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arXiv:hep-th/0110176 · Physical Review A

Renormalized path integral for the two-dimensional δ-function interaction

Horacio E. Camblong, CARLOS R. ORDÓÑEZ

Published May 17, 2002• 30 citations
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Abstract

A path-integral approach for \ensuremath{\delta}-function potentials is presented. Particular attention is paid to the two-dimensional case, which illustrates the realization of a quantum anomaly for a scale-invariant problem in quantum mechanics. Our treatment is based on an infinite summation of perturbation theory that captures the nonperturbative nature of the \ensuremath{\delta}-function bound state. The well-known singular character of the two-dimensional \ensuremath{\delta}-function potential is dealt with by considering the renormalized path integral resulting from a variety of schemes: dimensional, momentum-cutoff, and real-space regularization. Moreover, compatibility of the bound-state and scattering sectors is shown.

DOI
10.1103/physreva.65.052123

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A path-integral approach for \ensuremath{\delta}-function potentials is presented. Particular attention is paid to the two-dimensional case, which illustrates the realization of a quantum anomaly for a scale-invariant problem in quantum mechanics. Our treatment is based on an infinite summation of perturbation theory that captures the nonperturbative nature of the \ensuremath{\delta}-function bound state. The well-known singular character of the two-dimensional \ensuremath{\delta}-function potential is dealt with by considering the renormalized path integral resulting from a variety of schemes: dimensional, momentum-cutoff, and real-space regularization. Moreover, compatibility of the bound-state and scattering sectors is shown.
An expansion method in the path-integral formulation of quantum mechanics, as proposed in a previous paper, is extended to account for states with odd parity. The method is tested on the case of a one-dimensional attractive delta-function potential and the well-known solutions for both bound and scattering states are obtained analytically by an exact summation of the expansion series. The applicability of the method is discussed and it is shown how the energy spectrum could be determined by means of Stieltjes theory of moments.
Abstract The incorporation of two‐ and three‐dimensional Δ‐function perturbations into the path integral formalism is discussed. Contrary to the one‐dimensional case, a regularization prescription is needed due to the divergence of the Green function corresponding to a potential V, G(v) (x,y;E) (x, y ϵ ℝ2, ℝ3) for x→y. The known procedure to define proper self‐adjoint extensions for Hamiltonians with point‐interactions can be exploited to define the incorporation of δ‐function perturbations in the path integral. Several examples illustrate the formalism.
A path-integral approach for the computation of quantum-mechanical propagators and energy Green's functions is presented. Its effectiveness is demonstrated through its application to singular interactions, with particular emphasis on the inverse square potential — possibly combined with a delta-function interaction. The emergence of these singular potentials as low-energy nonrelativistic limits of quantum field theory is highlighted. Not surprisingly, the analog of ultraviolet regularization is required for the interpretation of these singular problems.
A system of two-species, one-dimensional fermions, with an attractive two-body interaction of the derivative-delta type, features a scale anomaly. In contrast to the well-known two-dimensional case with contact interactions, and its one-dimensional cousin with three-body interactions (studied recently by some of us and others), the present case displays dimensional transmutation featuring a power-law rather than a logarithmic behavior. We use both the Schrödinger equation and quantum field theory to study bound and scattering states, showing consistency between both approaches. We show that the expressions for the reflection (R) and the transmission (T) coefficients of the renormalized, anomalous derivative-delta potential are identical to those of the regular delta potential. The second-order virial coefficient is calculated analytically using the Beth–Uhlenbeck formula, and we make comments about the proper ϵB→0 (where ϵB is the bound-state energy) limit. We show the impact of the quantum anomaly (which appears as the binding energy of the two-body problem, or equivalently as Tan's contact) on the equation of state and on other universal relations. Our emphasis throughout is on the conceptual and structural aspects of this problem.