Common integrals in quantum field theory

Common integrals in quantum field theory are set of formulas that are useful for computation of various types in quantum field theory such as partition function, integrals of loop diagrams, etc.

The following Gaussian integrals are useful in calculating path integrals appearing in path integral formulation of quantum field theory:^[1][2] $${\displaystyle {\begin{aligned}\int _{-\infty }^{\infty }e^{-{1 \over 2}ax^{2}+Jx}\,dx&=\left({2\pi \over a}\right)^{1/2}\exp \left({J^{2} \over 2a}\right),&a,J\in \mathbb {C} ,\,\operatorname {Re} (a)>0\\\int _{-\infty }^{\infty }\exp \left(i\left({\frac {1}{2}}(a+i\varepsilon )x^{2}+Jx\right)\right)dx&=\left({2\pi i \over a+i\varepsilon }\right)^{1/2}\exp \left(-{\frac {i}{2}}{J^{2} \over a+i\varepsilon }\right),&a,J,\varepsilon \in \mathbb {R} ,\,\varepsilon \rightarrow 0^{+}\\\int \exp \left(\sum _{i,j=1}^{n}-{\frac {1}{2}}x_{i}A_{ij}x_{j}+J_{i}x_{i}\right)d^{n}x&={\sqrt {\frac {(2\pi )^{n}}{\det A}}}\exp \left({1 \over 2}\sum _{i,j=1}^{n}J_{i}A_{ij}^{-1}J_{j}\right),&A,J\in \mathbb {R} ,\,A_{ij}=A_{ji}{\text{ positive definite}}\\\int \exp \left(i\left(\sum _{i,j=1}^{n}{\frac {1}{2}}x_{i}(A+i\varepsilon I)_{ij}x_{j}+J_{i}x_{i}\right)\right)d^{n}x&={\sqrt {\frac {(2\pi )^{n}}{\det {(A+i\varepsilon I)}}}}\exp \left(-{\frac {i}{2}}\sum _{i,j=1}^{n}J_{i}(A+i\varepsilon I)_{ij}^{-1}J_{j}\right),&A,J,\varepsilon \in \mathbb {R} ,\,A_{ij}=A_{ji},\,\varepsilon \rightarrow 0^{+}\\\end{aligned}}}$$

As an example consider the integral^{[3]: 21‒22} $${\displaystyle \int \exp \left[\int d^{4}x\left(-{\frac {1}{2}}\varphi {\hat {A}}\varphi +J\varphi \right)\right]D\varphi }$$ where ${\displaystyle {\hat {A}}}$ is a Hermitian differential operator with positive spectra for convergence, ${\displaystyle \varphi }$ and J functions of spacetime, and ${\displaystyle D\varphi }$ indicates integration over all possible paths. In analogy with the matrix version of this integral the solution is $${\displaystyle \int \exp \left[\int d^{4}x\left(-{\frac {1}{2}}\varphi {\hat {A}}\varphi +J\varphi \right)\right]D\varphi \;\propto \;\exp \left({1 \over 2}\int d^{4}x\;d^{4}yJ(x)D(x-y)J(y)\right)}$$ where $${\displaystyle {\hat {A}}D(x-y)=\delta ^{4}(x-y)}$$ and D(x − y), called the propagator, is the inverse of ${\displaystyle {\hat {A}}}$, and ${\displaystyle \delta ^{4}(x-y)}$ is the Dirac delta function.

Similar arguments yield for ${\displaystyle {\hat {A}}}$ Hermitian differential operator of any spectra, where the ${\displaystyle \varepsilon \rightarrow 0^{+}}$ prescription, called Feynman prescription, is treated separately in the last step of all calculations $${\displaystyle \int \exp \left[i\int d^{4}x\left({\frac {1}{2}}\varphi ({\hat {A}}+i\varepsilon )\varphi +J\varphi \right)\right]D\varphi \;\propto \;\exp \left(-{i \over 2}\int d^{4}x\;d^{4}yJ(x)D_{\varepsilon }(x-y)J(y)\right).}$$

See Path-integral formulation of virtual-particle exchange for an application of this integral.

