In this report, we deliver a detailed introduction to the methods of path integration in the focu... more In this report, we deliver a detailed introduction to the methods of path integration in the focus of quantum mechanics. After an overview of the techniques of integration and the relationship to the familiar results of quantum mechanics such as the Schroedinger equation, we study some of the applications to mechanical systems with non-trivial degrees of freedom and discuss the advantages that path integration has as a formulation in studying these systems. To this end, in chapter 3, we introduce the topological concept of homotopy classes, and apply this to derive the spin statistics theorem, and discuss the phenomenon of parastatistics for systems constrained to dimension d<3. Following on from this, we calculate a general formula for the quantum mechanical propagator in terms of path integrals over different homotopy classes. The final chapter of this report studies the so called instanton solution as a way of describing quantum mechanical vacuum tunneling effects as a semi-classical problem. We conclude with discussing some applications in gauge field theory and describe the topological quality of the classical vacua of SU(2) Yang-Mills gauge theory; apply the path integral method to arrive at the theta-vacuum, and briefly say a few words about its far reaching consequences and applications.\\The content is aimed predominantly at a mathematical audience with a physical interest, moreover, we assume that the reader has a good grounding in topology and in both the classical and quantum mechanical theories. For completeness, essentials of these topics are reviewed in the appendices.
(NOTE: Previous version uploaded was an earlier draft accidentally uploaded. This is the correct final version)
In this report, we deliver a detailed introduction to the methods of path integration in the focu... more In this report, we deliver a detailed introduction to the methods of path integration in the focus of quantum mechanics. After an overview of the techniques of integration and the relationship to the familiar results of quantum mechanics such as the Schroedinger equation, we study some of the applications to mechanical systems with non-trivial degrees of freedom and discuss the advantages that path integration has as a formulation in studying these systems. To this end, in chapter 3, we introduce the topological concept of homotopy classes, and apply this to derive the spin statistics theorem, and discuss the phenomenon of parastatistics for systems constrained to dimension d<3. Following on from this, we calculate a general formula for the quantum mechanical propagator in terms of path integrals over different homotopy classes. The final chapter of this report studies the so called instanton solution as a way of describing quantum mechanical vacuum tunneling effects as a semi-classical problem. We conclude with discussing some applications in gauge field theory and describe the topological quality of the classical vacua of SU(2) Yang-Mills gauge theory; apply the path integral method to arrive at the theta-vacuum, and briefly say a few words about its far reaching consequences and applications.\\The content is aimed predominantly at a mathematical audience with a physical interest, moreover, we assume that the reader has a good grounding in topology and in both the classical and quantum mechanical theories. For completeness, essentials of these topics are reviewed in the appendices.
(NOTE: Previous version uploaded was an earlier draft accidentally uploaded. This is the correct final version)
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Papers by Josh Cork
(NOTE: Previous version uploaded was an earlier draft accidentally uploaded. This is the correct final version)
(NOTE: Previous version uploaded was an earlier draft accidentally uploaded. This is the correct final version)