The aim of the geometric quantization program is to describe a quantization procedure in terms of

Transcription

1 Thomas Mainiero Syed Asif Hassan Geometric Quantization 1 Introduction The aim of the geometric quantization program is to describe a quantization procedure in terms of natural geometric structures. To date, this program has succeeded in unifying various older methods of quantizing finite dimensional physical systems. The generalization to infinite-dimensional systems (for example, field theories) remains an active area of research. As such, we will restrict our attention the finite dimensional case. We will describe the basic construction procedure and the geometric structures involved, and for concreteness we will show the explicit details of the construction in the case of the n-dimensional harmonic oscillator. Despite its apparent simplicity, the harmonic oscillator is sufficiently rich a physical system to highlight the main points of the geometric quantization procedure while requiring us to grapple with some of the subtle issues which arise. In what follows we will primarily follow the exposition given by Woodhouse and Simms [1][2]. First we will provide a mathematical description of a classical physical system and define the conventions used in this paper. The basic object is a symplectic manifold: a pair (M, ω) with M a 2n dimensional manifold M equipped with a closed nondegenerate 2-form ω. We are then naturally concerned with morphisms which preserve this structure; in particular, these are the symplectomorphisms ρ : M 1 M 2 for symplectic manifolds (M 1, ω 1 ) and (M 2, ω 2 ) such that ρ ω 2 = ω 1. When appropriate we will concern ourselves with M an affine symplectic manifold and restrict our attention to the linear symplectomorphisms M M which form the group Sp(M) = Sp(2n, R). Without loss of generality for the following discussions, we will blur the distinction between an affine space and its associated vector space. Via Darboux s theorem, around any point m M a symplectic manifold M we can find a neighborhood U containing m and local coordinates p a, q a on U such that ω U = dp a dq a (1.1) 1

2 where the summation convention is employed and a, b,... {0,..., n}. The choice of such coordinates is far from unique; indeed, under any symplectomorphism ρ : M M fixing m, p a ρ and q a ρ also satisfy the above. 1 If M is a vector space V equipped with a complex structure, then there are particularly special subspaces W V on which ω W 0. If W is a maximal such subspace (i.e. it is not contained in any larger subspace on which the symplectic form also vanishes) we call W Lagrangian. It can be shown that Lagrangian subspaces always exist and are half the dimension of V. Given a set of Darboux coordinates on V, there are two particularly illuminating examples of Lagrangian subspaces: the space of constant p a ( position space ) and the space of constant q a ( momentum space ). That these are Lagrangian is clear from eq. 1.1, which is valid on all of V. When M is a general symplectic manifold, a Lagrangian submanifold L is a submanifold of M such that T m L T m M is a Lagrangian subspace for every m L. A Lagrangian submanifold is then a manifold of dimension half that of M. Along the lines of the examples given in the case when M is a vector space, position space or momentum space can be thought of Lagrangian submanifolds of M. However, the freedom in choice of Darboux coordinates shows that, heuristically, there is considerable freedom in what can be called momentum (position) space. The selection of a particular Lagrangian subspace is then a choice of what is momentum (position) space out of infinitely many possibilities. 2 This will become relevant when we discuss real polarizations. If we consider the complexification of the tangent bundle of M: T C M = T M C, then the symplectic form on M extends linearly on each T m M to T m M C. We can then talk about complex Lagrangian subspaces or submanifolds on which the extended form vanishes linearly. Complex Lagrangian subspaces of T m M C have half the complex-dimension of T m M C or equivalently, the same real-dimension as T m M. Since ω is closed (dω = 0), locally we can write ω = dθ (1.2) 1 Note that as far as Darboux s theorem is concerned we can relax the diffeomorphism condition to local diffeomorphism and take appropriate subsets of U 2 In fact, the set of (real) Lagrangian subspaces on a vector space has an induced manifold structure when thought of as a submanifold of the Grassmanian space of n-dimensional planes. The space of complex Lagrangian subspaces, which will arise later, also has a manifold structure. 2

3 where θ Ω 1 (M) is called a symplectic potential for ω. We will primarily be concerned with the case where ω is also exact so this relation is globally sensible, in particular when M is an affine space or a cotangent bundle. 3 In the case of M = T Q let α Ω 1 (Q) be a one-form on Q; we can also regard this one-form as a section α : Q T M, then there is a (unique) preferred choice of θ such that α θ = α. (1.3) In Darboux coordinates (p a, q a ) extending coordinates q a on the submanifold Q, θ = p a dq a. (1.4) One possible inconvenience of the symplectic potential (eq. 1.4) is that it is not invariant under general symplectic transformations (e.g. consider the transformation p a q a and q a p a ). The advantage of constructing an invariant symplectic potential is that we need not worry how it changes under a canonical (symplectic) change of coordinates, or even better how it changes under the Hamiltonian flow of some real-valued function (defined below). There is at least one choice of such a symplectic potential, defined invariantly by Y (ι X θ) = ω(x, Y ) for X, Y vector fields on M. In Darboux coordinates, we have a solution θ 0 = 1 2 (p adq a q a dp a ). (1.5) This choice of symplectic potential is useful when dealing with complex coordinates since it is real and also has a simple expression (eq. C.1) in terms of complex coordinates. The nondegeneracy of ω implies it defines an isomorphism ω : T M T M; hence for any smooth function f : M R, df is mapped to a unique vector field X f, called the Hamiltonian 3 More generally when [ω] = 0 as a class in H 1 (M; Z) we can also find such a global potential. 3

