The Schwinger Representation of a Group: Concept and Applications

Details The Schwinger Representation of a Group: Concept and Applications The Schwinger Representation of a Group: Concept and Applications

2005-05-03

arxiv.org/abs/quant-ph/0505012v1

Rev.Math.Phys. 18 (2006) 887-912

Physics

Quantum Physics

Latex, 17 pages

Scientific paper

10.1142/S0129055X06002802

The concept of the Schwinger Representation of a finite or compact simple Lie group is set up as a multiplicity-free direct sum of all the unitary irreducible representations of the group. This is abstracted from the properties of the Schwinger oscillator construction for SU(2), and its relevance in several quantum mechanical contexts is highlighted. The Schwinger representations for $SU(2), SO(3)$ and SU(n) for all $n$ are constructed via specific carrier spaces and group actions. In the SU(2) case connections to the oscillator construction and to Majorana's theorem on pure states for any spin are worked out. The role of the Schwinger Representation in setting up the Wigner-Weyl isomorphism for quantum mechanics on a compact simple Lie group is brought out.

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