Construction of self-adjoint Berezin–Toeplitz operators on Kähler manifolds and a probabilistic representation of the associated semigroups

Bodmann, Bernhard G.

doi:10.1016/s0393-0440(02)00191-2

Cited by 3 publications

(4 citation statements)

References 65 publications

(118 reference statements)

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“…(More precisely, Kontsevich's formula is an expansion of a certain Feynman integral at a saddle point, see Cattaneo and Felder [60].) Connections between Feynman path integrals, coherent states, and the Berezin quantization are discussed in Kochetov and Yarunin [152], Odzijewicz [189], Horowski, Kryszen and Odzijewicz [130], Klauder [149], Chapter V in Berezin and Shubin [33], Marinov [168], Charles [61], and Bodmann [41]. For a discussion of Feynman path integrals in the context of geometric quantization, see Gawedzki [98], Wiegmann [258], and Chapter 9 in the book of Woodhouse [263].…”

Section: Some Other Quantization Methodsmentioning

confidence: 99%

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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Section: Some Other Quantization Methodsmentioning

confidence: 99%

Quantization Methods: A Guide for Physicists and Analysts

2005

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“…On the path integral side, our approach is directly inspired by the Daubechies-Klauder [77,194,195,199,35,198] continuous-time regularised coherent state phase space approach to path integration, and the closely related Anastopoulos-Savvidou [11,12,13] analysis of decoherence functional in the Isham-Linden quantum histories approach. Both have shown that one can think of the underlying dynamical objects of respective theories (path integrals and decoherence functionals) as consisting of the hamiltonian evolution perturbed by the geometric structures on the space of quantum states.…”

Section: Local Quantum Information Dynamics In Algebraic and Path Int...mentioning

confidence: 99%

“…𝑝 2 `9 𝑞 2 q as the pinned Wiener measure (see [35] for a systematic mathematical treatment of these objects in terms of the Berezin-Toeplitz operators). The generalisation to a wide class of connections ∇ 𝐷 Ψ and local entropic priors (even if kept at the second order riemannian level of g 𝐷 ) asks for a systematic development of a technique of stochastic integration of random walks X on R 𝑛 associated with the Brownian motions on smooth manifolds ℳ, dim ℳ " 𝑛, that could systematically address the functional integration of the above geometric structures beyond the level of heuristic treatment that is standard for physicists.…”

Section: Curvature Measures Desynchronisation In the Multi-user Infer...mentioning

confidence: 99%

“…In case of the Hilbert space based histories, this change requires to use the Berry connection, while in the algebraic framework (as implemented systematically in Sections 2.5 and 3) this requires to use the connection ∇ 1{2 , corresponding to the parallel transport operator 𝑉 𝜑,𝜓 . 35…”

Section: Case Study: Algebraic Action Operator and The Limits Of Unit...mentioning

confidence: 99%

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Local quantum information dynamics

Kostecki

2016

Preprint

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This paper is intended to: 1) show how the local smooth geometry of spaces of normal quantum states over W ˚-algebras (generalised spaces of density matrices) may be used to substantially enrich the description of quantum dynamics in the algebraic and path integral approaches; 2) provide a framework for construction of quantum information theories beyond quantum mechanics, such that quantum mechanical linearity holds only locally, while the nonlocal multi-user dynamics exhibits some similarity with general relativity. In the algebraic setting, we propose a method of incorporating nonlinear Poisson and relative entropic local dynamics, as well as local gauge and local source structures, into an effective description of local temporal evolution of quantum states by using fibrewise perturbations of liouvilleans in the fibre bundle of Hilbert spaces over the quantum state manifold. We apply this method to construct an algebraic generalisation of Savvidou's action operator. In the path integral setting, motivated by the Savvidou-Anastopoulous analysis of the role of Kähler space geometry in the Isham-Linden quantum histories, we propose to incorporate local geometry by means of a generalisation of the Daubechies-Klauder coherent state phase space propagator formula. Finally, we discuss the role of Brègman relative entropy in the Jaynes-Mitchell-Favretti renormalisation scheme. Using these tools we show that: 1) the propagation of quantum particles (in Wigner's sense) can be naturally explained as a free fall along the trajectories locally minimising the quantum relative entropy; 2) the contribution of particular trajectories to the global path integral is weighted by the local quantum entropic prior, measuring user's lack of information; 3) the presence of nonlinear quantum control variables results in the change of the curvature of the global quantum state space; 4) the behaviour of zero-point energy under renormalisation of local entropic dynamics is maintained by local redefinition of information mass (prior), which encodes the curvature change. We conclude this work with a proposal of a new framework for nonequilibrium quantum statistical mechanics based on quantum Orlicz spaces, quantum Brègman distances and Banach Lie algebras.

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Essential aspects of Wiener-measure regularization for quantum-mechanical path integrals

Klauder

2005

Nonlinear Analysis: Theory, Methods & Applications

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Construction of self-adjoint Berezin–Toeplitz operators on Kähler manifolds and a probabilistic representation of the associated semigroups

Cited by 3 publications

References 65 publications

Quantization Methods: A Guide for Physicists and Analysts

Quantization Methods: A Guide for Physicists and Analysts

Local quantum information dynamics

Essential aspects of Wiener-measure regularization for quantum-mechanical path integrals

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