Quantum mechanics on homogeneous spaces

Doebner, H. D.; Tolar, J.

doi:10.1063/1.522604

Cited by 32 publications

(16 citation statements)

References 50 publications

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“…In this notation, the Hopf algebra structure is [18] 1= s e s In mathematical terms the algebra here is a cross product algebra. This is a more or less standard way to quantise particles moving under the action of a group [19] [20]. In nice cases the action ⊳ of G on M induces a metric with the particle then moving on geodesics.…”

Section: Bicrossproduct Hopf Algebrasmentioning

confidence: 99%

Duality principle and braided geometry

Majid

Strings and Symmetries

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We give an overview of a new kind symmetry in physics which exists between observables and states and which is made possible by the language of Hopf algebras and quantum geometry. It has been proposed by the author as a feature of Planck scale physics. More recent work includes corresponding results at the semiclassical level of Poisson-Lie groups and at the level of braided groups and braided geometry.

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Section: Bicrossproduct Hopf Algebrasmentioning

confidence: 99%

Duality principle and braided geometry

Majid

Strings and Symmetries

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“…Instead, our starting point is a natural choice of the quantum algebra of observables, the transformation group C -algebra A 1 = C (G; Q) 6,24,17,18] (see sect.…”

Section: Introductionmentioning

confidence: 99%

Deformations of Algebras of Observables and the Classical Limit of Quantum Mechanics

Landsman

1993

Rev. Math. Phys.

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The quantum algebra of observables of a particle moving on a homogeneous configuration space Q = G/H, the transformation group C*-algebra C* (G, G/H), is deformed into its classical counterpart C0 ((T*G)/H). The Poisson structure of the latter is obtained as the classical limit of the quantum commutator. The superselection sectors of both algebras describe the particle moving in an external Yang–Mills field. Analytical aspects of deformation theory, such as the nature of the limit ħ → 0, are studied in detail. A physically motivated convergence criterion in ħ is introduced. The Weyl–Moyal quantization formalism, and the associated use of Wigner distribution functions, is generalized from flat phase spaces T*ℝn to Poisson manifolds of the form (T*G)/H. The classical limit of quantum states as well as of superselection sectors is investigated. The former is handled by introducing the notion of a classical germ, generalizing coherent states. The latter is analyzed by studying the Jacobson topology on the primitive ideal space of a certain continuous field of C*-algebras, constructed from the classical and the quantum algebras of observables. The symplectic leaves of (T*G)/H are confirmed to be the correct classical analogue of the quantum superselection sectors of C*(G, G/H).

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“…The quantization theory of a particle moving on an arbitrary homogeneous manifold Q = G=H was initiated by Mackey 23], who replaced the canonical commutation relations as the basic object of study by systems of imprimitivity, and showed that such systems admit (unitarily) inequivalent representations, labeled by the dual H of H (that is, the set of equivalence classes of irreducible unitary representations of H). This work was extended in various directions in 5,12]; it should be mentioned, that the idea that a given classical system may admit a family of inequivalent quantizations was independently arrived at in the context of geometric quantization 28].…”

Section: Introductionmentioning

confidence: 99%

Induced Representations, Gauge Fields, and Quantization on Homogeneous Spaces

Landsman

1992

Rev. Math. Phys.

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We study representations of the enveloping algebra of a Lie group G which are induced by a representation of a Lie subgroup H, assuming that G/H is reductive. Such representations describe the superselection sectors of a quantum particle moving on G/H. It is found that the representatives of both the generators and the quadratic Casimir operators of G have a natural geometric realization in terms of the canonical connection on the principal H-bundle G. The explicit expression for the generators can be understood from the point of view of conservation laws and moment maps in classical field theory and classical particle mechanics on G/H. The emergence of classical geometric structures in the quantum-mechanical situation is explained by a detailed study of the domain and possible self-adjointness properties of the relevant operators. A new and practical criterion for essential self-adjointness in general unitary representations is given.

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Quantum mechanics on homogeneous spaces

Cited by 32 publications

References 50 publications

Duality principle and braided geometry

Duality principle and braided geometry

Deformations of Algebras of Observables and the Classical Limit of Quantum Mechanics

Induced Representations, Gauge Fields, and Quantization on Homogeneous Spaces

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