Renormalized Path Integral for the Two-Dimensional Delta-Function Interaction

Abstract

A path-integral approach for 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 delta-function bound state. The well-known singular character of the two-dimensional 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.

26 pages. The paper was significantly expanded and numerous equations were added for the sake of clarity; the main results and conclusions are unchanged

References in corpus (20)

• Towards a Perturbative Theory of Nuclear Forces
• Effective Field Theory of Nuclear Forces
• Singular Potentials and Limit Cycles
• Renormalization in Quantum Mechanics
• Renormalization of the Inverse Square Potential
• Non-perturbative regularization and renormalization: simple examples from non-relativistic quantum mechanics
• Quantum Anomaly in Molecular Physics
• Path Integration Via Summation of Perturbation Expansions and Applications to Totally Reflecting Boundaries, and Potential Steps
• Perturbative Renormalization in Quantum Mechanics
• Dimensional Transmutation and Dimensional Regularization in Quantum Mechanics. I. General Theory
• Path integral treatment of two- and three-dimensional delta-function potentials and application to spin-1/2 Aharonov-Bohm problem
• Dimensional Transmutation and Dimensional Regularization in Quantum Mechanics. II: Rotational Invariance
• Physical mechanisms generating spontaneous symmetry breaking and a hierarchy of scales
• The regulated four parameter one dimensional point interaction
• Feynman-Schwinger representation approach to nonperturbative physics
• Proper incorporation of self-adjoint extension method to Green's function formalism : one-dimensional ${\delta }^{{}^{\prime }}$-function potential case
• On Scale Invariance and Anomalies in Quantum Mechanics
• Self-Adjoint Extensions of the Pauli Equation in the Presence of a Magnetic Monopole
• Renormalized Contact Potential in Two Dimensions
• Local realizations of contact interactions in two- and three-body problems

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