It is not necessarily the case that the differential operator has appropriate spectral properties, for example, ${\displaystyle {\hat {A}}=\partial _{\mu }\partial ^{\mu }}$, has null eigenvalues, is non-invertible and hence, its propagator is not uniquely determined. However, addition of a small complex part ${\displaystyle i\varepsilon }$ removes any null eigenvalues as the spectra is necessarily complex making the operator invertible again. Such cases can also be treated by redefinition of fields and imposing appropriate boundary condition, which is equivalent to the Feynman ${\displaystyle i\varepsilon }$ prescription, which is also equivalent to analytically continuing to Euclidean theory and continuing back after computations using Wick rotations in a particular direction.^[2][4]

In quantum field theory n-dimensional integrals of the form $${\displaystyle \int _{-\infty }^{\infty }\exp \left(-{1 \over \hbar }f(q)\right)d^{n}q}$$ appear often. Here ${\displaystyle \hbar }$ is the reduced Planck constant and f is a function with a positive minimum at ${\displaystyle q=q_{0}}$. 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 $${\displaystyle \int _{-\infty }^{\infty }\exp \left[-{1 \over \hbar }\left(f\left(q_{0}\right)+{1 \over 2}\left(q-q_{0}\right)^{2}f^{\prime \prime }\left(q-q_{0}\right)+\cdots \right)\right]d^{n}q.}$$Here ${\displaystyle f^{\prime \prime }}$ 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. $${\displaystyle \int _{-\infty }^{\infty }\exp \left[-{1 \over \hbar }(f(q))\right]d^{n}q\approx \exp \left[-{1 \over \hbar }\left(f\left(q_{0}\right)\right)\right]{\sqrt {(2\pi \hbar )^{n} \over \det f^{\prime \prime }}}.}$$

A common integral is a path integral of the form $${\displaystyle \int \exp \left({i \over \hbar }S\left(q,{\dot {q}}\right)\right)Dq}$$ where ${\displaystyle S\left(q,{\dot {q}}\right)}$ is the classical action and the integral is over all possible paths that a particle may take. In the limit of small ${\displaystyle \hbar }$ 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 mechanics.

The Dirac delta distribution in spacetime can be written as a Fourier transform^[3]: 23$${\displaystyle \int {\frac {d^{4}k}{(2\pi )^{4}}}\exp(ik(x-y))=\delta ^{4}(x-y).}$$In general, for any dimension ${\displaystyle N}$$${\displaystyle \int {\frac {d^{N}k}{(2\pi )^{N}}}\exp(ik(x-y))=\delta ^{N}(x-y).}$$

Identifying two to two elastic scattering results of Quantum field theory with first Born approximation results from quantum mechanics in the relation ${\textstyle M_{\text{el.}}(p_{i}\rightarrow p_{f})=V(q=p_{f}-p_{i})}$ where incoming particle undergo elastic scattering, i.e. ${\textstyle p_{i}^{0}=p_{f}^{0}}$ against a heavy static particle. Thus, the form of potential is found by Fourier transform which is the Fourier inverse of propagator of the virtual exchange particle in the tree level.

While not an integral, the identity in three-dimensional Euclidean space $${\displaystyle -{1 \over 4\pi }\nabla ^{2}\left({1 \over r}\right)=\delta \left(\mathbf {r} \right)}$$where ${\textstyle r^{2}=\mathbf {r} \cdot \mathbf {r} }$, 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 $${\displaystyle \int {\frac {d^{3}k}{(2\pi )^{3}}}{\exp \left(i\mathbf {k} \cdot \mathbf {r} \right) \over k^{2}}={1 \over 4\pi r}.}$$

The Yukawa potential in three dimensions can be represented as an integral over a Fourier transform^{[3]: 26, 29} $${\displaystyle \int {\frac {d^{3}k}{(2\pi )^{3}}}{\exp \left(i\mathbf {k} \cdot \mathbf {r} \right) \over k^{2}+m^{2}}={e^{-mr} \over 4\pi r}}$$ where ${\textstyle r^{2}=\mathbf {r} \cdot \mathbf {r} ,\,k^{2}=\mathbf {k} \cdot \mathbf {k} }$.

