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.
Gaussian integrals
The following Gaussian integrals are useful in calculating path integrals appearing in path integral formulation of quantum field theory:
∫ − ∞ ∞ e − 1 2 a x 2 + J x d x = ( 2 π a ) 1 / 2 exp ( J 2 2 a ) , a , J ∈ C , Re ( a ) > 0 ∫ − ∞ ∞ exp ( i ( 1 2 ( a + i ε ) x 2 + J x ) ) d x = ( 2 π i a + i ε ) 1 / 2 exp ( − i 2 J 2 a + i ε ) , a , J , ε ∈ R , ε → 0 + ∫ exp ( ∑ i , j = 1 n − 1 2 x i A i j x j + J i x i ) d n x = ( 2 π ) n det A exp ( 1 2 ∑ i , j = 1 n J i A i j − 1 J j ) , A , J ∈ R , A i j = A j i positive definite ∫ exp ( i ( ∑ i , j = 1 n 1 2 x i ( A + i ε I ) i j x j + J i x i ) ) d n x = ( 2 π ) n det ( A + i ε I ) exp ( − i 2 ∑ i , j = 1 n J i ( A + i ε I ) i j − 1 J j ) , A , J , ε ∈ R , A i j = A j i , ε → 0 + {\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}}}
Integrals with differential operators in the argument As an example consider the integral
∫ exp [ ∫ d 4 x ( − 1 2 φ A ^ φ + J φ ) ] D φ {\displaystyle \int \exp \left[\int d^{4}x\left(-{\frac {1}{2}}\varphi {\hat {A}}\varphi +J\varphi \right)\right]D\varphi }
where A ^ {\displaystyle {\hat {A}}} is a Hermitian differential operator with positive spectra for convergence, φ {\displaystyle \varphi } and J functions of spacetime, and D φ {\displaystyle D\varphi } indicates integration over all possible paths. In analogy with the matrix version of this integral the solution is
∫ exp [ ∫ d 4 x ( − 1 2 φ A ^ φ + J φ ) ] D φ ∝ exp ( 1 2 ∫ d 4 x d 4 y J ( x ) D ( x − y ) J ( y ) ) {\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
A ^ D ( x − y ) = δ 4 ( x − y ) {\displaystyle {\hat {A}}D(x-y)=\delta ^{4}(x-y)}
and D(x − y), called the propagator, is the inverse of A ^ {\displaystyle {\hat {A}}} , and δ 4 ( x − y ) {\displaystyle \delta ^{4}(x-y)} is the Dirac delta function.
Similar arguments yield for A ^ {\displaystyle {\hat {A}}} Hermitian differential operator of any spectra, where the ε → 0 + {\displaystyle \varepsilon \rightarrow 0^{+}} prescription, called Feynman prescription, is treated separately in the last step of all calculations
∫ exp [ i ∫ d 4 x ( 1 2 φ ( A ^ + i ε ) φ + J φ ) ] D φ ∝ exp ( − i 2 ∫ d 4 x d 4 y J ( x ) D ε ( x − y ) J ( y ) ) . {\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, A ^ = ∂ μ ∂ μ {\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 i ε {\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 i ε {\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.
Integral approximation by the method of steepest descent In quantum field theory n-dimensional integrals of the form
∫ − ∞ ∞ exp ( − 1 ℏ f ( q ) ) d n q {\displaystyle \int _{-\infty }^{\infty }\exp \left(-{1 \over \hbar }f(q)\right)d^{n}q}
appear often. Here ℏ {\displaystyle \hbar } is the reduced Planck constant and f is a function with a positive minimum at q = q 0 {\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
∫ − ∞ ∞ exp [ − 1 ℏ ( f ( q 0 ) + 1 2 ( q − q 0 ) 2 f ′ ′ ( q − q 0 ) + ⋯ ) ] d n q . {\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 f ′ ′ {\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.
∫ − ∞ ∞ exp [ − 1 ℏ ( f ( q ) ) ] d n q ≈ exp [ − 1 ℏ ( f ( q 0 ) ) ] ( 2 π ℏ ) n det f ′ ′ . {\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 }}}.}
Integral approximation by the method of stationary phase A common integral is a path integral of the form
∫ exp ( i ℏ S ( q , q ˙ ) ) D q {\displaystyle \int \exp \left({i \over \hbar }S\left(q,{\dot {q}}\right)\right)Dq}
where S ( q , q ˙ ) {\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 ℏ {\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.
Fourier integrals
Dirac delta distribution The Dirac delta distribution in spacetime can be written as a Fourier transform ∫ d 4 k ( 2 π ) 4 exp ( i k ( x − y ) ) = δ 4 ( x − y ) . {\displaystyle \int {\frac {d^{4}k}{(2\pi )^{4}}}\exp(ik(x-y))=\delta ^{4}(x-y).} In general, for any dimension N {\displaystyle N}
∫ d N k ( 2 π ) N exp ( i k ( x − y ) ) = δ N ( x − y ) . {\displaystyle \int {\frac {d^{N}k}{(2\pi )^{N}}}\exp(ik(x-y))=\delta ^{N}(x-y).}
Fourier integrals for finding effective potential Identifying two to two elastic scattering results of Quantum field theory with first Born approximation results from quantum mechanics in the relation M el. ( p i → p f ) = V ( q = p f − p i ) {\textstyle M_{\text{el.}}(p_{i}\rightarrow p_{f})=V(q=p_{f}-p_{i})} where incoming particle undergo elastic scattering, i.e. p i 0 = p f 0 {\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.
Laplacian of 1/r While not an integral, the identity in three-dimensional Euclidean space
− 1 4 π ∇ 2 ( 1 r ) = δ ( r ) {\displaystyle -{1 \over 4\pi }\nabla ^{2}\left({1 \over r}\right)=\delta \left(\mathbf {r} \right)} where r 2 = r ⋅ r {\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
∫ d 3 k ( 2 π ) 3 exp ( i k ⋅ r ) k 2 = 1 4 π r . {\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}.}
Yukawa potential: the Coulomb potential with mass The Yukawa potential in three dimensions can be represented as an integral over a Fourier transform
∫ d 3 k ( 2 π ) 3 exp ( i k ⋅ r ) k 2 + m 2 = e − m r 4 π r {\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 r 2 = r ⋅ r , k 2 = k ⋅ k {\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.
Modified Coulomb potential with mass
∫ d 3 k ( 2 π ) 3 ( k ^ ⋅ r ^ ) 2 exp ( i k ⋅ r ) k 2 + m 2 = e − m r 4 π r [ 1 + 2 m r − 2 ( m r ) 2 ( e m r − 1 ) ] {\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.
Longitudinal potential with mass
∫ d 3 k