4 vector field of f. Explicitly, df + ι Xf ω = 0 (1.6) or in Darboux coordinates, X f = f p a q a f q a p a. (1.7) Each f then determines one-parameter family of diffeomorphisms ρ t : M M via X f. As L Xf ω = dι Xf ω + ι Xf dω = d(df) + ι Xf (0) = 0 each ρ t is a symplectomorphism. We will refer to the ρ t as the Hamiltonian flow of f. The Poisson bracket of two functions is given by {f 1, f 2 } = 2ω(X f1, X f2 ) = X f1 f 2 X f2 f 2 = 2X f1 f 2 (1.8) (note the factor of 2 here compared to the usual physics convention). This is essentially the Lie derivative of f 2 along the flow determined by f 1. The description of a physical system involves the specification of functions which correspond to physical observables. The Hamiltonian function h, typically representing the total energy of the system, encodes the dynamics via the Poisson bracket. That is, observables evolve along the Hamiltonian flow of the Hamiltonian function. More concretely, a point in phase space represents a state of the classical system. If the system is initially in a particular state, i.e. at a particular point in phase space, its subsequent states can be obtained by following the flow of X h from that point. Often the observables p and q are functions on M and also serve as coordinates for M. If the system follows the trajectory ( P (τ), Q(τ) ) then the tangent to the trajectory coincides with X h, dq dτ = h p (p,q)=(p,q) dp dτ = h (1.9) q (p,q)=(p,q) which are the standard Hamilton-Jacobi equations. 4

5 2 Complex Structure Some physical systems lend themselves to a description in terms of a phase space equipped with complex structure. For example, the Hamiltonian for the one-dimensional harmonic oscillator, h = 1 2 (p2 + q 2 ) (2.1) can be rewritten as h = 1 z z (2.2) 2 where z = p + iq and z = p iq. The phase space of the (n-dimensional) harmonic oscillator, a real 2n-dimensional symplectic vector space (V, ω), is apparently isomorphic to an n-dimensional complex vector space spanned by the coordinates z and z. This is possible because the original real phase space has a complex structure compatible with its symplectic form. In general, a linear map J : V V such that J 2 = id V, on a real (even dimensional) vector space V is said to be a complex structure. Given any such linear map on a 2n-dimensional vector space we can pass to an n-complex dimensional space V J by defining complex multiplication on a vector X as (x + iy)x = (x + yj)x. Here the subscript J is placed on V J to emphasize the dependence on the complex structure J. 4 If (V, ω) is a symplectic vector space and J Sp(V ) = Sp(2n, R) then J is said to be compatible with ω so that g(, ) = ω(, J ) 4 The space is n-complex dimensional as opposed to 2n complex dimensional as via our definition of complex multiplication, X and JX, two linearly independent vectors in V pass to the linearly dependent X and ix in V J 5

6 defines a symmetric, nondegenerate, bilinear form on V. Equivalently,, J = g(, ) + 2iω(, ) defines a hermitian inner product on V J. We say that J is positive if the metric g it defines is positive-definite. We will be mainly concerned with positive complex structures as the property of positive-definiteness on g is useful for defining properly normalizable wavefunctions after quantization. Furthermore, the complex structure on V is a choice; hence, we wish to study the space of all compatible complex structures L + V, a contractible space called the Lagrangian Grassmanian. As mentioned above, for each complex structure J we can define an n-dimensional complex vector space V J. Because of the arbitrariness of J, we would like to find a mechanism by which to compare V J for each choice of J; this can be done if we are able to embed the V J inside some larger space. Indeed, consider the complexification V C = V C, a 2n-dimensional complex vector space, where the symplectic and complex structures on V pass over to V C by linear extension. 5 J then splits V C into n-dimensional complex eigenspaces: V C = P J + P J on which it takes values +i and i (respectively); we then identify V J with the +i eigenspace P J. More explicitly we have the map ι J : V J V C X 1 (X ijx) 2 under which V J is mapped isomorphically to the subspace P J, and it is easily shown that P J is a complex-lagrangian subspace. 6 On the other hand, the process can be reversed to show that the selection of a complex-lagrangian subspace determines a complex structure J (not necessarily 5 J(a + ib)x = (a + ib)jx and ω((a + ib)x, Y ) = (a + ib)ω(x, Y ) 6 Note that this procedure constructs an identification of L + V with a complex submanifold of the n-grassmanian of V C, hence the name Lagrangian Grassmanian. 6

