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American Journal of Physics

Dynamical Applications of Bargmann Space III. Angular Momentum

John Martin Robinson, S. L. Trubatch

Published Oct 1, 1971• 1 citation
Publisher

Abstract

The coherent state representation is developed for angular momentum operators. As an example, the one-Boson Lee model is analyzed with its Fermi operators in Bargmann space. Finally, a synthesis is achieved by analyzing the three-level system with all operators in Bargmann space.
DOI
10.1119/1.1976601

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The various representations of rotations in physical space and Hilbert space are developed in detail. This leads to an in-depth treatment of the representation of states of well-defined spin and angular momentum in quantum systems. The peculiarities of the physics of spin— systems (spinors) is outlined. The tensorial character of representations is implicit in the treatment. The Schwinger and Bargmann representations are introduced in some detail; and this leads to SU(2) coherent states (which are important for more advanced group representation theory).
The monitoring of the three independent components of the angular momentum (or spin) of a quantum system by its environment that does not isolate any preferred orientation is modelled in two different ways. One describes the dynamics by the Lindblad equation generated by three independent angular momentum operators. The other uses iterated measurements of the ``phase-space'' point on the sphere in terms of the positive operator-valued measure generated by SU(2) coherent states. In contrast to the equivalent scenario on a flat phase space, these two models give rise to subtle differences. Specifically, it is shown that the two super-operators corresponding to the two decoherence models for angular momentum systems are commutative, but their eigenvalues are different. Hence although both models give rise to phase-space decoherence, their dynamical behaviours are not equivalent. In either model, we find that the characterisation of classicality as represented by the decay rates of the elements of the density matrix (i.e. decoherence) and that as represented by the positivity of the quasiprobability distribution are not equivalent for angular momentum systems.