We prove that its semiclassical limit does not depend on the choice of the complex structure. In other words, the moduli space of semisimple frobenius manifolds has a natural quantization from the perspective of quantum groups and poisson geometry the irregular atiyahbott construction 3. These maps are controlled in the semiclassical limit by two invariants of symplectic topology. While there are plenty of things to say about the quantum plane, and quantum groups, i think ill home in on my main topic at last. Spectral convergence in geometric quantizationthe case of toric. In classical and quantum mechanics, the geometric phase is a phase difference acquired over the course of a cycle, when a system is subjected to cyclic adiabatic processes, which results from the geometrical properties of the parameter space of the hamiltonian. Semiclassical limits of quantized coordinate rings. On a quasimorphism of hamiltonian diffeomorphisms and.
Geometric quantization and quantum mechanics jedrzej sniatycki. Lectures on the geometry of quantization math berkeley. We also prove a semiclassical version of bealss theorem, characterizing pseudodi. Semiclassical physics, or simply semiclassical refers to a theory in which one part of a system is described quantummechanically whereas the other is treated classically. It has given valuable and lasting insights into the relationship between classical and quantum systems, and continues to be a popular research topic. Konstantin zarembo nordic institute for theoretical physics. The book 29 treats further topics in symplectic geometry and mechanics, with special attention to the role of symmetry groups, a topic pretty much ignored in the present notes. Semiclassical properties of geometric quantization with. Semiclassical limit of joint distribution functions. This paper offers an expository account of some ideas, methods, and conjectures concerning quantized coordinate rings and their semiclassical limits, with a particular focus on primitive ideal spaces. I spectral geometry, starting withweyls lawon the asymptotic distribution of eigenvalues of laplacians, was in part motivated by.
Symplectic geometry, fourier transform, stationary phase, quantization of symbols, semiclassical defect measures, eigenvalues and eigenfunctions, exponential estimates for eigenfunctions, symbol calculus, quantum ergodicity and quantizing symplectic transformations. The semiclassical asymptotic in representation theory is the limit when all components of highest weights go to in nity at the same rate. Geometric quantization is often presented as a way to take a. In this limit many features of representation theory can be expressed in terms of geometry of coadjoint orbits. Semiclassical analysis harvard department of mathematics. In this book, eric heller introduces and develops this subject, demonstrating its power with many examples. Geometric quantization and semiclassical limits of pairings of tqft vectors. Firstly, we estimate the subfidelity and superfidelity of two such states, giving lower and upper bounds for their fidelity, when the underlying submanifolds are two lagrangian submanifolds intersecting transversally at a finite number of points, in the semiclassical limit. Lectures on the mathematics of quantum mechanics volume.
Of course any other ways whether physical or mathematical of thinking about and understanding semiclassical limits of quantum mechanics are also welcomed as answer. There has been a very strong development in the mathematical theory, mainly thanks to methods of microlocal analysis. Geometric quantization often produces not one hilbert space to represent the quantum states of a classical system but a whole family h s of hilbert spaces, and the. Perfect book for first reading in string theory that can be used as reference by its practitioners.
Geometric quantization is a method to overcome and answer these pro. Semiclassical approximation addresses the important relationship between quantum and classical mechanics. Sure, we can make quantization more systematic by limiting the class of. The semiclassical way to dynamics and spectroscopy. Geometric quantization often produces not one hilbert space to represent the quantum states of a classical system but a whole family h s of hilbert spaces, and the question arises if the spaces h. The resulting quantum geometrical states are hence represented by the 3d blocks of analytically continued chernsimons theory. In the setting of geometric quantization, we associate to any prequantum bundle automorphism a unitary map of the corresponding quantum space. Proofs may be found in any symplectic geometry book. Sl2,c chernsimons theory, flat connections, and four. Geometric quantization, semiclassical limits, and formal quantization. As an application we show how, on the level of order functions, quantization commutes with ex. In the book, the first reference, youll find the geometric prequantization construction and the quantization of elementary systems coadjoint orbits of poincare. A rigorous proof of the bohrvan leeuwen theorem in the semiclassical limit 2014. For the formalization of this method in symplectic geometry geometric quantization see at semiclassical state.
Geometric quantization birs banff international research station. Fractional gaussian fields in geometric quantization and. In mathematical physics, geometric quantization is a mathematical approach to defining a quantum theory corresponding to a given classical theory. Semiclassical limit of quantum mechanics stack exchange. From spectral geometry to geometry of noncommutative.
