Deformation Quantisation

This thesis discusses a lesser known formulation of quantum mechanics called deformation quantisation. This theory provides a more intuitive way to study quantum systems, in which confusing topics such as the relation between the operator commutator and the classical Poisson bracket and the concept of classical limit become very obvious. After a short introduction, Chapter 2 introduces the mathematical foundation of Hamiltonian classical mechanics, symplectic geometry, and states the Darboux Theorem. Chapter 3 develops the theory of Hochschild cohomology and gives a classification of the cohomology spaces of the algebra of smooth functions on a manifold. In Chapter 4, deformations of algebras are defined and results of Hochschild cohomology are used to prove lemmas about star products, which are smooth deformations of the algebra of smooth functions on a manifold. And Chapter 5 introduces the special case of the Moyal star product and shows how it can be used to obtain deformation...Show moreThis thesis discusses a lesser known formulation of quantum mechanics called deformation quantisation. This theory provides a more intuitive way to study quantum systems, in which confusing topics such as the relation between the operator commutator and the classical Poisson bracket and the concept of classical limit become very obvious. After a short introduction, Chapter 2 introduces the mathematical foundation of Hamiltonian classical mechanics, symplectic geometry, and states the Darboux Theorem. Chapter 3 develops the theory of Hochschild cohomology and gives a classification of the cohomology spaces of the algebra of smooth functions on a manifold. In Chapter 4, deformations of algebras are defined and results of Hochschild cohomology are used to prove lemmas about star products, which are smooth deformations of the algebra of smooth functions on a manifold. And Chapter 5 introduces the special case of the Moyal star product and shows how it can be used to obtain deformation quantisation. In this chapter, it is also shown that deformation quantisation is completely equivalent to the Hilbert space formalism which is traditionally taught in undergraduate studies, and the simple harmonic oscillator is treated as an example. Finally, Chapter 6 summarises the possible benefits and drawbacks of teaching deformation quantisation instead of the Hilbert space formalism and lists some avenues of further studyShow less

Leiden University Student Repository

Documents

In Collections

Deformation Quantisation