Exactly solvable supersymmetric quantum mechanics

In an abstract framework, we present a class of supersymmetric quantum mechanics whose eigenvalue problem is (in part) exactly solvable. In concrete realizations, the class includes supersymmetric quantum mechanical models associated with one-dimensional or radial Schrodinger operators with potentials of a special type, called “shape-invariant potentials” in the physics literature.

A general class of infinite dimensional Dirac operators and path integral representation of their index

The unitary transformations which convert an abstract Dirac operator into an ‘‘even’’ (resp. ‘‘odd’’) operator are determined. The problem is formulated and solved completely within the general setup of supersymmetric quantum mechanics. This leads to some apparently new applications in relativistic quantum mechanics, where the transformations are known as the Foldy–Wouthuysen (resp. Cini–Touschek) transformations. The scattering theory for abstract Dirac operators is discussed and the utility of the general theory is illustrated by proving existence of relativistic Mo/ller operators for scattering from long‐range magnetic fields.

Normal forms of an abstract Dirac operator and applications to scattering theory

A formalism comprising and extending quasifree second quantization of charged bosons and fermions is presented. The second quantization of one‐particle observables leads to current superalgebras, and a super‐Schwinger term shows up. Anticommuting parameters are introduced in order to construct super‐Bogoliubov transformations mixing bosons and fermions. As an application, representations of Lie superalgebras are given which are semidirect products of extensions of affine Kac–Moody algebras and the Virasoro algebra, and of the super‐Virasoro algebra.

A superversion of quasifree second quantization. I : Charged particles

A formulation of supersymmetric quantum mechanics in terms of sesquilinear forms is presented. Otherwise occurring rather complicated domain questions are solved in a simple and natural way. The authors formulate the axioms and describe a number of models. Supersymmetry transformations are implemented as superunitary transformations in a space defined over a Grassmann algebra, generalised Bogoliubov transformations acting in that space are also formulated.

http://iopscience.iop.org/article/10.1088/0305-4470/20/13/030/pdf

L Pittner

Papers

Supersymmetric quantum mechanics defined as sesquilinear forms