Infinitesimal Symmetries in Covariant Quantum Mechanics

Abstract

We discuss the Lie algebras of infinitesimal symmetries of the main covariant geometric objects of covariant quantum mechanics: the time form, the hermitian metric, the upper quantum connection, the quantum lagrangian. Indeed, these infinitesimal symmetries are generated, in a covariant way, by the Lie algebra of time preserving conserved special phase functions. Actually, this Lie algebra of special phase functions generates also the Lie algebra of infinitesimal symmetries of the main classical objects: the time form and the cosymplectic 2-form. A natural output of the classification of the quantum symmetries is a covariant approach to quantum operators and to quantum currents associated with special phase functions.