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arXiv:hep-th/9402110 · Journal of Physics A Mathematical and General

delta '-function perturbations and Neumann boundary conditions by path integration

C Grosche

Published Feb 7, 1995• 29 citations
arXivHTMLPDF

Abstract

delta '-function perturbations and Neumann boundary conditions are incorporated into the path integral formalism. The starting point is consideration of the path integral representation for the one-dimensional Dirac particle together with a relativistic point interaction. The nonrelativistic limit yields either the usual delta -function or a delta '-function perturbation; making their strengths infinitely repulsive one respectively obtains either Dirichlet or Neumann boundary conditions in the path integral.

DOI
10.1088/0305-4470/28/3/007

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delta '-function perturbations and Neumann boundary conditions are incorporated into the path integral formalism. The starting point is consideration of the path integral representation for the one-dimensional Dirac particle together with a relativistic point interaction. The nonrelativistic limit yields either the usual delta -function or a delta '-function perturbation; making their strengths infinitely repulsive one respectively obtains either Dirichlet or Neumann boundary conditions in the path integral.
#delta#'-function perturbations and Neumann boundary conditions are incorporated into the path integral formalism. The starting point is the consideration of the path integral representation for the one dimensional Dirac particle together with a relativistic point interaction. The non-relativistic limit yields either a usual #delta#-function or a #delta#'-function perturbation; making their strengths infinitely repulsive one obtains Dirichlet, respectively Neumann boundary conditions in the path integral. (orig.)
A proposal is given, of how to implement point interactions and boundary conditions in the path integral. The starting point is a path-integral formulation of a Dirac particle in one dimension. The implementation of a point interaction yields, by means of a perturbation expansion, the corresponding Green function for a relativistic particle. In the non-relativistic limit several cases can be distinguished depending on which of the four possibilities of the point interaction has been chosen. By a proper combination of the various possibilities of implementing the point interaction, the whole range of the four-parameter family of boundary conditions for point interactions can be exploited, in the relativistic case as well as in the non-relativistic limit. In addition, making the strength of the point interaction infinitely repulsive yields boundary conditions at finite points on the real line. In particular, Dirichlet and Neumann boundary conditions emerge. The method is illustrated with some examples.
Abstract A wide class of boundary problems in quantum mechanics is discussed by using path integrals. This includes motion in half‐spaces, radial boxes, rings, and moving boundaries. As a preparation the formalism for the incorporation of δ‐function perturbations is outlined, which includes the discussion of multiple δ‐function perturbations, δ‐function perturbations along perpendicular lines and planes, and moving δ‐function perturbations. The limiting process, where the strength of the δ‐function perturbations gets infinitely repulsive, has the effect of producing impenetrable walls at the locations of the δ‐function perturbations, i.e. a consistent description for boundary problems with Dirichlet boundary‐condition emerges. Several examples illustrate the formalism.
The formulation of the relativistic spinless path integral on the general affine space is presented. For the one dimensional space, the Duru-Kleinert (DK) method and the δδδ-function perturbation technique are applied to solve the relativistic path integrals of the smooth potential and the Dirichlet boundary condition problems, respectively.
We have developed a unified perspective for boundary conditions concerned with (I) a direct relativistic treatment of a \ensuremath{\delta}-function potential and (II) the \ensuremath{\delta}-function limit of solutions for relativistic treatment of peaked potential. We have presented a critical discussion of our findings, especially in the context of (i) certain conditions related very importantly to this unified perspective and (ii) the difficulty in making a choice between the approaches (I) and (II) on the basis of mathematical analyses alone.
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.