7 positive) on V. The whole description above can be applied to a general symplectic manifold (M, ω) where the vector space V is replaced with the fibers of T M and V C is replaced with the fibers of T C M = T M C. A compatible complex structure J is defined as an assignment of a linear map J m : T m M T m M on each fiber and is pointwise compatible with ω. Furthermore, we require an integrability condition on J, so that our spaces V J pass over to complex manifolds M J with holomorphic coordinate charts. Any such symplectic manifold (M, ω) with a compatible, integrable J is called a Kähler manifold. As we will see later, it can be an advantage to have the complex description available since the subset of classical observables which are easily represented as quantum observables is tightly constrained, and the constraints are different in the real and complex cases. In the case of the harmonic oscillator, when working in complex variables the construction of an operator corresponding to the Hamiltonian is straightforward, whereas when working in real variables the construction is much more difficult. The analogy that should be kept in mind is that of real vs. complex representations of a group; sometimes it is easier to work with complex representations. 3 Prequantization 3.1 Preliminaries The first step in the geometric quantization program is to find geometric structures which reproduce the essential features of a quantum mechanical system based on a classical system. In broad terms, the state of a quantum system is no longer a single point in the phase space M but rather a complexvalued function over M (the wave function ) which when squared represents the probability of observing the system in a particular classical configuration. Observables are then no longer simply real-valued functions over M but instead are symmetric operators 7 which map wavefunctions to wavefunctions. As usual in quantum mechanics we equip the space of wavefunctions with an inner product 7 An operator ˆf is symmetric if ψ, ˆfψ = ˆfψ, ψ where ψ, ψ are wavefunctions and, is the inner product. When the inner product is defined in the standard way (section 3.3) a symmetric operator is Hermitian and has real eigenvalues, which is the physical motivation for this requirement. 7

8 which allows us to compare physical information contained in wavefunctions and introduce the probabilistic interpretation of quantum mechanics. Usually we take a completion of our wavefunction space under our inner product and arrive at a separable Hilbert space. 8 The inner product is invariant under a pointwise change of phase on the space of wavefunctions; the physical content of our theory is invariant under local changes in phase (i.e. change in phase is a gauge symmetry) Dirac Quantization Conditions In order to construct the relevant space of quantum states, we will first look at how to pass from classical observables to quantum observables, in other words from a real-valued function f : M R on the classical phase space to an operator ˆf on some quantum mechanical space to be determined. By placing some basic requirements on the general prescription f ˆf we will deduce what quantum mechanical spaces ˆf can act on naturally. In particular, the requirements placed on f ˆf are three principles given by Dirac: 1. the map f ˆf is linear 2. if f is constant, then ˆf is a multiplication operator 3. Poisson brackets map to commutators, {f 1, f 2 } = f 3 [ ˆf 1, ˆf 2 ] = i ˆf 3 These requirements motivate the following prescription. Wavefunctions φ are sections of a complex line bundle B M. Functions on M are mapped to operators via ( ˆf = i X f i ) ι X f θ + f (3.1) where θ is some symplectic potential and acts also as a connection 1-form on the line bundle, so that the first two terms together form a U(1) covariant derivative. 8 In the infinite dimensional case there are subtle analytical issues that arise when dealing with operators on the full Hilbert space. Tools such as Schwarz spaces and the rigged Hilbert space construction overcome these difficulties but these are technical issues that are outside the confines of our discussion. 9 Furthermore, in physics, the probabilistic interpretation requires us to normalize wavefunctions; hence, we can extend our gauge symmetry to multiplication by an arbitrary complex number, thus, the true quantum mechanical phase space is the space of equivalence classes of wavefunctions under this gauge symmetry: the projective Hilbert space (the space of rays in the Hilbert space). This larger gauge symmetry is exactly what we obtain for wavefunctions over the complexified classical phase space. 8

9 This construction may seem artificial at first, but becomes more clear if we first try to make a more obvious choice, that is to take wavefunctions to be in L 2 (M), square integrable complex functions on M, and operators to be given by ˆf = i X f. 10 This satisfies the first condition but not the second, so we must add on the third term of eq. (3.1). Now the third condition fails, but by adding on the middle term for some choice of symplectic potential θ, all three conditions hold. However, the need choose a symplectic potential to quantize is unsatisfactory: we could add an arbitrary closed form to θ and achieve a different quantization of f. But salvation comes in the form of gauge symmetry. 11 Take a contractible open cover of M; then on any open set U in our cover, the choice of θ is unique up to transformations θ θ + dλ for λ a real-valued function on U. Under such transformations ˆf ˆf = ˆf ι Xf dλ. On the other hand, a quick calculation shows that ˆf ( ) [ ] e iλ/ φ = e iλ/ ˆf (φ) + ι Xf dλφ = e iλ/ ˆf (φ). So if under θ θ + dλ we also impose φ e iλ/ φ, we recover invariance under choice of the symplectic potential. We have a U(1) gauge invariance, which suggests that our our wavefunctions φ are actually sections of a principal U(1) bundle (a line bundle), as suggested. In fact, with this interpretation and noting that X f φ = dφ(x f ) we see that eq. (3.1) takes the form ˆf = i Xf + f (3.2) 10 Here X f, which acts on real-valued functions on M, is trivially extended to acting on complex-valued functions. As before, there are subtle analytical questions pertaining to whether X f is bounded on L 2 (M), but these will not be expounded upon. 11 For this argument we follow [1] closely 9

