Superselection rules, quantisation by parts and point interactions
Abstract
This thesis is concerned with the origination and development of mathematical tools, formalisms and idealised physical models, for the analysis of superselection rules and point interactions in quantum theory. We begin by presenting a unified Hilbert space formalism, capable of describing classical, standard quantum and mixed quantum systems in Hilbert space. Superselection rules enable us to construct this unified theory. Incorporating classical systems into a Hilbert space formalism improves our resolution of the boundary between the classical and the quantum, and enables us to compare and contrast a variety of classical, quantum and mixed quantum systems. This comparison yields a new definition of classical systems, as we find the compatibility of a set of observables is no longer a sufficient condition to define a classical system. We find that we can distinguish the three types of systems if we move to discuss their dynamical behaviour. An interesting consequence of casting classical systems into Hilbert space is that we are forced to make a distinction between the operator which generates the dynamics for the system, and the Hamiltonian. We are compelled to make this distinction because we find that the operator which generates the dynamics is not an observable of the system. We also discuss equilibrium and inequilibrium systems and the possibilities of including environmental effects in the formalism. r We develop a novel and rigorous approach to deal with single point interactions in one dimension. We combine the disciplines of selfadjoint extension theory and transfer matrix analysis, to produce a strong analytical quantisation tool. This tool provides a catagorisation of a wide spectrum of possible point interactions in one-dimension. We show that the behaviour of a system with only one single point interaction can be extremely diverse. In particular, new possible behaviours are predicted. Superconducting systems and their inherent superselection rules, can also be admirably handled by this approach. We then move on to analyse a variety of interesting geometries with this method, including quantum dot systems. We find that our new quantisation tool again predicts new behaviour. We present a path space approach to quantum theory, which can be seen to be significantly more intuitive than the standard approach in some cases. The example presented is a ring containing a single point interaction. In this case the path space approach provides an interpretation of flow around the ring and a natural choice for the momentum operator. The path space approach also inherently possesses the required superselection rule when we consider a superconducting system with this geometry. These insights are not available in the traditional approach. Finally we present a discussion of the five distinct origins of superselective behaviour evident in this thesis, and draw conclusions as to whether a definitive origin of this phenomenon may be found in quantum theory.
Type
Thesis, PhD Doctor of Philosopy
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