A. Stern, I. Yakushin
Lie-Poisson Deformation of the Poincare Algebra
The paper is published: J. Math. Phys. 37, 4 (1996) 2053-2070

MSC:

17B37 Quantum groups and related deformations, See also {16W30, 81R50, 82B23}

58F06 Geometric quantization (applications of representation theory), See also {22E45, 81S10}

81R50 Quantum groups and related algebraic methods, See Also {16W30, 17B37}

81V80 Quantum optics

Abstract: 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.


Keywords: Poisson brackets, Poisson-Lie groups, quadratic Poisson structures, deformed Poincare algebra, spinning and spinless particles