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Concept# Hubbard–Stratonovich transformation

Summary

The Hubbard–Stratonovich (HS) transformation is an exact mathematical transformation invented by Russian physicist Ruslan L. Stratonovich and popularized by British physicist John Hubbard. It is used to convert a particle theory into its respective field theory by linearizing the density operator in the many-body interaction term of the Hamiltonian and introducing an auxiliary scalar field. It is defined via the integral identity
:
\exp \left{ - \frac{a}{2} x^2 \right} =
\sqrt{\frac{1}{2 \pi a}} ; \int_{-\infty}^\infty
\exp \left[ - \frac{y^2}{2 a} - i x y \right] , dy,
where the real constant a > 0. The basic idea of the HS transformation is to reformulate a system of particles interacting through two-body potentials into a system of independent particles interacting with a fluctuating field. The procedure is widely used in polymer physics, classical particle physics, spin glass theory, and electronic structure theory.
Calculation of resulting fi

Official source

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