10 where = d i 1 θ is the connection on our line bundle. Thus, the symplectic form behaves as a connection 1-form and our line-bundle has curvature 1 dθ = 1 ω. To prequantize a classical theory with symplectic manifold (M, ω) we then select a line bundle B M with curvature ω and lift the classical observables f : M R to operators on sections of our bundle via eq. (3.2). However, this prequantization procedure, as suggested by its name, is only a naïve first step toward quantization. As will be discussed later, some sections of our prequantum bundle violate the uncertainty principle; so we must place further constraints on the allowable sections. However, before the proper machinery is developed, there is one other missing puzzle piece: the L 2 norm on our wavefunctions. 3.3 Gauge Symmetry and the L 2 norm Because (M, ω) is a symplectic manifold there is a natural choice of volume form (up to an overall constant), the Liouville form ω n. We can choose the constant so that the rescaled volume form ɛ is unitless, ɛ = 1 (2π ) n ω n. In local Darboux coordinates ɛ = 1 (2π ) n dp 1 dp n dq 1 dq n. We then can construct an L 2 norm on prequantum bundle sections φ, φ given by φ, φ = (φ, φ )ɛ M Where (φ, φ ) is a pointwise pairing such that (φ, φ ) Ω 0 (M), i.e. it is a function on M, and it is gauge-invariant. A suitable choice (in some local trivialization) is (φ, φ ) = φφ φe iλ/ e iλ/ φ = φφ (3.3) 10

11 which is invariant under U(1) gauge transformations θ θ + dλ (so it makes sense globally on M). The measure ɛ is invariant since ω is, so the L 2 norm thus defined is U(1) gauge-invariant. The situation is more complicated when we work on a complexified manifold[3], since the gauge symmetry group is enlarged from U(1) to C {0} (multiplication by nonzero complex numbers rather than just phases). While ω is still real, θ may be complex since it transforms as θ θ + dλ + i dσ when φ e (iλ σ)/ φ. The transformation properties of φ now spoil the gauge invariance of the inner product previously defined, since φφ φe ( iλ σ)/ e (iλ σ)/ φ = φφe 2σ/. (3.4) What we need is a new measure ρɛ where ρ is a function that has the necessary transformation properties to cancel the unwanted factor. If we impose symmetry of ˆf ( ˆfφ, φ = φ, ˆfφ ) for f real, we obtain the necessary condition 12 d ln ρ = 2 Im(θ) (3.5) which fixes ρ for a given gauge θ and establishes that Im(θ) must be exact for ρ to exist. (Note that the reality of ω only ensures that Im(θ) is closed.) What this amounts to is that there must exist a choice of gauge in which θ is real. In such a gauge we have a natural choice ρ = 1 so that the inner product coincides with our previous definition. Since Im(θ) Im(θ) + dσ, d ln ρ d ln ρ + 2 dσ = d ln (ρe 2σ/ ), ρ ρe 2σ/ (3.6) so ρ has precisely the desired transformation behavior, hence the inner product φ, φ = (φ, φ )ρɛ (3.7) M is gauge-invariant. 12 Note that [3] and [1] use opposite sign conventions in the third Dirac quantization condition. Here we follow the conventions of [1]. 11

12 In practice, we will encounter situations where wavefunctions take a simple form (e.g. holomorphic or anti-holomorphic) in a special gauge, but it is most convenient to evaluate inner products in a different gauge where θ is real (for example θ 0 as defined in eqs. 1.5, C.1) so that ρ = 1. Thus, we can restrict ourselves to sections of B that have finite L 2 norm (as it is defined above); the completion of this subspace would form a Hilbert space, i.e. a possible model for our quantum state space. 3.4 Weil s Integrality Condition Not all symplectic manifolds (M, ω) may admit line bundles B M with curvature 1 ω. Indeed, by analyzing the holonomies induced by the connection one finds a necessary condition on the symplectic form ω called Weil s Integrality condition. We start by choosing a line bundle B M with curvature 1 ω and choose a connection 1-form h 1 θ defined on some local trivialization over U M. As shown in appendix A, the holonomy operator around some closed curve γ : [0, 1] U is given in this trivialization as multiplication by ( ) i ξ = exp θ. γ Now, suppose γ is the boundary of some two-surface Σ M, then via Stoke s theorem, as dθ = ω ( ) ( ) i i exp θ = exp ω. γ Σ Furthermore, suppose γ is bounded by a second surface Σ M such that Σ Σ form a closed orientable surface (so Σ = γ and Σ = γ taking into account orientations). Then, ( ) ( i exp θ = exp i ) ω γ Σ 12

