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Lectures on the Geometry of Quantization

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Lectures on the Geometry of Quantization Lectures on the Geometry of Quantization Sean Bates Alan Weinstein Department of Mathematics Department of Mathematics Columbia University University of California New York, NY 10027 USA Berkeley, CA 94720 USA [email protected] [email protected] 1 Contents 1 Introduction: The Harmonic Oscillator 5 2 The WKB Method 8 2.1 Some Hamilton-Jacobi preliminaries . 8 2.2 The WKB approximation . 11 3 Symplectic Manifolds 17 3.1 Symplectic structures . 17 3.2 Cotangent bundles . 28 3.3 Mechanics on manifolds . 32 4 Quantization in Cotangent Bundles 36 4.1 Prequantization . 36 4.2 The Maslov correction . 41 4.3 Phase functions and lagrangian submanifolds . 46 4.4 WKB quantization . 56 5 The Symplectic Category 64 5.1 Symplectic reduction . 64 5.2 The symplectic category . 76 5.3 Symplectic manifolds and mechanics . 79 6 Fourier Integral Operators 83 6.1 Compositions of semi-classical states . 83 6.2 WKB quantization and compositions . 87 7 Geometric Quantization 93 7.1 Prequantization . 93 7.2 Polarizations and the metaplectic correction . 98 7.3 Quantization of semi-classical states . 109 8 Algebraic Quantization 111 8.1 Poisson algebras and Poisson manifolds . 111 8.2 Deformation quantization . 112 8.3 Symplectic groupoids . 114 A Densities 119 B The method of stationary phase 121 C Cechˇ cohomology 124 D Principal T~ bundles 125 2 Preface These notes are based on a course entitled “Symplectic geometry and geometric quantization” taught by Alan Weinstein at the University of California, Berkeley, in the fall semester of 1992 and again at the Centre Emile Borel (Institut Henri Poincar´e)in the spring semester of 1994. The only prerequisite for the course (and for these notes) was a knowledge of the basic notions from the theory of differentiable manifolds (differential forms, vector fields, transversality, etc.). The aim of the course was to give students an introduction to the ideas of microlocal analysis and the related symplectic geometry, with an emphasis on the role which these ideas play in formalizing the transition between the mathematics of classical dynamics (hamiltonian flows on symplectic manifolds) and that of quantum mechanics (unitary flows on Hilbert spaces). There already exist many books on the subjects treated here, but most of them provide too much detail for the reader who just wants to find out what the subject is about. These notes are meant to function as a guide to the literature; we refer to other sources for many details which are omitted here, and which can be bypassed on a first reading. The pamphlet [63] is in some sense a precursor to these notes. On the other hand, a much more complete reference on the subject, written at about the same time, is [28]. An earlier work, one of the first to treat the connections between classical and quantum mechanics from a geometric viewpoint, is [41]. 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. For more extensive treatment of the PDE aspects of the subject, we refer to [43] for a physics-oriented presentation and to the notes [21] and the treatises [32], [46], and [56]. For “geometric quantization”, one may consult [35], [53], [54], [60] or [71]. For classical mechanics and symplectic geometry, we suggest [1], [2], [6], [8], [25], [38], [59]. Finally, two basic references on quantum mechanics itself are [13] and [20]. Although symplectic geometry is like any field of mathematics in having its definitions, theorems, etc., it is also a special way of looking at a very broad part of mathematics and its applications. For many “symplecticians”, it is almost a religion. A previous paper by one of us [64] referred to “the symplectic creed”.1 In these notes, we show how symplectic geometry arises from the study of semi-classical solutions to the Schr¨odingerequation, and in turn provides a geometric foundation for the further analysis of this and other formulations of quantum mechanics. These notes are still not in final form, but they have already benefitted from the comments 1We like the following quotation from [4] very much: In recent years, symplectic and contact geometries have encroached on all areas of mathemat- ics. As each skylark must display its comb, so every branch of mathematics must finally display symplectisation. In mathematics there exist operations on different levels: functions acting on numbers, operators acting on functions, functors acting on operators, and so on. Symplec- tisation belongs to the small set of highest level operations, acting not on details (functions, operators, functors), but on all the mathematics at once. Although some such highest level operations are presently known (for example, algebraisation, Bourbakisation, complexification, superisation, symplectisation) there is as yet no axiomatic theory describing them. 3 and suggestions of many readers, especially Maurice Garay, Jim Morehead, and Dmitry Roytenberg. We welcome further comments. We would like to thank the Centre Emile Borel and the Isaac Newton Institute for their hospitality. During the preparation of these notes, S.B. was supported by NSF graduate and postdoctoral fellowships in mathematics. A.W. was partially supported by NSF Grants DMS-90-01089 and 93-01089. 