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arXiv:1411.4127

A group theoretical approach to quantum theory of the interaction for a non-relativistic spin-0 particle

Giuseppe Nisticò

Published Nov 15, 2014• 0 citations
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Abstract

We show a possibility of extending the group theoretical approach which deductively establish the quantum theory of a free particle, developed in particular by Mackey, to the interacting particle case.

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We derive the representation theory of SU(2)SU(2)SU(2) from the expository theory of Lie groups and Lie algebras. Based on this, the mathematics of non-relativistic quantum mechanics of a spin 12\frac{1}{2}21​ particle are described from a representation-theoretic perspective, and are extended to many particle systems.
The diffculties of relativistic particle theories formulated my means of canonical quantization, such as Klein-Gordon and Dirac theories, ultimately led theoretical physicists to turn on quantum field theory to model elementary particle physics. The aim of the present work is to pursue a method alternative to canonical quantization that avoids these dfficulties. In order to guarantee this result, the present approach is constrained to be developed deductively from physical principles. The physical principles assumed for a free particle consist of the symmetry properties of the particle with respect to the Poincaré group and of the transformation properties of the position observable, expressed by means of a suitably conceived notion of quantum transformation. In so doing, the effectiveness of group theoretical methods is exploited. Our work has pointed out the necessity of new classes of irreducible representations of the Poincaré group the theory can be based on. For spin 0 particle, four inequivalent theories are completely determined, that differ from Klein-Gordon theory for the physical interpretation but also for the mathematical structures. In particular, two of these theories requires irreducible representations in the new classes. The present lack of the explicit transformation properties of position with respect to boosts forbids the complete determination of non zero spin particle theories. A particular form of these transformation properties was in past adopted by Jordan and Mukunda. We check its consistency within the present approach; it is found that for spin 1/2 particles there is only one consistent theory; it is unitarily related to Dirac theory, but, once again, it requires the new classes of irreducible representations.
establish the Quantum Theory of a free particle for which Galileian transformations form a symmetry group, are extended to the case of an interacting particle. In doing so, the obstacles caused by loss of symmetry are overcome. In this approach, specific forms of the wave equation of an interacting particle, including the equation derived from the minimal coupling principle, are implied by particular first-order invariance properties that characterize the interaction with respect to specific subgroups of Galileian transformations; moreover, the possibility of yet unknown forms of the wave equation is left open.