13 so we must have ( i 1 = exp ω + i Σ ( ) i = exp. Σ Σ ω Σ ω ) Which enforces 1 ω Z. 2π Σ Σ Thus, we arrive at the necessary condition the integral of ω over any closed oriented surface in M is an integral multiple of 2π. In other words, ω defines an integral cohomology class (2π ) 1 [ω] H 2 (M; Z) (where H 2 (M; Z) is identified with its image in H 2 (M; R)); this is called the integrality condition. Furthermore, it can be shown this condition is also sufficient for the existence of a line bundle B M with curvature 1 ω. In fact, when the integrality condition holds, there is a whole family of such line bundles whose isomorphism classes are parameterized by the cohomology group H 1 (M, T) for T = U(1) the circle group. This puts a strong constraint on the symplectic manifolds (M, ω) which we can prequantize. For the cases of M affine H 1 (M, R) = 0 so we can find prequantum bundles, and more generally in the case of M = T Q the natural symplectic structure ω is exact so [ω] = 0 and we can always prequantize such a space. 4 Polarization 4.1 Real Polarizations From a physical perspective the wavefunctions constructed in the previous section have a major problem: for any prequantum bundle B M we can find unit-norm, square-integrable sections which are arbitrarily localized around a given m = (p, q) M, such sections interpreted as wave- 13

14 functions lead to a clear violation of the uncertainty principle. 13 The source of the problem can be traced back to choosing our space of wavefunctions to be complex functions of both position and momentum without any additional constraints. In basic quantum mechanics wavefunctions are usually square-integrable functions of either position or momentum. With this in mind, the basic strategy for remedying this problem should be to restrict the space of wavefunctions to those which are constant along some directions in M. For example, if we want functions only of position, we might demand that they be covariantly constant along the momentum directions (the span of / p a in some coordinate system). Evidently the geometric structure we want is a distribution P of dimension 1 dim M. Furthermore, we would like our distribution to be 2 integrable. Indeed, motivated by our example of P being the distribution of momentum directions, in each neighborhood of m M we would like to find a coordinate system q 1,, q n, p 1,, p n with the surfaces of constant p a spanning our leaves. Finally, from the expression ω = dp a dq a, we see that the constant momentum distribution satisfies ω Pm = 0 where m M and P m T m M, so we expect to choose distributions of Lagrangian subspaces. This motivates the following definition. Definition: A (real) polarization P T M of M is a smooth, integrable, distribution such that P m is Lagrangian for each m M. In other words, P is a foliation of M by Lagrangian submanifolds. On M = T Q, the constant momenta example of a polarization discussed above is called the vertical polarization. Our definition is further motivated by the following proposition which shows all polarizations look like the vertical polarization (at least locally)[4, 5, 1]. Proposition (Kostant-Weinstein): Suppose that P is a real polarization of M and Q a Lagrangian submanifold of M which intersects each leaf transversally (T m Q + P m = T m M for every m Q), then there is a symplectomorphism ρ of some neighborhood of Q in M onto some neighborhood of the zero section of T Q under which Q maps to the zero section and P is mapped to the vertical polarization. Furthermore, if the leaves of P are simply connected and geodesically complete, ρ can be extended to all of M, making the identification Q (zero section) and 13 Mathematically the source of this problem lies with the fact that our prequantization construction furnished a reducible representation of the Heisenberg group. What we need is an irreducible representation. 14

15 P (vertical polarization) global. 4.2 Kähler Polarizations It is always not physically necessary, however, to restrict wavefunctions so that they are functions of only positions or momenta. Indeed, the so-called coherent states (app. C) for the harmonic oscillator are functions of both q and p that satisfy the uncertainty principle and span the entire quantum state space. More precisely, if we define z = q + ip And z = q ip such coherent states are of the form ( ψ(z, z) = φ(z) exp z z 4 ). In order to encompass such states when we pass from prequantization to quantization, we consider complex polarizations. That is, we select complex-lagrangian integrable distributions on the complexified bundle T C M. In this paper we will not be concerned with general complex polarizations on arbitrary symplectic manifolds, but a particular type: Kähler polarizations. Definition: A Kähler Polarization P T C M is a smooth, integrable, distribution such that P m is complex-lagrangian for each m M, P m P m = {0}, and P determines a positive integrable complex structure J on T M (compatible with the symplectic form). To understand the latter condition we note the discussion in section 2. A choice of complex- Lagrangian subspace P m (T m M) C uniquely determines a compatible complex structure J m : T m M T m M. The requirement of integrability is so that we can identify M with a complex submanifold M J of T C M and positivity ensures that the induced hermitian inner product on M J is positive-definite. Note that we can only find Kähler polarizations for a restricted class of symplectic manifolds (M, ω), i.e. Kähler manifolds. In particular, we must be able to realize M as a complex manifold. However, if we are given such a 2n real-dimensional Kähler manifold M and equipped with a 15

