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Abstract. We analyze the spectral structure of the one dimensional quantum mechanical system with point interaction, which is known to be parametrized by the group U (2). Based on the classification of the interactions in terms of symmetries, we show, on a general ground, how the fermion-boson duality and the spectral anholonomy recently discovered can arise. A vital role is played by a hidden su(2) formed by a certain set of discrete transformations, which becomes a symmetry if the point interaction belongs to a distinguished U (1) subfamily in which all states are doubly degenerate. Within the U (1), there is a particular interaction which admits the interpretation of the system as a supersymmetric Witten model.

The quantum dynamics of a free particle on a circle with point interaction is described by a U (2) family of self-adjoint Hamiltonians. We provide a classification of the family by introducing a number of subfamilies and thereby analyze the spectral structure in detail. We find that the spectrum depends on a subset of U (2) parameters rather than the entire U (2) needed for the Hamiltonians, and that in particular there exists a subfamily in U (2) where the spectrum becomes parameter-independent. We also show that, in some specific cases, the WKB semiclassical approximation becomes exact (modulo phases) for the system.

Abstract. It has been known for some time that there are many inequivalent quantizations possible when the configuration space of a system is a coset space G/H. Viewing this classical system as a constrained system on the group G, we show that these inequivalent quantizations can be recovered from a generalization of Dirac's approach to the quantization of such a constrained system within which the classical first class constraints (generating the H-action on G) are allowed to become anomalous (second class) when quantizing. The resulting quantum theories are characterized by the emergence of a Yang-Mills connection, with quantized couplings, and new 'spin' degrees of freedom. Various applications of this procedure are presented in detail: including a new account of how spin can be described within a pathintegral formalism, and how on S 4 chiral spin degrees of freedom emerge, coupled to a BPST instanton.

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