Hubbard–Stratonovich transformation

Summary

The Hubbard–Stratonovich transformation writes the exponential of a quadratic expression as a Gaussian integral over an auxiliary variable. The integrand is an exponential that is linear in the original variables. In a partition function this replaces an interaction between particles or spins by the coupling of each of them to a fluctuating auxiliary field. The particles then become independent for each fixed value of the field. The remaining integral over the field can be treated by saddle-point methods or numerically. It is the standard route from interacting models to field theories and to mean-field theory.