16 complex structure J there are two god-given 14 Kähler polarizations. Indeed, the map M M J T C M is equivalent to choosing the so-called holomorphic polarization P = span ( / z a ) where the local holomorphic coordinates z a (a = 1,, n) of M J are defined in appendix B. We could instead choose the anti-holomorphic polarization P = span ( / z a ). As indicated in section 2, if the phase space in question is a symplectic vector space (V, ω), we can immediately utilize such Kähler polarizations. 4.3 Real vs. Kähler Polarizations Despite their more direct physical interpretation, it turns out that real polarizations are less convenient to deal with than Kähler polarizations. The first problem arises when attempting to define a finite inner product of wavefunctions (polarized sections of B); this will lead to the introduction of half-forms, and we will defer that topic until section 6. The second problem with real polarizations is that the class of allowed observables, those that preserve the polarization, is so tightly constrained that it does not include the Hamiltonian of the harmonic oscillator, nor any Hamiltonian with a kinetic term (a term quadratic in momenta). This rather serious shortcoming can be addressed via a mechanism called pairing which allows one to relate observables and wavefunctions adapted to different polarizations, but we will not pursue this subject here. Instead, we will primarily work with Kähler polarizations, since in that case the Hamiltonian for the harmonic oscillator is immediately quantizeable. 5 Quantization 5.1 Holomorphic Quantization Assume that we have chosen a prequantum bundle B M where our phase space is a Kähler manifold M with complex structure J and let us choose the holomorphic Kähler polarization P of M. In order to eliminate the extra degrees of freedom that make the sections of B M unphysical 14 for certain values of god 16

17 we only consider sections that are covariantly constant along P, i.e. we require Xs = 0, X tangent to P ; (5.1) such sections are called polarized. The hope is that the space of polarized sections will define the appropriate quantum state space. We will work in the gauge such that the symplectic potential/connection 1-form θ is adapted to P (eq. B.4). If we select a local trivialization of some neighborhood U, then we have a choice of unit section s and we can write any local section s over U as s = φ(z a, z a ) s as θ is adapted to P then ι Xθ = 0 and the polarized sections must satisfy 0 = Xs = ( Xφ iι Xθφ ) s but X = α(z, z) / z a for some complex valued function α, so we must have φ z a =0. Thus, in the θ gauge, the polarized sections are of the form s = φ(z) s, i.e. they are holomorphic functions over U. Now all relevant observables f : M R are quantized to operators ˆf which preserve the polarized sections. Noting that X ( ) ˆfs = ˆf Xs i [ X,Xf] s we see that ˆf preserves polarized sections iff [ X, X f ] is tangent to P. Since X f is a real-vector field ( X f = X f ) this condition is equivalent to [X, X f ] being tangent to P, i.e. the flow of f must preserve P. This is a small class of observables. 17

18 So small, in fact, that it is easy to exhibit them all. If an observable f preserves P, ( ) [ L Xf z a = X f, ] z a = ( 2 ) ( z a (X f 2 ) f ) = 2i z a z b z b 2i f z a z b z b and since observables must be real we also have 2 f z a z b 2 f z a z b = 0 (5.2) = 0 so the general form of an observable is f = 1 2 i w az a 1 2 iwa z a U abz a z b + c (5.3) where w and c are complex constants, and U ab = Ūba. The operator ˆf constructed from such a general f is given in appendix B. In the case of the harmonic oscillator the Hamiltonian becomes the operator ĥ = z a z a (5.4) acting on functions of z. Taking the one-dimensional case, the eigenfunctions are homogeneous polynomials in z of degree N, and the corresponding eigenvalues of ĥ are N. This is close to the right answer, but it is missing the zero point energy; the eigenvalues should be (N + 1 ). To fix 2 this we need to redefine our wavefunctions to include half-forms, which we will do in section Real Quantization For real polarizations on a symplectic vector space if we impose the condition that the flow of X f corresponding to observables f : M R preserve the polarization, ( ) [ L Xf = X f, p a p a ] = ( 2 ) ( f 2 ) (X f ) = p a p a p b q b f p a q b p b 2 f p a p b = 0 (5.5) so that the only allowed observables are of the form f = v a (q)p a + u(q). (5.6) In particular we are allowed observables at most linear in p, hence we cannot even utilize standard Hamiltonian kinetic terms that arise in physics. 18

