Perturbative Renormalization in Quantum Mechanics

Some quantum mechanical potentials, singular at short distances, lead to ultraviolet divergences when used in perturbation theory. Exactly as in quantum field theory, but much simpler, regularization and renormalization lead to finite physical results, which compare correctly to the exact ones. The Dirac delta potential, because of its relevance to triviality, and the AharonovBohm potential, because of its relevance to anyons, are used as examples here. PACS numbers: 03.65.-w, 03.65.Ca, 03.65.Ge, 03.65.Nk. Bitnet PALAS@EBUBECM1 Bitnet ROLF@EBUBECM1 An important instrument of present days physics, quantum field theory, is permeated by short distance singularities, which are thoroughly understood in the framework of regularized and renormalized perturbation theory. Our non-perturbative understanding, mainly lattice-bound, is not so firm, and exact solutions for physically relevant theories are basically absent. Quantum mechanics does not usually have short distance singularities, but they show up if the potential is singular enough (but not too much: the Hamiltonian should be bounded from below and self-adjoint). Then regularization and renormalization consistently cure the short distance singularities and lead to physical results independent of the precise regulator and independent of the precise renormalization scheme. Furthermore, renormalized perturbation theory reproduces the exact solutions for physical magnitudes. The two problems we have chosen to study are best considered in two dimensions. First, the Dirac delta, zero-range or contact interaction, because already its exact solution is most conveniently obtained by regulating and renormalizing, because one can perform perturbation theory to all orders, and because of its relevance to triviality [1]. Second, the Aharonov-Bohm potential, because it perturbatively induces a new interaction absent in the exact setting and because of being at the foundation of anyon physics [2], [3]. Recall that the Schrödinger equation (throughout this letter we will use 2M = h̄ = 1) for positive energies E = k is equivalent to the LippmanSchwinger equation