Common integrals in quantum field theory: Difference between revisions

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There are '''common integrals in quantum field theory''' that appear repeatedly.<ref>{{cite book| author=A. Zee| title=Quantum Field Theory in a Nutshell| publisher= Princeton University| year=2003 | isbn=0-691-01019-6}} pp. 13-15</ref> These integrals are all variations and generalizations of [[gaussianGaussian integral|Gaussian integrals]]s to the complex plane and to multiple dimensions. Other integrals can be approximated by versions of the gaussianGaussian integral. Fourier integrals are also considered.

==Variations on a simple gaussianGaussian integral==

===Gaussian integral===

The first integral, with broad application outside of quantum field theory, is the [[gaussianGaussian integral]].

:<math> G \equiv \int_{-\infty}^{\infty} e^{-{1 \over 2} x^2}\,dx</math>

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:<math> \int_{-\infty}^{\infty} e^{-{1 \over 2} x^2}\,dx = \sqrt{2\pi}. </math>

===Slight generalization of the gaussianGaussian integral===

:<math> \int_{-\infty}^{\infty} e^{-{1 \over 2} a x^2}\,dx = \sqrt{2\pi \over a} </math>

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:<math> \int_{-\infty}^{\infty} \exp\left( -{1 \over 2} a x^2 + iJx\right ) dx = \left ( {2\pi \over a } \right ) ^{1\over 2} \exp\left( -{ J^2 \over 2a }\right ) </math>

is proportional to the [[Fourier transform]] of the gaussianGaussian where {{mvar|J}} is the [[conjugate variables|conjugate variable]] of {{mvar|x}}.

By again completing the square we see that the Fourier transform of a gaussianGaussian is also a gaussianGaussian, but in the conjugate variable. The larger {{mvar|a}} is, the narrower the gaussianGaussian in {{mvar|x}} and the wider the gaussianGaussian in {{mvar|J}}. This is a demonstration of the [[uncertainty principle]].

This integral is also known as the [[Hubbard-Stratonovich transformation]] used in field theory.