19 6 Half forms and the Metaplectic Correction 6.1 Motivation for half-forms: The real quantization We will sketch the case of quantization by real-polarizations to motivate the discussion of half-forms; however, we will not go into the details of the construction as it is not our main point of focus. Assume we have a prequantum bundle B M with the space of sections Γ(B) and we choose a real polarization P, to quantize this space we proceed as in the complex case: by considering those sections which are covariantly constant along P X s = 0; under completion this space creates the quantum space Γ P (B) Γ(B). Assume the space of leaves for our polarization Q = M/P forms a Hausdorff manifold and we have chosen a polarization which looks globally like the vertical polarization. 15 For simplicity assume the polarization P and M satisfy the properties of the Kostant-Weinstein proposition. If we choose a θ-gauge adapted to the polarization P, and P is identified with the vertical polarization, then essentially we are picking out sections which are given locally by functions of the q-directions (i.e. parameters on the space of leaves Q) and are constant along the leaves (the p-directions). In other words, s Γ P (B) can be written locally on any neighborhood U M in the θ gauge as s = ψ(q) s, where s is the unit section. This is analogous to standard quantum mechanics where our wavefunctions are only functions of position coordinates q. However, because each leaf of the vertical polarization is a non-compact affine space, and our sections are constant along each leaf, then any such non-vanishing section is certainly not square integrable with respect to the L 2 norm defined on the prequantum Hilbert space. Thus, we must attempt to introduce a new norm on the space Γ P (B). More precisely we can modify the bundle B slightly so that there is a naturally induced norm on covariantly constant sections Γ P ( ˆB) of the twisted ˆB. Indeed, as our sections are locally functions on Q, we would 15 We could choose real polarizations for which the leaves form compact spaces, but in that situation P would look only locally like the vertical polarization. We will not consider this situation for our discussion 19

20 like them to satisfy an L 2 norm on Q of the form s, s = Q (s, s )ɛ Q = Q ψψɛ Q where ɛ Q is some n-form on Q. However, there is no natural choice of measure for Q as there was for M (where the symplectic form introduced a natural choice). Instead of making an arbitary choice, we allow (s, s ) to define an n-form on Q (or better yet, a density) in a natural way so that it can be integrated. We can do this and still integrate expressions of the form ψψ if we try twisting Γ P (B) by tensoring with some line bundle, creating new sections ŝ = s ν = sν, ŝ = s ν = s ν where s is in the original Γ P (B) and ν is some section of a line bundle. If we let (ŝ, ŝ ) = (s, s ) νν (6.1) then the inner product s, s = Q (ŝ, ŝ ) (6.2) makes sense if νν is a density form on Q (equivalently a volume form if Q is orientable). In other words, ν is the square root of a volume form on Q. Hence, we need to construct some half-form bundle. 6.2 The Half-form bundle The construction of half-forms here follows that of [5]. To begin constructing such forms, we must be a bit more precise about what we need. Firstly, note that as s Γ P (B) is a section of a complex line bundle, we can safely generalize to νν a complex n-form, i.e. a section of Λ n TC Q. To begin constructing such a half-form bundle we will first start with the case Q a vector space W. Let B(W ) be the bundle of frames over W, a right GL(n, R) torsor. Then a complex n-form α over W 20

21 is defined by a function α : B(W ) C such that for any section b of B(W ) and A GL(n, R) α(ba) = det(a)α(b) and a complex density ρ over W is a function B(W ) C such that ρ(ba) = det(a) ρ(b). We would like to define ν such that ν 2 is a complex n-form and νν is a complex density. To do this, we need ν(ba) = ( det A ) ν(b) (6.3) Of course, this is ill-defined as det(a) has has two possible branches. The solution to this problem is to consider ν not as a function on B(W ), but as a function on some space MB(W ) that is a right G-torsor with G a double cover of GL(n, R): the metalinear group ML(n, R). Just as the function z 1/2 lifts to a single-valued holomorphic function on a double cover of C, the square-root of the determinant function lifts to a single-valued 16 function χ : ML(n, R) C such that under the projection π : ML(n, R) GL(n, R) we have for B ML(n, R) χ(b)χ(b) = det(πb) χ(b)χ(b) = det(πb). In order to pass back to B(W ) and GL(n, R), we need a double cover r : MB(W ) B(W ) that commutes with the double cover π : ML(n, R) GL(n, R) in the following manner: MB(W ) ML(n, R) MB(W ) r π r B(W ) GL(n, R) 16 Holomorphic on the complexified space ML(n, R) B(W ). 21

22 With this construction, then for A ML(n, R) and b MB(W ), we use ν(bb) = χ(b)ν(b) (6.4) as our definition of half-forms. The space of all such forms forms a vector space Λ 1/2 W over C and each element ν satisfies the correct requirements: ν 2 is an n-complex volume form on W and νν is a complex density. The construction above passes immediately to a general manifold Q by bundlizing. We work with the principal GL(n, R) bundle BQ, a principal ML(n, R) bundle MBQ, the commutative diagram passes to a commutative bundle diagram and the construction above holds on each tangent space T C Q. The only difficulties that arise are topological obstructions to the definition of MBQ [5]. Assuming there are no such obstructions, however, we end up with the bundle of half-forms Λ 1/2 TC Q whose sections transform under metalinear transformations via (6.4). 6.3 The Metaplectic Group The half-form construction on some Q = M/P solves the most difficult part of the construction of the appropriate ν. However, a subtle issue still remains. The sections s are on a bundle over M, while the sections of Λ 1/2 TC Q are over Q. However, in order to define a tensor product of B with another bundle and talk about sections s ν, we need both bundles to be over the same base space. Furthermore, our construction was explicitly dependent on the polarization P (via the definition of Q); we would like to have a better picture of half-forms that could possibly allow for comparison between polarizations. The basic idea is straightforward: we would like to somehow embed the M L(n, R) bundle MBQ Q into a metaplectic bundle of frames MB(M) over M. The half-forms, defined in terms of their action on MBQ can then be extended to general frames over M by taking them to be constant on metaplectic transformations which do not modify the embedding of M BQ. The embedding should cover the natural embedding of the GL(n, R) bundle BQ Q into the Sp(2n, R) bundle of frames SpB(M). So it is not hard to believe the so-called metaplectic group Mp(2n, R) is taken to be the double cover of the symplectic group. Thus half-forms are intimately 22

