Common Integrals in Quantum Field Theory - Gaussian Integrals in Higher Dimensions

Gaussian Integrals in Higher Dimensions

The one-dimensional integrals can be generalized to multiple dimensions.

$\int \exp\left( - \frac 1 2 x \cdot A \cdot x +J \cdot x \right) d^nx = \sqrt{\frac{(2\pi)^n}{\det A}} \exp \left( {1\over 2} J \cdot A^{-1} \cdot J \right)$

Here is a real symmetric matrix.

This integral is performed by diagonalization of with an orthogonal transformation

$D_{ }^{ } = O^{-1} A O = O^T A O$

where is a diagonal matrix and is an orthogonal matrix. This decouples the variables and allows the integration to be performed as one-dimensional integrations.

This is best illustrated with a two-dimensional example.

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