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Renormalized Path Integral for the Two-Dimensional Delta-Function Interaction
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.Comment: 26 pages. The paper was significantly expanded and numerous equations
were added for the sake of clarity; the main results and conclusions are
unchange