 Hubbard–Stratonovich transformation

A Gaussian auxiliary-field identity replacing a quadratic expression in a composite field by a linear coupling to a new field. Completing the square in $M^2/(2U)-Mj$ yields an exponential $e^{U{j}^2/2}$ after integrating $M$, with normalized Gaussian prefactors. It converts suitable quartic field interactions into bilinear couplings and permits fermionic fields to be integrated out.

The Hubbard-Stratonovich transformation is a mathematical technique commonly used in theoretical physics, particularly in the fields of many-body physics and quantum field theory. It is used to simplify the analysis of interacting systems by transforming products of exponentials into more manageable forms involving auxiliary fields. ### Context In statistical mechanics and quantum field theory, one often encounters partition functions or path integrals involving quadratic forms, particularly in the context of fermionic or bosonic systems.