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.
Subscribe to:
Post Comments (Atom)
Lamenting the destruction of science in the USA
I continue to follow the situation in the USA concerning the future of science with concern. Here are some of the articles I found most info...

-
Is it something to do with breakdown of the Born-Oppenheimer approximation? In molecular spectroscopy you occasionally hear this term thro...
-
I welcome discussion on this point. I don't think it is as sensitive or as important a topic as the author order on papers. With rega...
-
Nitrogen fluoride (NF) seems like a very simple molecule and you would think it would very well understood, particularly as it is small enou...
No comments:
Post a Comment