2011/08/01 by S. G. Low, Stephen G. Low, Peter Jarvis +3
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #Fundamental representation #Geometry #Group (periodic table) #Heisenberg group #Lie algebra #Lorentz group #Lorentz transformation #Mathematical physics #Mathematics #Nonlinear Waves and Solitons #Physics #Poincaré group #Projective representation #Projective test #Pure mathematics #Quantum #Quantum dynamics #Quantum mechanics #Relativistic quantum mechanics #Representation theory of the Lorentz group #Symmetry group #Symmetry in quantum mechanics #math-ph #math.MP #quant-ph
paper · pdf · doi:10.1016/j.aop.2011.10.010
published as Annals of Physics 327 (2012), pp. 74-101
arxiv created 2011/08/01 · openalex publication_date 2011/11/06 · arxiv updated 2012/01/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 its nonrelativistic limit in terms of the Galilei group, are fundamental examples. Interestingly, neither of these symmetries includes 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 projective representation of the abelian group of translations on phase space. A theorem involving the automorphism group shows that the maximal symmetry that leaves invariant the Heisenberg commutation relations are essentially projective representations 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 Hamilton's equations. The projective representations of these groups are calculated using the Mackey theorems for the general case of a nonabelian normal subgroup.