Some strange features of the Galilei group. (English) Zbl 1262.81065

Gołubowska, Barbara; Kovalchuk, Vasyl; Martens, Agnieszka; Rozko, Ewa Eliza; Sławianowski, Jan J.

J. Geom. Symmetry Phys. 26, 33-59 (2012).

Summary: We discuss certain strange properties of the Galilei group, connected first of all with the property of mechanical energy-momentum covector to be an affine object, rather than a linear one. Its affine transformation rule is interesting in itself and dependent on the particle mass. At the quantum level, this means obviously that we deal with the projective unitary representation of the group rather than with the usual representation. The status of mass is completely different than in relativistic theory, where it is a continuous eigenvalue of the Casimir invariant. In the Galilei framework, it is a parameter characterizing the factor of the projective representation, in the sense of V. Bargmann. This “pathology” from the relativistic point of view is nevertheless very interesting and it underlies the Weyl-Wigner-Moyal-Ville approach to quantum mechanics.

MSC:

81R05	Finite-dimensional groups and algebras motivated by physics and their representations
22E70	Applications of Lie groups to the sciences; explicit representations
20C25	Projective representations and multipliers
22D10	Unitary representations of locally compact groups
20C35	Applications of group representations to physics and other areas of science
70A05	Axiomatics, foundations