%0 Journal Article
%A Low, Stephen
%A Jarvis, P. D.
%A Campoamor Stursberg, Otto-Rudwig
%T Projective representations of the inhomogeneous Hamiltongroup: Noninertial symmetry in quantum mechanics
%D 2012
%@ 0003-4916
%U https://hdl.handle.net/20.500.14352/43746
%X Symmetries in quantum mechanics are realized by the projective representations of the Lie group as physical states are defined only up to a phase. A cornerstone theorem shows that these representations are equivalent to the unitary representations of the central extension of the group. The formulation of the inertial states of special relativistic quantum mechanics as the projective representations of the inhomogeneous Lorentz group, and itsnonrelativistic limit in terms of the Galilei group, are fundamental examples. Interestingly, neither of these symmetries include the Weyl–Heisenberg group; the hermitian representations of its algebra are the Heisenberg commutation relations that are a foundation of quantum mechanics. The Weyl–Heisenberg group is a one dimensional central extension of the abelian group and its unitary representations are therefore a particular projectiverepresentation of the abelian group of translations on phasespace. A theorem involving the automorphism group shows thatthe maximal symmetry that leaves the Heisenberg commutationrelations invariant is essentially a projective representation of the inhomogeneous symplectic group. In the nonrelativistic domain, we must also have invariance of Newtonian time. This reduces the symmetry group to the inhomogeneous Hamilton group that is a local noninertial symmetry of the Hamilton equations. The projective representations of these groups are calculated usingthe Mackey theorems for the general case of a nonabelian normalsubgroup.
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