Common integrals in quantum field theory

Common integrals in quantum field theory are all variations and generalizations of Gaussian integrals to the complex plane and to multiple dimensions. Other integrals can be approximated by versions of the Gaussian integral. Fourier integrals are also considered. The first integral, with broad application outside of quantum field theory, is the Gaussian integral. In physics the factor of 1/2 in the argument of the exponential is common. Note: Thus we obtain where we have scaled and In general Note that the integrals of exponents and odd powers of x are 0, due to odd symmetry. This integral can be performed by completing the square: Therefore: The integral is proportional to the Fourier transform of the Gaussian where J is the conjugate variable of 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 a is, the narrower the Gaussian in x and the wider the Gaussian in J. This is a demonstration of the uncertainty principle. This integral is also known as the Hubbard–Stratonovich transformation used in field theory. 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) We now assume that a and J may be complex. Completing the square By analogy with the previous integrals 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. See Fresnel integral. The one-dimensional integrals can be generalized to multiple dimensions. Here A is a real positive definite symmetric matrix. This integral is performed by diagonalization of A with an orthogonal transformation where D is a diagonal matrix and O is an orthogonal matrix. This decouples the variables and allows the integration to be performed as n one-dimensional integrations. This is best illustrated with a two-dimensional example.