Common integrals in quantum field theory are all variations and generalizations of Gaussian integrals to the complex plane and to multiple dimensions.
is proportional to the Fourier transform of the Gaussian where J is the conjugate variable of x.
This result is valid as an integration in the complex plane as long as a is non-zero and has a semi-positive imaginary part.
The one-dimensional integrals can be generalized to multiple dimensions.
This integral is performed by diagonalization of A with an orthogonal transformation
Substitution of the eigenvalues back into the eigenvector equation yields
With the two-dimensional example it is now easy to see the generalization to the complex plane and to multiple dimensions.
In analogy with the matrix version of this integral the solution is
See Path-integral formulation of virtual-particle exchange for an application of this integral.
In quantum field theory n-dimensional integrals of the form
is the reduced Planck constant and f is a function with a positive minimum at
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
is the classical action and the integral is over all possible paths that a particle may take.
In this approximation the integral is over the path in which the action is a minimum.
The Dirac delta distribution in spacetime can be written as a Fourier transform[1]: 23
While not an integral, the identity in three-dimensional Euclidean space
is a consequence of Gauss's theorem and can be used to derive integral identities.
This identity implies that the Fourier integral representation of 1/r is
The Yukawa potential in three dimensions can be represented as an integral over a Fourier transform[1]: 26, 29
See Static forces and virtual-particle exchange for an application of this integral.
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.
Note that in the small m limit the integral reduces to
For large distance, the integral falls off as the inverse cube of r
The angular integration of an exponential in cylindrical coordinates can be written in terms of Bessel functions of the first kind[4][5]: 113
For applications of these integrals see Magnetic interaction between current loops in a simple plasma or electron gas.
For an application of this integral see Two line charges embedded in a plasma or electron gas.
For applications of this integral see Magnetic interaction between current loops in a simple plasma or electron gas.
The two-dimensional integral over a magnetic wave function is[6]: §11.4.28
For an application of this integral see Charge density spread over a wave function.