What is the relation between renormalization and self-adjoint extension?

Question

What is the relation between renormalization and self-adjoint extension? It seems that a renormalization scheme can be rigorously treated mathematically using the self-adjoint extension theory.

Is there any good reference on it?

Answer 1

Quantum dynamics is commonly known to be generated by self-adjoint operators. Therefore in order to properly define the dynamics of a system it is necessary to introduce a suitable self-adjoint Hamiltonian operator.

In quantum field theories, this task is extremely difficult, because the formal operators that emerge quantizing a classical field theory are---borrowing a term used by Glimm---plagued with divergences. In practice, apart from exceptionally easy situations, interacting Hamiltonians of QFT make sense only as quadratic forms, and their domain of definition as operators is only the vector zero.

The first task is then to obtain at least a densely defined symmetric operator that describes the system. This is already extremely difficult, and can be achieved only in few situations. This operation requires the subtraction from the operator of infinite quantities, but often also a renormalization of the vectors of the Hilbert space: the final space will be, in general, different from the original one.

After all that (most of the times it has been done independently, and with different tools such as functional integration), it is also necessary to prove that such densely defined operator is self-adjoint, or it has self-adjoint extensions. The "easy way" is to prove that it is bounded from below, so it has at least one self-adjoint extension.

Suppose you have boundedness from below, then you have a well-defined renormalized dynamics for the system. Now it is suitable that the aforementioned dynamics is also unique, because in principle different extensions of the Hamiltonian generate different dynamics with different properties. In order to do that, it is necessary either to prove essential self-adjointness of the operator, or that it is bounded from below and the corresponding form is closed.

This is the ambitious program of the Constructive QFT started in the sixties---some of the most influential contributors were Wightman, Glimm, Jaffe, Nelson---and it has achieved some definitive result only in few dimensions (at most 2+1), and in simple situations. Most of the time it is already a great achievement to obtain a renormalized densely defined symmetric operator!

In this brief description, I have only outlined the non-perturbative approach to renormalization that is related to your question on self-adjointness. To sum it up: the problems with renormalization of the Hamiltonians of QFT are very difficult already at the fundamental level of "definition of a symmetric operator". Existence of self-adjoint extensions is a subsequent question, in some sense unrelated (because boundedness from below can be proved "a priori" giving cut off independent informations about the regularized operators). Good references on constructive quantum field theory are the books by Glimm and Jaffe, e.g. their collected papers.

Answer 2

A great elementary reference is this paper by Essin and Griffiths, which just considers the simple problem of one quantum-mechanical particle in a one-dimensional potential $V(x) \propto -1/x^2$. This simple setup is enough to introduce quite sophisticated ideas like renormalization, anomalous symmetry breaking, and self-adjoint extensions.

First, they demonstrate that the Hamiltonian is not (naively) bounded below and so not "nice." Second, they do renormalization the way a physicist would do it - they introduce a regulator on the bare Hamiltonian, calculate observables in terms of the bare couplings and the regulator, then send the regulator to infinity while simultaneously modifying the bare couplings in such a way as to keep the regulated physically observable quantities fixed. They also show that the system has a quantum anomaly that spontaneously breaks scale invariance, just like QED. Third, they renormalize the Hamiltonian the way a mathematician would do it: they show that the Hamiltonian is Hermitian but not self-adjoint, and that there exists a one-parameter family of self-adjoint extensions to the naive Hilbert space. One must specify a particular choice of self-adjoint extension in this family in order to uniquely specify the Hamiltonian. The "physical observable" quantities are defined to be those that do not depend on the choice of self-adjoint extension. Physically, the choice of self-adjoint extension is almost precisely equivalent to choosing the value of the UV regulator in the physicist's treatment. So strictly speaking, the Hamiltonian is only uniquely defined (as a self-adjoint operator) if you specify a cutoff, but the things we care about turn out to be cutoff-independent.

I don't know whether it's still the case in quantum field theory RG that different choices of UV cutoff correspond to different self-adjoint extensions of the Hamiltonian. I'm not sure if anyone knows for sure.