Observables, Symmetries and Constraints
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The notion of observable is one of the key concepts of a physical theory. We introduce a definition of observables within the framework of ensembles on configuration space, based on the idea of associating observables with generators of canonical transformations acting on the phase space of the fundamental variables P and S. These ensemble observables encompass both classical and quantum observables. Remarkably, for classical observables the Poisson bracket of the ensemble observables is isomorphic to the usual bracket on standard classical phase space, while for quantum observables it is isomorphic to the commutator in Hilbert space. We show that the formalism allows for the generalisation of certain quantum concepts, such as eigenstates, eigenvalues, weak values and transition probabilities, to arbitrary configuration ensembles. We discuss also systems with symmetries, in particular examples which involve representations of the Galilean group for the case of a free particle and rotations defined on discrete configuration spaces. Finally, we generalise and reinterpret quantum superselection rules in terms of constraints on observables.
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