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Perturbative Renormalization in Quantum Mechanics

Cristina Manuel, R. Tarrach

Published Jan 1, 1993• 45 citations
OpenAlex

Abstract

Some quantum mechanical potentials, singular at short distances, lead to ultraviolet divergences when used in perturbation theory. Exactly as in quantum field theories, 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 Aharonov-Bohm potential, because ot its relevance to anyons, are used as examples here.

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Some quantum mechanical potentials, singular at short distances, lead to ultraviolet divergences when used in perturbation theory. Exactly as in quantum field theories, 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 Aharonov-Bohm potential, because ot its relevance to anyons, are used as examples here.
As in quantum field theory, for some singular two-body potentials, the momentum-space perturbative treatment of the three-dimensional quantum NNN-body problem is shown to exhibit ultraviolet divergences. Renormalization of these models leads to a scale and finite observables. For a Dirac delta potential we carry on this renormalization for two- and three-body systems. A new divergence, and hence a new scale, emerges for each added particle. For $N\ensuremath{\ge}3$ there are no divergences in one and two dimensions. The physical implications of these results are discussed.
In bosonic and perturbative calculations for quantum mechanical anyon systems a regularization and renormalization procedure, analogous to those used in field theory, is necessary. I examine the reliability and the physical interpretation of the most commonly used bosonic end regularization procedures. I then use the regularization procedure with the most transparent physical intepretation to derive some bosnic end perturbation theory results on anyon spectra, including a three-anyon ground state energy.
We study conformal quantum mechanics by first considering the perturbative SSS-matrix in various dimensions. The model has two couplings and we study perturbatively the degree of ultraviolet divergences arising in the interplay between the two couplings. We then focus on the inverse square potential in one spatial dimension and compute the beta function to arbitrarily perturbative and non-perturbative orders. This we do in both the bound state sector and scattering sector. We provide explicit, exact and infinite series results of the first few non-perturbative orders.
Bosonic end perturbative calculations for quantum mechanical anyon systems require a regularization. I regularize by adding a specific delta\\deltadelta-function potential to the Hamiltonian. The reliability of this regularization procedure is verified by comparing its results for the 2-anyon in harmonic potential system with the known exact solutions. I then use the delta\\deltadelta-function regularized bosonic end perturbation theory to test some recent conjectures concerning the unknown portion of the many-anyon spectra.
We study perturbation theory in certain quantum mechanics problems in one dimension in which the perturbing potential diverges at some points, even though the energy eigenvalues are smooth functions of the coefficient of the potential. We discuss some of the unusual techniques which are required to obtain perturbative expansions of the energies in such cases. These include a point-splitting prescription with a δ′′ function for expanding around the Dirichlet (fermionic) limit of the δ function potential, and performing a similarity transformation to a non-Hermitian potential in the Calogero–Sutherland model. As an application of the first technique, we study the ground state of the δ function Bose gas near the fermionic limit.
We show how to derive fixed-point Hamiltonians in quantum mechanics from a proposed renormalization group invariance approach that relies in a subtraction procedure at a given energy scale. The scheme is valid for arbitrary interactions that originally contain point-like singularities (as the Dirac-delta and/or its derivatives). One example of diagonalization of a fixed-point Hamiltonian illustrates the procedure for an interaction that contains regular and singular parts. In another example, the fixed point Hamiltonian is derived for the case of higher order singularities in the interaction.