On reproducing kernels and quantization of states

Odzijewicz, Anatol

doi:10.1007/bf01229456

Cited by 68 publications

(61 citation statements)

References 12 publications

(30 reference statements)

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“…Some representative references are by Odzijewicz [188] [189] [190] [191] and Ali [4] [6] [11]. We begin with a quick review of the symplectic geometry of the projective Hilbert space.…”

Section: Coherent State Quantizationmentioning

confidence: 99%

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Quantization Methods: A Guide for Physicists and Analysts

2005

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This survey is an overview of some of the better known quantization techniques (for systems with finite numbers of degrees-of-freedom) including in particular canonical quantization and the related Dirac scheme, introduced in the early days of quantum mechanics, Segal and Borel quantizations, geometric quantization, various ramifications of deformation quantization, Berezin and Berezin–Toeplitz quantizations, prime quantization and coherent state quantization. We have attempted to give an account sufficiently in depth to convey the general picture, as well as to indicate the mutual relationships between various methods, their relative successes and shortcomings, mentioning also open problems in the area. Finally, even for approaches for which lack of space or expertise prevented us from treating them to the extent they would deserve, we have tried to provide ample references to the existing literature on the subject. In all cases, we have made an effort to keep the discussion accessible both to physicists and to mathematicians, including non-specialists in the field.

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“…Some representative references are by Odzijewicz [188] [189] [190] [191] and Ali [4] [6] [11]. We begin with a quick review of the symplectic geometry of the projective Hilbert space.…”

Section: Coherent State Quantizationmentioning

confidence: 99%

“…See Odzijewicz [188], p. 584, for some remarks and physical motivation for studying equations of this type. Some results on the dependence µ → K µ are in Pasternak-Winiarski [199].…”

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confidence: 99%

Quantization Methods: A Guide for Physicists and Analysts

2005

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“…Then, using Eqs. (11) and the Jacobi identities we can calculate all polynomials P ij;kl ({E ij ; R 1 }) = [V ij , V + kl ] in specfications of CR (6c) by the u(2) adjoint actions on P = P 11;11 (. .…”

Section: Dual Algebraic Pairs and Polynomial Lie Algebras In Multibosmentioning

confidence: 99%

“…has the E(B GI )-path integral form 10,11 . Its calculation in the stationary phase approximation 10 determines U 0 H ({I j }; t).…”

Section: Dual Algebraic Pairs and Polynomial Lie Algebras In Multibosmentioning

confidence: 99%

Dual Algebraic Pairs and Polynomial Lie Algebras in Quantum Physics: Foundations and Geometric Aspects

Karassiov

2004

Contemporary Geometry and Related Topics

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We discuss some aspects and examples of applications of dual algebraic pairs (G 1 , G 2 ) in quantum many-body physics. They arise in models whose Hamiltonians H have invariance groups G i . Then one can take G 1 = G i whereas another dual partner G 2 = g D is generated by G i invariants, possesses a Lie-algebraic structure and describes dynamic symmetry of models; herewith polynomial Lie algebrasĝ = g D appear in models with essentially nonlinear Hamiltonians. Such an approach leads to a geometrization of model kinematics and dynamics.

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“…Recently reproducing kernels of Bergman type have been used to define the quantization of classical states in holomorphic models of quantum field theory (see [11], [7] or [8]). Earlier they have been appeard in studies of wave and Dirac equations (see [2]).…”

Section: Introductionmentioning

confidence: 99%