39.5 Higher order operators and renormalization
Concept links · terms present in this machine draft; source roles are unverified: eigenvalue · Lie algebra · Lie algebra representation
We have generally restricted ourselves to considering only products of basis elements of the Heisenberg Lie algebra (position and momentum in the finite dimensional case, fields in the infinite dimensional case) of degree less than or equal to two, since it is these that after quantization have an interpretation as the operators of a Lie algebra representation. In the finite dimensional case one can consider higher-order products of operators, for instance systems with Hamiltonian operators of higher order than quadratic. Unlike the quadratic case, typically no exact solution for eigenvectors and eigenvalues will exist, but various approximation methods may be available. In particular, for Hamiltonians that are quadratic plus a term with a small parameter, perturbation theory methods can be used to compute a power-series approximation in the small parameter. This is an important topic in physics, covered in detail in the standard textbooks.
The standard approach to quantization of infinite dimensional systems is to begin with “regularization”, somehow modifying the system to only have a finite dimensional phase space, for instance by introducing cutofs that make the possible momenta discrete and finite. One quantizes this theory by taking the state space and canonical commutation relations to be the unique ones for the Heisenberg Lie algebra, somehow dealing with the calculational dificulties in the interacting case (non-quadratic Hamiltonian).
One then tries to take a limit that recovers the infinite dimensional system. Such a limit will generally be quite singular, leading to an infinite result, and the process of manipulating these potential infinities is called “renormalization”. Techniques for taking limits of this kind in a manner that leads to a consistent and physically sensible result typically take up a large part of standard quantum field theory textbooks. For many theories, no appropriate such techniques are known, and conjecturally none are possible. For others there is good evidence that such a limit can be successfully taken, but the details of how to do this remain unknown, with for instance a $1 million Millenium Prize ofered for showing rigorously this is possible in the case of Yang-Mills gauge theory (the Hamiltonian in this case will be discussed in chapter 46).
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