See Static forces and virtual-particle exchange for an application of this integral. In the small m limit the integral reduces to ⁠1/4πr⁠.

$${\displaystyle \int {\frac {d^{3}k}{(2\pi )^{3}}}\left(\mathbf {\hat {k}} \cdot \mathbf {\hat {r}} \right)^{2}{\frac {\exp \left(i\mathbf {k} \cdot \mathbf {r} \right)}{k^{2}+m^{2}}}={\frac {e^{-mr}}{4\pi r}}\left[1+{\frac {2}{mr}}-{\frac {2}{(mr)^{2}}}\left(e^{mr}-1\right)\right]}$$ where the hat indicates a unit vector in three dimensional space.

Note that in the small m limit the integral goes to the result for the Coulomb potential since the term in the brackets goes to 1.

$${\displaystyle \int {\frac {d^{3}k}{(2\pi )^{3}}}\mathbf {\hat {k}} \mathbf {\hat {k}} {\frac {\exp \left(i\mathbf {k} \cdot \mathbf {r} \right)}{k^{2}+m^{2}}}={1 \over 2}{\frac {e^{-mr}}{4\pi r}}\left(\left[\mathbf {1} -\mathbf {\hat {r}} \mathbf {\hat {r}} \right]+\left\{1+{\frac {2}{mr}}-{2 \over (mr)^{2}}\left(e^{mr}-1\right)\right\}\left[\mathbf {1} +\mathbf {\hat {r}} \mathbf {\hat {r}} \right]\right)}$$ where the hat indicates a unit vector in three dimensional space. Note that in the small m limit the integral reduces to $${\displaystyle {1 \over 2}{1 \over 4\pi r}\left[\mathbf {1} -\mathbf {\hat {r}} \mathbf {\hat {r}} \right].}$$

$${\displaystyle \int {\frac {d^{3}k}{(2\pi )^{3}}}\left[\mathbf {1} -\mathbf {\hat {k}} \mathbf {\hat {k}} \right]{\exp \left(i\mathbf {k} \cdot \mathbf {r} \right) \over k^{2}+m^{2}}={1 \over 2}{e^{-mr} \over 4\pi r}\left\{{2 \over (mr)^{2}}\left(e^{mr}-1\right)-{2 \over mr}\right\}\left[\mathbf {1} +\mathbf {\hat {r}} \mathbf {\hat {r}} \right]}$$

In the small mr limit the integral goes to $${\displaystyle {1 \over 2}{1 \over 4\pi r}\left[\mathbf {1} +\mathbf {\hat {r}} \mathbf {\hat {r}} \right].}$$For large distance, the integral falls off as the inverse cube of r $${\displaystyle {\frac {1}{4\pi m^{2}r^{3}}}\left[\mathbf {1} +\mathbf {\hat {r}} \mathbf {\hat {r}} \right].}$$For applications of this integral see Darwin Lagrangian and Darwin interaction in a vacuum.

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^{[5][6]: 113} $${\displaystyle \int _{0}^{2\pi }{d\varphi \over 2\pi }\exp \left(ip\cos(\varphi )\right)=J_{0}(p)}$$ and $${\displaystyle \int _{0}^{2\pi }{d\varphi \over 2\pi }\cos(\varphi )\exp \left(ip\cos(\varphi )\right)=iJ_{1}(p).}$$

For applications of these integrals see Magnetic interaction between current loops in a simple plasma or electron gas.

