Abstract
Group representations play a central role in theoretical physics. In particular, in quantum mechanics unitary — or, in general, projective unitary — representations implement the action of an abstract symmetry group on physical states and observables. More specifically, a major role is played by the so-called square integrable representations. Indeed, the properties of these representations are fundamental in the definition of certain families of generalized coherent states, in the phase-space formulation of quantum mechanics and the associated star product formalism, in the definition of an interesting notion of function of quantum positive type, and in some recent applications to the theory of open quantum systems and to quantum information.
Highlights
Symmetries and group representations are fundamental in modern science
E.g., due to Wigner’s theorem on symmetry transformations [1,2,3,4], unitary group representations [5,6] play a central role in quantum theory
We argue that the usefulness of square integrable representations is mainly due to certain ‘orthogonality relations’ generalizing Schur’s orthogonality relations for compact topological groups [6]
Summary
Symmetries and group representations are fundamental in modern science. E.g., due to Wigner’s theorem on symmetry transformations [1,2,3,4], (projective) unitary group representations [5,6] play a central role in quantum theory. The harmonic analysis associated with a square integrable projective representation of the group of translations on phase space — the Weyl system — allows one to capture, in a very elegant and effective way, some peculiarities of a quantum system versus a classical one. This field of research is still in constant progress and there is room for new investigations that do not fall within the traditional range of applications of abstract harmonic analysis to theoretical physics; consider, e.g., some new applications to the theory of open quantum systems.
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