Broken dynamical symmetries in quantum mechanics and phase transition phenomena

Abstract

This thesis describes applications of dynamical symmetries to problems in quantum mechanics and many-body physics where the latter is formulated as a Euclidean scalar field theory in d-space dimensions. By invoking the concept of a dynamical symmetry group a unified understanding of apparently disparate results is achieved. In the case of the quantum mechanical problem - a parametric oscillator with time-dependent parameters - the relevant dynamical symmetry group permits an explanation of the spectrum and an interpretation of the time-dependence as a symmetry breaking phenomenon. The breaking generator is found to transform as a (6 + 6) of SU(3). In order to provide a foundation for our later discussion, the 0(N) nonlinear sigma - model is reviewed as a field theory in the context of elementary particle physics and statistical mechanics. Spontaneous breaking of the internal 0(N) symmetry implies the existence of (N-l) Goldstone modes. The effective Hamiltonian for these Goldstohe modes furnishes a description of i) low energy pion interactions in elementary particle physics and ii) the statistical mechanics of the classical 0(N) - invariant Heisenberg model of ferromagnetic systems. Loop diagrams in the effective Hamiltonian reveal that infra-red singularities are induced which inhibit a phase transition below two dimensions i.e. the critical temperature, Tc , moves to zero as the spatial dimension is lowered to 2. Proceeding by analogy with the nonlinear sigma model we regard the Euclidean group as a dynamical symmetry applicable to critical phenomena in Ising-like systems with discrete internal symmetry. The Euclidean invariance is spontaneously broken below T by the existence of an interfacial boundary between thermodynamic phases in the Ising system. The corresponding Goldstone modes are identified with surface fluctuations in the boundary. An effective Hamiltonian is found in which the full Euclidean symmetry is nonlinearly realized on the Goldstone fields. In this way, several seemingly unrelated results for Ising systems can be understood in the same formalism. The ultra-violet renormalizability of these Hamiltonians above one-dimension, can be related to the absence of a phase transition in one-dimension. Also these effective Hamiltonians indicate the presence of a universal essential singularity at a first order phase transition in Ising systems.

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