Quantum mechanical symmetries and topological invariants

Abstract

We give the definition and explore the algebraic structure of a class of quantum symmetries, called topological symmetries, which are generalizations of supersymmetry in the sense that they involve topological invariants similar to the Witten index. A topological symmetry (TS) is specified by an integer n > 1, which determines its grading properties, and an n-tuple of positive integers (m 1 , m 2 ,. .. , m n). We identify the algebras of supersymmetry, p = 2 parasupersymmetry, and fractional supersymmetry of order n with those of the Z 2-graded TS of type (1, 1), Z 2-graded TS of type (2, 1), and Z n-graded TS of type (1, 1,. .. , 1), respectively. We also comment on the mathematical interpretation of the topological invariants associated with the Z n-graded TS of type (1, 1,. .. , 1). For n = 2, the invariant is the Witten index which can be identified with the analytic index of a Fredholm operator. For n > 2, there are n independent integer-valued invariants. These can be related to differences of the dimension of the kernels of various products of n operators satisfying certain conditions.

A relation betweenZ3-graded symmetry and shape invariant supersymmetric systems

“…In this article, the focus is on a particular property of every shape invariant SUSY QM system, with unbroken supersymmetry. Specifically, we shall demonstrate that each shape invariant SUSY QM system can constitute a Z 3 -graded symmetric quantum mechanical system [19,20]. Topological symmetries in the spirit of references [19,20], are symmetries that generalize the concept of the Witten index and therefore provide useful insights to further develop the algebraic structure of SUSY QM systems.…”

“…Specifically, we shall demonstrate that each shape invariant SUSY QM system can constitute a Z 3 -graded symmetric quantum mechanical system [19,20]. Topological symmetries in the spirit of references [19,20], are symmetries that generalize the concept of the Witten index and therefore provide useful insights to further develop the algebraic structure of SUSY QM systems. We shall establish the result that shape invariant systems can constitute such Z 3 -graded systems and therefore, since shape invariance is actually imposed as an ad-hoc relation without having a deeper structural-algebraic reason, our results might shed some light towards the problem of finding a deeper explanation of the occurrence of shape invariance.…”

“…Before we present the essentials of a Z 3 -graded topological symmetry, we briefly explain what a generalized Z k -graded symmetry [19,20] for a quantum mechanical system is, and also try to explain where the terminology topology comes from, always based on the references [19,20]. In order to do so, we shall need some definitions.…”

“…In order to do so, we shall need some definitions. Following [19,20], let  be the Hilbert space of a quantum system and let n an integer and , ,..., n…”

“…In this section we shall demonstrate that each shape invariant supersymmetric quantum system can constitute a Z 3 -graded system. In order to make the article self-contained, we shall present the necessary information on the Z 3 symmetry, following closely [19,20].…”