The following is a useful redefinition used in calculations:

${\displaystyle {\frac {1}{a_{0}a_{1}a_{2}\cdots a_{n}}}=\Gamma (n+1)\int _{0}^{1}{d}z_{1}\int _{0}^{z_{1}}{d}z_{2}\cdots \int _{0}^{z_{n-1}}{d}z_{n}{\frac {1}{[a_{0}+(a_{1}-a_{0})z_{1}+\cdots (a_{n}-a_{n-1})z_{n}]^{n+1}}}}$

where increasing powers in the denominator is possible ${\textstyle {\frac {1}{a\cdots }}\rightarrow {\frac {1}{a^{k}\cdots }}}$ by operating ${\textstyle {\frac {1}{(k-1)!}}\left(-{\frac {\partial }{\partial a}}\right)^{k-1}}$ on either side.

${\displaystyle {\frac {1}{A_{1}^{\alpha _{1}}\cdots A_{n}^{\alpha _{n}}}}={\frac {\Gamma (\alpha _{1}+\dots +\alpha _{n})}{\Gamma (\alpha _{1})\cdots \Gamma (\alpha _{n})}}\int _{0}^{1}d^{n}x_{1}{\frac {\delta (1-\sum _{k=1}^{n}x_{k})\;x_{1}^{\alpha _{1}-1}\cdots x_{n}^{\alpha _{n}-1}}{\left(\sum _{k=1}^{n}x_{k}A_{k}\right)^{\sum _{k=1}^{n}\alpha _{k}}}}}$

This is referred to as Feynman Parametrization.

The following integrals are commonly used results in the calculation of loop integrals in cutoff regularization, where Minkowski dot product is used between vectors and ${\displaystyle \lim _{\epsilon \rightarrow 0^{+}}}$ is evaluated:^[2]$${\displaystyle {\begin{aligned}\int {d}^{4}k{\frac {1}{\left(k^{2}-s+i\varepsilon \right)^{n}}}&=i\pi ^{2}(-1)^{n}{\frac {\Gamma (n-2)}{\Gamma (n)}}{\frac {1}{s^{n-2}}},&n\geq 3,\\\int {d}^{4}k{\frac {k^{\mu }}{\left(k^{2}-s+i\varepsilon \right)^{n}}}&=0,&n\geq 3,\\\int {d}^{4}k{\frac {k^{\mu }k^{\nu }}{(k^{2}-s+{i}\varepsilon )^{n}}}&={i}\pi ^{2}(-1)^{n+1}{\frac {\Gamma (n-3)}{2\Gamma (n)}}{\frac {g^{\mu \nu }}{s^{n-3}}},&n\geq 4,\\\int {d}^{4}p{\frac {1}{(p^{2}+2pq+t+{i}\varepsilon )^{n}}}&={i}\pi ^{2}{\frac {\Gamma (n-2)}{\Gamma (n)}}{\frac {1}{(t-q^{2})^{n-2}}},&n\geq 3,\\\int {d}^{4}p{\frac {p^{\mu }}{(p^{2}+2pq+t+{i}\varepsilon )^{n}}}&=-{i}\pi ^{2}{\frac {\Gamma (n-2)}{\Gamma (n)}}{\frac {q^{\mu }}{(t-q^{2})^{n-2}}},&n\geq 3,\\\int {d}^{4}p{\frac {p^{\mu }p^{\nu }}{(p^{2}+2pq+t+{i}\varepsilon )^{n}}}&={i}\pi ^{2}{\frac {\Gamma (n-3)}{2\Gamma (n)}}{\frac {[2(n-3)q^{\mu }q^{\nu }+(t-q^{2})g^{\mu \nu }]}{(t-q^{2})^{n-2}}},&n\geq 4.\end{aligned}}}$$

