Completing the square

When studying quantum many-body theory, sometimes one gets lost in all the indices, functional integrals, Feynman diagrams, ...
Then one can lose sight of the fact that some techniques are really just the same as in simple mathematics. Examples include the method of steepest descent and cumulant expansions.

In basic algebra, a simple exercise is to complete the square in a quadratic equation, i.e. to make use of the following identity.

Suppose one has the following Hamiltonian. If describes a field q that couples linearly to a different field s, with a coupling constant s.

 Now if we complete the square and do a displacement of the field q we are left with the new Hamiltonian.

This now describes a free field q (i.e. non-interacting) and there is an attractive self-interaction of the field s with coupling constant a^2.

A related example is the Hubbard-Stratonovich_transformation. This allows one to introduce a new field that couples to the original field and then ``integrate out" the original field to leave a new interacting field theory. Two important and related examples are the following.

1. The Ising model is equivalent to a Landau theory for a scalar field (order parameter) and so they are in the same universality class. There is a nice treatment of this in Negele and Orland

2. Introduction of a superconducting order parameter to describe a fermion system with an attractive four-fermion interaction in the Cooper channel. There is a natural generalisation to superfluid 3He. I first encountered this approach in a book by Popov.