In this work we consider non-relativistic quantum mechanics, obtained from a classical configuration space Q of indistinguishable particles. Following an approach proposed in [8], wave functions are regarded as elements of suitable projective modules over C(Q). We take furthermore into account the G-Theory point of view (cf. [4, 10]) where the role of group action is particularly emphasized. As an example illustrating the method, the case of two particles is worked out in detail. Previous works (cf. [2,3]) aiming at a proof of a spin-statistics theorem for non-relativistic quantum mechanics are re-considered from the point of view of our approach, enabling us to clarify several points.
A recently proposed, new construction of the Standard Model based on the graded Lie algebra SU (2|1) is analyzed in some depth. The essential ingredient is an algebraic superconnection which incorporates both the gauge fields and the Higgs fields and whose curvature automatically leads to a spontaneously broken realization of the theory. The mechanism of hiding the original algebraic structure is unorthodox and is due to the specific, "noncommutative" realization of SU (2|1). The model is characterized by a constant background supercurvature which is invariant under arbitrary, constant SU (2|1) gauge transformations. This background field whose effect is analogous to the action of a constant magnetic field on a spherical atom, is traced back to the differential in the space of (super)matrices by means of which the supercurvature is constructed. The same background field is responsible for the fact that the ground state has no more than the U (1) e.m. symmetry of electromagnetism, the SU (2)L × U (1) symmetry of the Standard Model being recovered only after "backshifting" the Higgs fields. Thus, the Higgs mechanism receives a new and geometrical interpretation.
Within a geometric and algebraic framework, the structures which are related to the spin-statistics connection are discussed. A comparison with the BerryRobbins approach is made. The underlying geometric structure constitutes an additional support for this approach. In our work, a geometric approach to quantum indistinguishability is introduced which allows the treatment of singlevaluedness of wave functions in a global, model independent way.
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