| year=1965| isbn=0486-61272-4| url-access=registration| url=https://archive.org/details/handbookofmathe000abra}} Section </ref>{{rp|at=§11.4.44</ref>}}

For <math> mr \ll 1 </math>, we have<ref name="Jackson"/>Jackson, {{rp|p. =116</ref>}}

<math display="block">K_0 (mr) \to -\ln \left( {mr \over 2}\right) + 0.5772.</math>

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===Integration over a magnetic wave function===

The two-dimensional integral over a magnetic wave function is<ref name="AbramowitzStegun"/>Abramowitz and Stegun, Section {{rp|at=§11.4.28</ref>}}

<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>