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Journal of Physics A Mathematical and General

Wigner rotations, Bargmann invariants and geometric phases

Narasimhaiengar Mukunda, P. K. Aravind, Richard M Simon

Published Feb 19, 2003• 23 citations
Publisher

Abstract

The concept of the ‘Wigner rotation’, familiar from the composition law of (pure) Lorentz transformations, is described in the general setting of Lie group coset spaces and the properties of coset representatives. Examples of Abelian and non-Abelian Wigner rotations are given. The Lorentz group Wigner rotation, occurring in the coset space SL (2, R )/ SO (2) ≃ SO (2, 1)/ SO (2), is shown to be an analytic continuation of a Wigner rotation present in the behaviour of particles with nonzero helicity under spatial rotations, belonging to the coset space SU (2)/ U (1) ≃ SO (3)/ SO (2). The possibility of interpreting these two Wigner rotations as geometric phases is shown in detail. Essential background material on geometric phases, Bargmann invariants and null phase curves, all of which are needed for this purpose, is provided.

DOI
10.1088/0305-4470/36/9/312

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The concept of the ‘Wigner rotation’, familiar from the composition law of (pure) Lorentz transformations, is described in the general setting of Lie group coset spaces and the properties of coset representatives. Examples of Abelian and non-Abelian Wigner rotations are given. The Lorentz group Wigner rotation, occurring in the coset space SL (2, R )/ SO (2) ≃ SO (2, 1)/ SO (2), is shown to be an analytic continuation of a Wigner rotation present in the behaviour of particles with nonzero helicity under spatial rotations, belonging to the coset space SU (2)/ U (1) ≃ SO (3)/ SO (2). The possibility of interpreting these two Wigner rotations as geometric phases is shown in detail. Essential background material on geometric phases, Bargmann invariants and null phase curves, all of which are needed for this purpose, is provided.
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We develop the broadest possible generalization of the well known connection between quantum-mechanical Bargmann invariants and geometric phases. The key concept is that of null phase curves in quantum-mechanical ray and Hilbert spaces. Examples of such curves are developed. Our generalization is shown to be essential for properly understanding geometric phase results in the cases of coherent states and of Gaussian states. Differential geometric aspects of null phase curves are also briefly explored.
We present a theory of the geometric phase based logically on the Bargmann invariant of quantum mechanics, and null phase curves in ray space, as the fundamental ingredients. Null phase curves are themselves defined entirely in terms of the (third order) Bargmann invariant, and it is shown that these are the curves natural to geometric phase theory, rather than geodesics used in earlier treatments. The natural symplectic structure in ray space is seen to play a crucial role in the definition of the geometric phase. Logical consistency of the formulation is explicitly shown, and the principal properties of geometric phases are deduced as systematic consequences.
Bargmann invariants and null phase curves are known to be important ingredients in understanding the essential nature of the geometric phase in quantum mechanics. Null phase manifolds in quantum-mechanical ray spaces are submanifolds made up entirely of null phase curves, and so are equally important for geometric phase considerations. It is shown that the complete characterization of null phase manifolds involves both the Riemannian metric structure and the symplectic structure of ray space in equal measure, which thus brings together these two aspects in a natural manner.
This note summarizes the physics and mathematics of Lorentz transformations for massless particles, specifically for photons. We provide a complete analytical derivation of Wigner's little group matrix and a closed formula for the calculation of Wigner's angle for arbitrary Lorentz transformations. Our derivation highlights the geometrical content of the sequence of little group transformations leading to Wigner's matrix and links it to classical theorems in spherical trigonometry.
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