4 1 Introduction: The Harmonic Oscillator In these notes, we will take a “spiral” approach toward the quantization problem, beginning with a very concrete example and its proposed solution, and then returning to the same kind of problem at progressively higher levels of generality. Specifically, we will start with the harmonic oscillator as described classically in the phase plane R2 and work toward the problem of quantizing arbitrary symplectic manifolds. The latter problem has taken on a new interest in view of recent work by Witten and others in the area of topological quantum field theory (see for example [7]). The classical picture The harmonic oscillator in 1 dimension is described by Newton’s differential equation: mx¨ = −kx. By a standard procedure, we can convert this second-order ordinary differential equation into a system of two first-order equations. Introducing the “phase plane” R2 with position and momentum coordinates (q, p), we set q = x p = mx,˙ so that Newton’s equation becomes the pair of equations: p q˙ = p˙ = −kq. m If we now introduce the hamiltonian function H : R2 → R representing the sum of kinetic and potential energies, p2 kq2 H(q, p) = + 2m 2 then we find ∂H ∂H q˙ = p˙ = − ∂p ∂q These simple equations, which can describe a wide variety of classical mechanical systems with appropriate choices of the function H, are called Hamilton’s equations.2 Hamilton’s equations define a flow on the phase plane representing the time-evolution of the classical system at hand; solution curves in the case of the harmonic oscillator are ellipses centered at the origin, and points in the phase plane move clockwise around each ellipse. We note two qualitative features of the hamiltonian description of a system: 1. The derivative of H along a solution curve is dH ∂H ∂H = q˙ + p˙ = −p˙q˙ +q ˙p˙ = 0, dt ∂q ∂p 2If we had chosenx ˙ rather than mx˙ as the second coordinate of our phase plane, we would not have arrived at this universal form of the equations. 5 i.e., the value of H is constant along integral curves of the hamiltonian vector field. Since H represents the total energy of the system, this property of the flow is interpreted as the law of conservation of energy. ∂H ∂H 2. The divergence of the hamiltonian vector field XH = (q, ˙ p˙) = ( ∂p , − ∂q ) is ∂2H ∂2H ∇ · X = − = 0. H ∂q ∂p ∂p ∂q Thus the vector field XH is divergence-free, and its flow preserves area in the phase plane. The description of classical hamiltonian mechanics just given is tied to a particular coordinate system. We shall see in Chapter 3 that the use of differential forms leads to a coordinate- free description and generalization of the hamiltonian viewpoint in the context of symplectic geometry. The quantum mechanical picture In quantum mechanics, the motion of the harmonic oscillator is described by a complex- valued wave function ψ(x, t) satisfying the 1-dimensional Schr¨odinger equation: ∂ψ 2 ∂2ψ k i = − ~ + x2ψ. ~ ∂t 2m ∂x2 2 Here, Planck’s constant ~ has the dimensions of action (energy × time). Interpreting the right hand side of this equation as the result of applying to the wave function ψ the operator 2 2 def ~ ∂ k Hˆ = − + m 2 , 2m ∂x2 2 x 2 where mx2 is the operator of multiplication by x , we may rewrite the Schr¨odinger equation as ∂ψ i = Hψ.ˆ ~ ∂t A solution ψ of this equation does not represent a classical trajectory; instead, if ψ is normalized, i.e. Z ψ∗ψ = 1, R then its square-norm ρ(x, t) = |ψ(x, t)|2 is interpreted as a probability density for observing the oscillator at the position x at time t. The wave function ψ(x, t) itself may be viewed alternatively as a t-dependent function of x, or as a path in the function space C∞(R, C). From the latter point of view, Schr¨odinger’s equation defines a vector field on C∞(R, C) representing the time evolution of the quantum system; a wave function satisfying Schr¨odinger’s equation then corresponds to an integral curve of the associated flow. Like Hamilton’s equations in classical mechanics, the Schr¨odinger equation is a general form for the quantum mechanical description of a large class of systems. 6 Quantization and the classical limit The central aim of these notes is to give a geometric interpretation of relationships between the fundamental equations of classical and quantum mechanics.
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