A similar set of relations are used in dimensional regularization as follows, where ${\displaystyle n>D/2}$ and ${\displaystyle \lim _{\epsilon \rightarrow 0^{+}}}$ is evaluated:^[7]$${\displaystyle {\begin{aligned}\int {\frac {{d}^{D}k}{\left(k^{2}-s+{i}\varepsilon \right)^{n}}}&={i}\pi ^{D/2}(-1)^{n}{\frac {\Gamma (n-D/2)}{\Gamma (n)}}{\frac {1}{s^{n-D/2}}},\\\int {d}^{D}k{\frac {k^{\mu }}{\left(k^{2}-s+{i}\varepsilon \right)^{n}}}&=0,\\\int {d}^{D}k{\frac {k^{\mu }k^{\nu }}{\left(k^{2}-s+{i}\varepsilon \right)^{n}}}&={i}\pi ^{D/2}(-1)^{n+1}{\frac {\Gamma (n-D/2-1)}{2\Gamma (n)}}{\frac {g^{\mu \nu }}{s^{n-D/2-1}}},\\\int {d}^{D}k{\frac {k^{2}}{\left(k^{2}-s+{i}\varepsilon \right)^{n}}}&={i}\pi ^{D/2}(-1)^{n+1}{\frac {\Gamma (n-D/2-1)}{2\Gamma (n)}}{\frac {D}{s^{n-D/2-1}}},\\\int {d}^{D}k{\frac {(k^{2})^{2}}{\left(k^{2}-s+{i}\varepsilon \right)^{n}}}&={i}\pi ^{D/2}(-1)^{n}{\frac {\Gamma (n-D/2-1)}{4\Gamma (n)}}{\frac {D(D+2)}{s^{n-D/2-1}}},\\\int {d}^{D}k{\frac {k^{\mu }k^{\nu }k^{\rho }k^{\sigma }}{\left(k^{2}-s+{i}\varepsilon \right)^{n}}}&={i}\pi ^{D/2}(-1)^{n}{\frac {\Gamma (n-D/2-1)}{4\Gamma (n)}}{\frac {[g^{\mu \nu }g^{\rho \sigma }+g^{\mu \rho }g^{\nu \sigma }+g^{\mu \sigma }g^{\nu \rho }]}{s^{n-D/2-1}}}.\end{aligned}}}$$

$${\displaystyle \int _{0}^{\infty }{k\;dk \over k^{2}+m^{2}}J_{0}\left(kr\right)=K_{0}(mr).}$$

See Abramowitz and Stegun.^{[8]: §11.4.44}

For ${\displaystyle mr\ll 1}$, we have^[6]: 116 $${\displaystyle K_{0}(mr)\to -\ln \left({mr \over 2}\right)+0.5772.}$$

For an application of this integral see Two line charges embedded in a plasma or electron gas.

The integration of the propagator in cylindrical coordinates is^[5] $${\displaystyle \int _{0}^{\infty }{k\;dk \over k^{2}+m^{2}}J_{1}^{2}(kr)=I_{1}(mr)K_{1}(mr).}$$

For small mr the integral becomes $${\displaystyle \int _{o}^{\infty }{k\;dk \over k^{2}+m^{2}}J_{1}^{2}(kr)\to {1 \over 2}\left[1-{1 \over 8}(mr)^{2}\right].}$$

For large mr the integral becomes $${\displaystyle \int _{o}^{\infty }{k\;dk \over k^{2}+m^{2}}J_{1}^{2}(kr)\to {1 \over 2}\left({1 \over mr}\right).}$$

For applications of this integral see Magnetic interaction between current loops in a simple plasma or electron gas.

In general, $${\displaystyle \int _{0}^{\infty }{k\;dk \over k^{2}+m^{2}}J_{\nu }^{2}(kr)=I_{\nu }(mr)K_{\nu }(mr)\qquad \Re (\nu )>-1.}$$

The two-dimensional integral over a magnetic wave function is^{[8]: §11.4.28} $${\displaystyle {2a^{2n+2} \over n!}\int _{0}^{\infty }{dr}\;r^{2n+1}\exp \left(-a^{2}r^{2}\right)J_{0}(kr)=M\left(n+1,1,-{k^{2} \over 4a^{2}}\right).}$$

Here, M is a confluent hypergeometric function. For an application of this integral see Charge density spread over a wave function.

• Relation between Schrödinger's equation and the path integral formulation of quantum mechanics