Time Dependent Conserved Charges and their Gauging -- A Modest Case Study in Shared Memory of Victor --

Abstract

There exist instances of dynamical systems possessing symmetry transformations of which the conserved Noether charges generating these symmetries feature an explicit time dependence in their functional representation over phase space. The generators of such symmetries certainly do not commute with the Hamiltonian, and yet these charges are conserved observables for the classical and quantised dynamics. Furthermore within the Hamiltonian formalism and in the case of global symmetries such charges may be gauged to allow for arbitrary time dependent symmetry transformations, simply by extending the Hamiltonian to include the Noether charges as first-class constraints. An explicit illustration of these issues is presented in a simple and most familiar model that applies also to the constant gravitational force. This note draws its primary motivation from the quest towards a theory for quantum gravity, in wanting to understand better the tension existing between the local Equivalence Principle of the gravitational interaction and the fundamental principles of Quantum Mechanics by considering the formulation of quantum systems relative to reference frames that are inertial or noninertial, and thus accelerated relative to one another through arbitrary time dependent spatial translations.