Canonical group quantisation on one-dimensional complex projective space

In this thesis we study the idea of quantisation approach to study the mathematical formalism of quantum theory with the intent to relate it with the idea of geometry of quantum states, particularly, Isham’s group-theoretic quantisation technique to quantise compact manifold. The core of the discussions is based upon the Isham’s quantisation programme and the compact classical phase space S2 andCP1. In Chapter 2, we review some of the literature that give some motivations to our investigation and also of those closely related to our present work. In Chapter 3, we emphasize on reviewing several mathematical ingredients needed and also the idea of Isham’s group-theoretic quantisation method and discussed some insights to further the investigation in the subsequent chapter. Chapter 4 consists of the author’s original contributions to the thesis. In this chapter, by using the aforementioned technique proposed in Chapter 3, we quantise the systems on one-dimensional complex projective space which is topologically homeomorphic to two-dimensional sphere. These two topological spaces are regarded as the underlying compact phase spaces for which there is no longer a cotangent bundle structure. These spaces have natural symplectic structure that allows one to use them for quantisation. The crucial part is to identify canonical group that acts on the phase space. The first phase is completed by finding all the algebras related to the groups. With the canonical groups SO(3) and SU(2) found, we complete the quantisation process by finding representations of the canonical groups for CP1. It is also discussed that Isham’s group-theoretic quantisation can be used for quantising complex projective spaces in general and study the complex projective space from group theoretical aspects for infinite-dimensional Hilbert space. Finally, Chapter 5 is a conclusion, in this chapter we summarise all our work and suggest some idea for future research.

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