Lie-Poisson Deformation of the Poincaré Algebra

Details Lie-Poisson Deformation of the Poincaré Algebra Lie-Poisson Deformation of the Poincaré Algebra

1995-05-25

arxiv.org/abs/q-alg/9505030v1

Physics

High Energy Physics

High Energy Physics - Theory

Latex file, 26 pages

Scientific paper

10.1063/1.531498

We find a one parameter family of quadratic Poisson structures on ${\bf R}^4\times SL(2,C)$ which satisfies the property {\it a)} that it is preserved under the Lie-Poisson action of the Lorentz group, as well as {\it b)} that it reduces to the standard Poincar\'e algebra for a particular limiting value of the parameter. (The Lie-Poisson transformations reduce to canonical ones in that limit, which we therefore refer to as the `canonical limit'.) Like with the Poincar\'e algebra, our deformed Poincar\'e algebra has two Casimir functions which we associate with `mass' and `spin'. We parametrize the symplectic leaves of ${\bf R}^4\times SL(2,C)$ with space-time coordinates, momenta and spin, thereby obtaining realizations of the deformed algebra for the cases of a spinless and a spinning particle. The formalism can be applied for finding a one parameter family of canonically inequivalent descriptions of the photon.

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