Galilean covariant Lie algebra of quantum mechanical observables

Olímpia Lombardi, Mario Castagnino, Juan Sebastián Ardenghi

Róbert Hermann

Daniel Canarutto

In the context of Covariant Quantum Mechanics for a spin particle, we classify the "quantum vector fields", i.e. the projectable Hermitian vector fields of a complex bundle of complex dimension 2 over spacetime. Indeed, we prove that the Lie algebra of quantum vector fields is naturally isomorphic to a certain Lie algebra of functions of the classical phase space, called "special phase functions". This result provides a covariant procedure to achieve the quantum operators generated by the quantum vector fields and the corresponding observables described by the special phase functions.

Kalyanapuram Rangachari Parthasarathy

The aim of the present chapter is to obtain concrete expressions for observables like position, linear momentum, orbital and spin angular momenta, energy etc., through an investigation of the projective unitary antiunitary (p.u.a.) representations of the groups under which the description of the quantum mechanical system is assumed to be covariant. (See sections 1.4 and 1.5). Special emphasis is laid on the Galilean group and the Poincare group. (See sections 4.2–4.5). The basic feature of these groups is that they are semidirect products of a vector group and a Lie group which acts on the vector group as nonsingular linear transformations.

Yair Shapira

Yair Shapira