43 research outputs found

    Geometry of 2d spacetime and quantization of particle dynamics

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    We analyze classical and quantum dynamics of a particle in 2d spacetimes with constant curvature which are locally isometric but globally different. We show that global symmetries of spacetime specify the symmetries of physical phase-space and the corresponding quantum theory. To quantize the systems we parametrize the physical phase-space by canonical coordinates. Canonical quantization leads to unitary irreducible representations of SO↑(2.1)SO_\uparrow (2.1) group.Comment: 12 pages, LaTeX2e, submitted for publicatio

    Particle dynamics on hyperboloid and unitary representation of SO(1,N) group

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    We analyze particle dynamics on NN dimensional one-sheet hyperboloid embedded in N+1N+1 dimensional Minkowski space. The dynamical integrals constructed by SO↑(1,N)SO_\uparrow (1,N) symmetry of spacetime are used for the gauge-invariant Hamiltonian reduction. The physical phase-space parametrizes the set of all classical trajectories on the hyperboloid. In quantum case the operator ordering problem for the symmetry generators is solved by transformation to asymptotic variables. Canonical quantization leads to unitary irreducible representation of SO↑(1,N)SO_\uparrow (1,N) group on Hilbert space L2(SN−1)L^2(S^{N-1}).Comment: 12 pages, LaTeX2e, no figure

    Quantization of the AdS3{\rm AdS}_3 Superparticle on OSP(1∣2)2/SL(2,R){\rm OSP}(1|2)^2/{\rm SL}(2,\mathbb{R})

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    We analyze AdS3{\rm AdS}_3 superparticle dynamics on the coset OSP(1∣2)×OSP(1∣2)/SL(2,R){\rm OSP}(1|2) \times {\rm OSP}(1|2)/{\rm SL}(2,\mathbb{R}). The system is quantized in canonical coordinates obtained by gauge invariant Hamiltonian reduction. The left and right Noether charges of a massive particle are parametrized by coadjoint orbits of a timelike element of osp(1∣2)\frak{osp}(1|2). Each chiral sector is described by two bosonic and two fermionic canonical coordinates corresponding to a superparticle with superpotential W=q−m/qW=q-m/q, where mm is the particle mass. Canonical quantization then provides a quantum realization of osp(1∣2)⊕osp(1∣2)\frak{osp}(1|2)\oplus\frak{osp}(1|2). For the massless particle the chiral charges lie on the coadjoint orbit of a nilpotent element of osp(1∣2)\frak{osp}(1|2) and each of them depends only on one real fermion, which demonstrates the underlying κ\kappa-symmetry. These remaining left and right fermionic variables form a canonical pair and the system is described by four bosonic and two fermionic canonical coordinates. Due to conformal invariance of the massless particle, the osp(1∣2)⊕osp(1∣2)\frak{osp}(1|2)\oplus\frak{osp} (1|2) extends to the corresponding superconformal algebra osp(2∣4)\frak{osp}(2|4). Its 19 charges are given by all real quadratic combinations of the canonical coordinates, which trivializes their quantization.Comment: 25+1 pages; v2: minor changes, references added and updated; v3: minor changes, one reference added, matches published versio

    Oscillator quantization of the massive scalar particle dynamics on AdS spacetime

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    The set of trajectories for massive spinless particles on AdSN+1AdS_{N+1} spacetime is described by the dynamical integrals related to the isometry group SO(2,N). The space of dynamical integrals is mapped one to one to the phase space of the NN-dimensional oscillator. Quantizing the system canonically, the classical expressions for the symmetry generators are deformed in a consistent way to preserve the so(2,N)so(2,N) commutation relations. This quantization thus yields new explicit realizations of the spin zero positive energy UIR's of SO(2,N) for generic NN. The representations as usual can be characterized by their minimal energy α\alpha and are valid in the whole range of α\alpha allowed by unitarity.Comment: Latex, 14 pages, version to appear in PL
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