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When geometric quantization is applied to a manifold using a real polarization which is “nice enough”, a result of Śniatycki says that the quantization can be found by counting certain objects, called Bohr-Sommerfeld leaves. Subsequently, several authors have taken this as motivation for counting Bohr-Sommerfeld leaves when studying the quantization of manifolds which are less “nice”. In this paper, we examine the quantization of compact symplectic manifolds that can locally be modelled by a toric manifold, using a real polarization modelled on fibres of the moment map. We compute the results directly, and obtain a theorem similar to Śniatycki’s, which gives the quantization in terms of counting Bohr-Sommerfeld leaves. However, the count does not include the Bohr-Sommerfeld leaves which are singular. Thus the quantization obtained is different from the quantization obtained using a Kähler polarization.

In the last decade, the development of new ideas in quantum theory, including geometric and deformation quantization, the non-Abelian Berry's geometric factor, super- and BRST symmetries, non-commutativity, has called into play the geometric techniques based on the deep interplay between algebra, differential geometry and topology. The book aims at being a guide to advanced differential geometric and topological methods in quantum mechanics. Their main peculiarity lies in the fact that geometry in quantum theory speaks mainly the algebraic language of rings, modules, sheaves and categories. Geometry is by no means the primary scope of the book, but it underlies many ideas in modern quantum physics and provides the most advanced schemes of quantization.

This volume contains the refereed proceedings of two symposia on symplectic geometry and quantization problems which were held in Japan in July 1993. The purpose of the symposia was to discuss recent progress in a range of related topics in symplectic geometry and mathematical physics, including symplectic groupoids, geometric quantization, noncommutative differential geometry, equivariant cohomology, deformation quantization, topological quantum field theory, and knot invariants. The book provides insight into how these different topics relate to one another and offers intriguing new problems. Providing a look at the frontier of research in symplectic geometry and quantization, this book is suitable as a source book for a seminar in symplectic geometry.

Based on lectures given at the renowned Villa de Leyva summer school, this book provides a unique presentation of modern geometric methods in quantum field theory. Written by experts, it enables readers to enter some of the most fascinating research topics in this subject. Covering a series of topics on geometry, topology, algebra, number theory methods and their applications to quantum field theory, the book covers topics such as Dirac structures, holomorphic bundles and stability, Feynman integrals, geometric aspects of quantum field theory and the standard model, spectral and Riemannian geometry and index theory. This is a valuable guide for graduate students and researchers in physics and mathematics wanting to enter this interesting research field at the borderline between mathematics and physics.

The purpose of the present memoir is two-fold. First, we obtain a non-abelian localization theorem when M is any even dimensional compact manifold : following an idea of E. Witten, we deform an elliptic symbol associated to a Clifford bundle on M with a vector field associated to a moment map. Second, we use this general approach to reprove the [Q,R] = 0 theorem of Meinrenken-Sjamaar in the Hamiltonian case, and we obtain mild generalizations to almost complex manifolds. This non-abelian localization theorem can be used to obtain a geometric description of the multiplicities of the index of general

This book contains eleven refereed research papers on deformation quantization by leading experts in the respective fields.These contributions are based on talks presented on the occasion of the meeting between mathematicians and theoretical physicists held in Strasbourg in May 2001.

This book is a collection of expository articles from the Center of Mathematics at Notre Dame's 2011 program on quantization. Included are lecture notes from a summer school on quantization on topics such as the Cherednik algebra, geometric quantization, detailed proofs of Willwacher's results on the Kontsevich graph complex, and group-valued moment maps. This book also includes expository articles on quantization and automorphic forms, renormalization, Berezin-Toeplitz quantization in the complex setting, and the commutation of quantization with reduction, as well as an original article on derived Poisson brackets. The primary goal of this volume is to make topics in quantization more accessible to graduate students and researchers.

Two types of randomness are associated with a mixed quantum state: the uncertainty in the probability coefficients of the constituent pure states and the uncertainty in the value of each observable captured by the Born’s rule probabilities. Entropy is a quantification of randomness, and we propose a spin-entropy for the observables of spin pure states based on the phase space of a spin as described by the geometric quantization method, and we also expand it to mixed quantum states. This proposed entropy overcomes the limitations of previously-proposed entropies such as von Neumann entropy which only quantifies the randomness of specifying the quantum state. As an example of a limitation, previously-proposed entropies are higher for Bell entangled spin states than for disentangled spin states, even though the spin observables are less constrained for a disentangled pair of spins than for an entangled pair. The proposed spin-entropy accurately quantifies the randomness of a quantum state, it never reaches zero value, and it is lower for entangled states than for disentangled states.

Quantum mechanics postulates wave–particle duality and assigns amplitudes of the form eiS/ℏ, yet no existing formulation explains why physical observables depend only on the phase of the action. Here we show that if the quantum of action ℏgeom is finite, the classical action manifold R becomes compact under the identification S≡S+2πℏgeom, yielding a U(1) action space on which only modular action is observable. Wave interference then follows as a geometric necessity: a finite action quantum forces physical amplitudes to live on a circle, while the classical limit arises when the modular spacing 2πℏgeom becomes negligible compared with macroscopic actions. We formulate this as a compact-action theorem. Chronon Field Theory (ChFT) provides the physical origin of ℏgeom: its causal field Φμ carries a quantized symplectic flux ∮ω=ℏgeom, making Planck’s constant a geometric topological invariant rather than an imposed parameter. Within this medium, the Real–Now–Front (RNF) supplies a local reconstruction rule that reproduces the structure of the Feynman path integral, the Schrödinger evolution, the Born rule, and macroscopic definiteness as consequences of geometric compatibility rather than supplemental postulates. Phenomenologically, identifying the electron as the minimal chronon soliton—carrying the fundamental unit of symplectic flux—links its spin, charge, and stability to topological properties of the chronon field, yielding concrete experimental signatures. Thus the compact-action/RNF framework provides a unified geometric origin for quantum interference, measurement, and matter, together with falsifiable predictions of ChFT.