Spin in Classical Systems

Sengupta, Anindya (2014) Spin in Classical Systems. Masters thesis, Indian Institute of Science Education and Research Kolkata.

PDF (MS dissertation of Anindya Sengupta) Anindya_Sengupta_09MS039.pdf - Submitted Version Restricted to Repository staff only Download (687kB)

Abstract

The aim of the present study is to understand the notion of spin in a classical context. We want to study classical dynamical systems that have spin. Such systems have constraint relations between phase space variabes. In order to understand this, the first chapter starts with a systematic study of constraints and their classifications and then discusses the Dirac-Bergmann theory. Then we define spring-mass equivalent models of field theories. The possibility of the spin information being encoded in nontrivial spring connections is discussed. Then we study generalized classical dynamics with the phase space manifold being a Grassmann algebra. In this generalized classical dynamics, the Poisson brackets between spin variables are of the same form as the commutation relations between spin operators in quantum mechanics. In particular, the classical model for a spin 1/2 particle produces the Pauli-Dirac theory upon canonical quantization. The Dirac equation is obtained after quantizing the constraint imposed on the phase space variables in this model. Following the discussion of classical spin in terms of Grassmann variables, we study how classical dynamics without Grassmann variables can produce the same spin algebra. Finally, we come to the canonical realizations of the Poincare group. In quantum field theory, irreducible unitary representations of the Poincare group are often identified with the notion of particles. The classical counterpart of such an approach would be to identify the irreducible canonical realizations of the Poincare group. Finally, we round up by establishing the connection between spin and statistics in the classical context.

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