Introduction coadjoint orbits geometric quantizationthe irreducible unitary representations of su2conclusions the coadjoint representation 1 introduction 2 coadjoint orbits. In the second chapter, i will describe the quantization process before givingtwo examples, namely of the cotangentbundle of r, derivingthe. In mathematical physics, geometric quantization is a mathematical approach to defining a quantum theory corresponding to a given classical theory. In particular we give a new interpretation about previous results of the author in order to build an asymptotics quantum. The theory of geometric quantization has been developed independently by b.
For unconstrained systems, any polarization on the. Such systems are quite natural when we analyze coupled bosons and fermions. L with curvature rl satisfying the following prequantization. Department of physics, university of colorado, boulder, colorado 80309, united states abstract geometric quantization is an attempt at using the di. In this article we discuss the geometric quantization on a certain type of infinite dimensional superdisk.
Geometric quantization of completely integrable hamiltonian. Jedrzej sniatycki geometric quantization and quantum. In this section we will examine the quantization procedures usefull to pass from a phase space, generally a symplectic manifold, to an hilbert space. Lectures on geometric quantization lecture notes in. Finally, the last two sections examine the connection between functorial geometric quantization and the kostantsouriau theory, culminating in a heuristic. The precise structure needed on the constraint surface of the full phase space to obtain a polarization on the reduced phase space is displayed in a form that is particularly convenient for applications. He is currently professor of mathematics at wadham college, university of oxford he was treasurer of the london mathematical society until 2009 and was president of clay mathematics institute he has frequently collaborated with david j.
The associated quantum algebra consists of functions affine in action coordinates. In this theory the additional structure needed for quantization is explicitely defined in terms of the geometry of the phase space, and it consists of a prequantization, a polarization and a metaplectic structure 3. In this project we introduce the general idea of geometric quantization and demonstratehow to apply the process on a few examples. Part of the lecture notes in physics book series lnp, volume 53 chapters table of contents 9 chapters about about this book. In this article we develop tools to compute the geometric quantization of a symplectic manifold with respect to a regular lagrangian foliation via sheaf cohomology.
Box 41882, 1009 db dam mster a the netherlands 1 this is a preliminary ersion v not tended in for distribution outside this ho scol. Topics included are coherent states, geometric quantization, phase space quantization, deformation and products and berrys phase. Measures in the geometric quantization of field theories. This is a generalization of the procedure for building quantum mechanics from classical mechanics. David borthwick, alejandro uribe, almost complex structures and geometric quantization arxiv. Mathematical foundations of geometric quantization a. The kostantsouriau geometric quantization theory is applied to the problem of constructing a generally covariant quantum field theory. A quantization method, generalizing the usual canonical one. This paper is about geometric quantization of the hitchin system. The ideas have proved useful in pure mathematics, notably in representation theory, as well as in theoretical physics. Geometric quantization is an attempt at using the differentialgeometric ingredients of classical phase spaces regarded as symplectic manifolds in order to define a corresponding quantum theory. In this theory the additional structure needed for quantization is explicitely defined in terms of the geometry of the phase space, and it consists of a prequantization, a polarization and a.
I am grateful for using the notes of alexandre kirillov 1 in an intensive manner for both sections about coadjoint orbits. Quantization schemecomputations and examples bertram kostant. In case the manifold is equipped with the hamiltonian action of a lie group. A prequantum line bundle with connection on m is a line bundle l m equipped with a connection. Pdf geometric quantization is an attempt at using the. Quantization is an attempt to determine the quantum theory of a physical system from the knowledge of its classical theory. It has given valuable and lasting insights into the relationship between classical and quantum systems, and continues to be a popular research topic. Geometric quantization of hamiltonian flows and the gutzwiller trace. A practical and scientific approach to deep sky imaging, second edition. For geometric quantization, one may consult 35, 53, 54, 60 or. Geometric quantization oxford mathematical monographs. We provide geometric quantization of this cotangent bundle where the quantum constraint serves. Geometric quantization of the hitchin system international. Geometric quantization is an attempt at using the differential geometric ingredients of classical phase spaces regarded as symplectic manifolds in order to define a corresponding quantum theory.
Oct 10, 2004 geometric quantization of relativistic hamiltonian mechanics sardanashvily, g. Geometric quantization and quantum mechanics jedrzej. The lectures notes of stefan vandoren 6 where useful for both section 5 and. The geometric approach to quantization was introduced by konstant and souriau more than 20 years ago. Geometric quantization, berezintoeplitz quantization and povm. One problem i run into is that, when trying to repeat the usual derivation of the path integral, we want to insert a complete set of states, but typically here the hilbert space. An introduction to the orbit method universiteit utrecht. First, we define an example of a superhomogeneous manifold. We want to recast the fundamentals of the kostantsouriau theory of geometric quantization 12, taking for the 4symplectic manifold the prolongation lgomof the bundle of linear frames of spacetime mwith the 4symplectic twoform d we restrict attention to the essentials of the initial prequantization procedure. Geometric quantization of real polarizations via sheaves eva miranda and francisco presas abstract. To date, this program has succeeded in unifying various older methods of quantizing. Generally, the process of geometric quantization is applicable to other. S1 and this is the subject of the closing chapter 7. Symplectic geometry and geometric tization quan 1 matthias blau 2 nikhefh p.
Functorial geometric quantization is the subject of section 5, and in section 6 van hoves theorem is used to prove the functorial nonexistence theorem. The aim of the conference was to find common elements between quantization and coherent states, and quantization on poisson manifolds. Its often assumed that something like this is possible, for example, when working with chernsimons theory, but rarely explained in detail. Introduction generally speaking, geometric quantization is a scheme for associating hilbert spaces to symplectic manifolds.