We prove that its semiclassical limit does not depend on the choice of. Besides, landau pointed out the existence of another. This book develops the basic methods, including the wkbmethod, stationary phase and hpseudodifferential operators. In this report, the theory of geometric quantization and its uses in physics are described. H 2 m, z on the moduli space of equivalence classes of differential star products on m, defdiffm. As an application, we show that eigenvectors of these operators are concentrated near the level sets of these trace functions, and obtain asymptotic. Gianfausto dellantonio lectures on the mathematics of quantum mechanics february 12, 2015 mathematical department, universita sapienza rome. We introduce a minimalistic notion of semiclassical quantization and use it to prove that the convex hull of the semiclassical spectrum of a quantum system given by a collection of. It is a kind of preliminary version of the hilbert space we get when we quantize the classical system whose phase space is x. Spectral asymptotics in the semiclassical limit by m. The geometric approach to quantization was introduced by konstant and souriau more than 20 years ago. Quantization and semiclassical analysis 5 discrete and the corresponding eigenfunctions form a basis of l2x.
As an application, we show that eigenvectors of these operators are concentrated near the level sets of these trace functions, and obtain asymptotic estimates of pairings of such eigenvectors. For su2 the semiclassical asymptotic of 6j symbols was computed by ponzano and regge in. My main research interests are berezintoeplitz operators, that one encounters one studying the semiclassical limit of geometric quantization, and classical and quantum integrable systems, in particular semitoric systems. Semiclassical approaches have been successful in explaining several phenomena which had become paradigms. In the first half of the book, heller covers relevant aspects of classical mechanics, building from them the semiclassical way through the semiclassical limit of. Citeseerx document details isaac councill, lee giles, pradeep teregowda. The semiclassical contribution to the density of states 35. For instance in wave optics this yields the shortwavelength limit of the geometrical optics approximation. Lie groups originate as the semiclassical limit of drinfeldjimbo quantum groups1. Semiclassical and large quantum number limits of the. The geometric quantization of a symplectic manifold endowed with a prequantum bundle and a metaplectic structure is dened by means of an integrable complex structure. The geometric quantization of a symplectic manifold endowed with a prequantum bundle and a metaplectic structure is defined by means of an integrable complex structure. For su2 the semiclassical asymptotic of 6jsymbols was computed by ponzano and regge in 16.
In general, it incorporates a development in powers of plancks constant, resulting in the classical physics of power 0, and. In the first half of the book, heller covers relevant aspects of classical mechanics, building from them the semiclassical way through the semiclassical limit of the feynman path integral. From spectral geometry to noncommutative geometry i one of the pillars of alain connes program ofnoncommutative geometry ncg, specially itsmetric and di erential geometric aspects, isspectral geometry. The book covers amazingly large amount of topics with due attention to detail and logical rigour. The quantum mechanics axiom states that the set of eigenvalues usually called the spectrum of a quantum observable is exactly the set of possible values that can be obtained in a measurement. Classical limit and quantization of hamiltonian systems. Citeseerx semiclassical geometry of quantum line bundles. It attempts to carry out quantization, for which there is in general no exact recipe, in such a way that certain analogies between the classical theory and the quantum theory remain manifest. In this paper we show how deformation quantization of line bundles over a poisson manifold m produces a canonical action. This book contains a revised and expanded version of the lecture notes of two seminar series given during the academic year 197677 at the department of. The poisson bracket has a number of useful properties which we list below. This wkb method makes sense for a more general class of wave equations. Archive ouverte hal bounds for fidelity of semiclassical.
In chapter40we will rederive these formulas for the case of scattering in open systems. As an application, we show that eigenvectors of these operators are concentrated near the level sets of these trace functions, and obtain asymptotic estimates of pairings of. For more extensive treatment of the pde aspects of the subject. A rigorous proof of the bohrvan leeuwen theorem in the. Let t be a torus with lie algebra t and let m be a. Geometric quantization, semiclassical limits, and formal. A correspondence between threedimensional flat connections and constant curvature fourdimensional simplices is used to give a novel quantization of geometry via complex sl2,c chernsimons theory. Semiclassical limits of geometric quantization and the graded equivariant todd class. Geometric quantization uvafnwi universiteit van amsterdam. In this historical sense, semiclassical radiation theory strikes at the very roots of quantum mechanics. A potential well with a sufficiently strong attractive inversesquare tail supports an infinite dipole series of bound states, but the limit of.
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