The purpose of the present book is to complement the preceding ones by including new developments of the theory and emphasizing the computations leading to results in quantum mechanics. Nicholas michael john woodhouse cbe born 27 february 1949 is a british mathematician woodhouse completed his undergraduate degree at oxford. So we go ahead with our goal that we can reduce to the case of studying the geometric quantization of t. The basic idea the setting of the hamiltonian version of classical newtonian mechanics is the phase space position and momentum, which is a symplectic manifold. However, its a bit like a power tool you have to be an expert to operate it without running the risk of seriously injuring your brain. Lectures on the geometry of quantization math berkeley. Download fulltext pdf download fulltext pdf quantum logic and geometric quantization article pdf available in journal of quantum information science 701 march 2017 with 49 reads. Reliable information about the coronavirus covid19 is available from the world health organization current situation, international travel.
It is a procedure for constructing a quantum field theory starting from a classical field theory. We quantize a kahler form on the hitchin moduli space which is half the first kahler form defined by hitchin by considering the quillen bundle as the prequantum line bundle and modifying the quillen metric using the higgs field so that the curvature is proportional to the kahler form. Ritter if anyone can suggest any others, i would be greatly appreciative. For example, the similarity between the heisenberg. Geometric prequantization on the spin bundle based on. A discussion of path integral in the framework of geometric quantization can be found in the standard book. In geometric quantization applied to physical systems, we describe the choices. Geometric quantization is a marvelous tool for understanding the relation between classical physics and quantum physics. On geometric quantization of classical systems sciencedirect. The largen limit of a system like that corresponds to a certain superhomogeneous space. The books of souriau 1970 and simms and woodhouse 1976 present the theory of geometric quantization and its relationship to quantum mech anics.
Geometric quantization of relativistic hamiltonian mechanics. Oclcs webjunction has pulled together information and resources to assist library staff as they consider how to handle coronavirus. Woodhouse was appointed commander of the order of the british empire cbe in the 2020 new year honours for services to mathematics. The classical phase space doesnt have to be a cotangent bundle, and one has a general framework to talk about different. In physics, quantization quantisation is the process of transition from a classical understanding of physical phenomena to a newer understanding known as quantum mechanics. To date, this program has succeeded in unifying various older meth. Aug 19, 2002 we provide geometric quantization of a completely integrable hamiltonian system in the actionangle variables around an invariant torus with respect to polarization spanned by almosthamiltonian vector fields of angle variables. We discuss how to construct a line bundleover the symplectic manifold with diracs quantization conditions and how to determine if we are able to quantize a system with the help of weils integrability condition. Geometric quantization of symplectic vector spaces.
Oct 21, 2015 2 is the standard text on geometric quantization which is much more comprehsive. I will try to give an overview of what geometric quantization is. The section on geometric quantization could not have been written without the book of nicholas woodhouse 7. In what follows we will primarily follow the exposition given by woodhouse and simms 12.
Numerous and frequentlyupdated resource results are available from this search. We obtain a set of its nonequivalent representations in the separable prehilbert. Steinberg, geometric quantization and multiplicities of group representations, invent. Geometric prequantization on the spin bundle based on n. Quantenmechanik geometry mechanics quantization quantum mechanics. Atiyah, geometry of yangmills fields fermi lectures, scuola normal superior, pisa 1979. We assume that m is a phase space and h an hilbert space yielded by a quantization scheme. I was wondering how one obtains geometric quantization from a path integral. Nicholas woodhouse, geometric quantization and quantum field theory in curved spacetimes, reports on mathematical physics 12.
In this paper we consider the set of all experimental propositions of m and we look for a model of quantum logic in relation to the quantization of the base manifold m. We quantize a kahler form on the hitchin moduli space which is half the first kahler form defined by hitchin by considering the quillen bundle as the prequantum line bundle and modifying the quillen metric using the higgs field so that the curvature is proportional to the. Aug 21, 2001 in this article we discuss the geometric quantization on a certain type of infinite dimensional superdisk. In particular, we want to clarify the mathematical aspects related to the geometrical structures involved in this theory. The typical example of this is the cotangent bundle of a. Buy lectures on geometric quantization lecture notes in physics 53 on free shipping on qualified orders lectures on geometric quantization lecture notes in physics 53. Student seminar on geometric quantization max planck. Path integral and geometric quantization physics stack exchange. 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. Path integrals are defined for finite dimensional hilbert spaces, too. Geometric quantization 1 introduction the aim of the geometric quantization program is to describe a quantization procedure in terms of natural geometric structures. Geometric quantization a good overview of which is provided by the books of woodhouse, 14, and sniatycki, is basically a replacement. The problem of obtaining the quantum theory of systems with first class constraints is discussed in the context of geometric quantization. Symplectic geometry and mechanics a useful reference is simms and woodhouse, lectures on geometric quantization, available online.
Woodhouse the geometric approach to quantization was introduced by konstant and souriau more than 20 years ago. Im searching for a good primer on geometric quantization. Twistor theory and geometric quantization springerlink. Let be a complex projective compact manifold and a kahler form. Hall, quantum theory for mathematicians, springer 20 2 n.
30 457 1372 1022 240 185 1184 1059 913 1000 1513 1102 152 111 849 740 604 509 15 497 85 1413 801 87 1049 817 1298 556 1108 1430 271 681 207 673 572 1047 701 442 1321 303 1244 1136 948