23 related to the metaplectic group rather than the symplectic group; this leads to the interesting insight that perhaps it is the metaplectic group (the double cover of the symplectic group) that acts naturally in quantum mechanics rather than the symplectic group itself. Thus, the half-form bundle Λ 1/2 TC Q Q can alternatively be viewed as a bundle over M which we will refer to as δ P (referring to the polarization P in the construction of Q). 6.4 The Real Quantization Via the above construction, the bundle long sought is given as ˆB = B δ P M and, the innerproduct given by (6.1) and (6.2) is now appropriately defined on sections of ˆB. One thing remains before complete quantization: extending the covariant derivative to ˆB. In fact, the covariant derivative can be partially extended to complex n-forms µ over M, constant along P (ι X µ = 0 for all X tangent to the leaves of P ) [1] via X µ =ι X dµ, X tangent to the leaves of P. By using this and the fact that ν δ P defines such a complex n-form ν 2, then we can define a covariant derivative on δ P via the Leibnitz rule X ν 2 = 2( X ν)ν. Similarly covariant differentiation of ŝ = sν is also given by application of the Leibnitz rule. Quantized sections are then square-integrable sections of ˆB which are covariantly constant along P. 6.5 Half forms in the Kähler Quantization The requirement of half-forms in the real quantization lead to natural actions of the two-sheeted covering groups ML(n, R) and Mp(2n, R) on our quantum sections. It is only natural to expect that these groups, and the half-forms they act on, naturally arise in any polarization. Indeed, the metaplectic group rears its head immediately in the case of Kähler polarizations. We will sketch 23

24 how this occurs, taking mostly from the discussion in [1]. Consider the Kähler quantization of a symplectic vector space V of dimension 2n. The quantization procedure above introduced a space of square integrable sections H J for each Kähler polarization corresponding to the positive complex structure J. Now we can embed each H J in the space of all sections of B M. Recall that we can quantize a classical observable f : M R as an operator ˆf on the space of all sections, but only a restricted class of f lift to an ˆf whose flow preserves our H J. We will put this observation in a different light: the flow of f : V R is a symplectomorphism on V and the lift of our flow (the flow of ˆf thought of as a vector field on the sections) furnishes a representation of such symplectomorphisms on the space of sections. However, because the flow of ˆf does not in general preserve H J, this subspace does not generally furnish a representation space for symplectomorphisms with respect to our quantization procedure. However, we can attempt to rectify the situation by flowing a point in H J along the vector field in the space of sections defined by ˆf, and continually projecting back onto H J over every infinitesimal time-step. It turns out this furnishes a projective representation of symplectomorphisms on V. In fact, via the relationship between projective representations and representations of central extensions (and covering groups), this representation is related to the metaplectic group (more precisely, it is a representation of metaplectomorphisms ). Half-forms also pop into the picture. The assignment of a Hilbert space H J to every positive complex structure J L + V leads to a vector-bundle E L + V whose fiber over J is H J. If each H J is thought of as a subspace of the total space of sections of B M, there is an orthogonal projection operator between two fibers H J and H J, to first order this creates a connection on our bundle E. One can show this connection is projectively flat, i.e. the holonomy on our space is multiplication by a complex phase factor. Hence, all fibers can be identified via our holonomy into a single Hilbert space E. By quantizing linear symplectic transformations, we have a representation of the symplectic group Sp(2n, R) on the entire space of sections E; however because our holonomy is multiplication by a phase factor, this descends to a projective representation of the symplectic group on V. Just as hinted above, this is a representation of the metaplectic group Mp(2n, R). If we want a representation of the symplectic group, however, we must make the holonomy of 24

25 our line bundle E F completely trivial. This can be done by twisting the bundle (via tensor producting each fiber with some other space) to give the connection vanishing curvature. This is where the half-forms arise: by changing each fiber to H J δ PJ (i.e. tensor H J with the halfforms adapted to the polarization P J ) we obtain a bundle E with vanishing curvature (and trivial holonomy). The identification space E is then a representation space for Sp(2n, R). It should also be noted that, besides just giving an abstractly beautiful result, the half-form modification H J H J δ PJ gives the correct ground-state energy of the harmonic oscillator 1 2, a result that is not achieved with the representation of the harmonic oscillator Hamiltonian on the space of sections H J (see app. B). 7 Conclusion/Outlook We have described enough of the geometric quantization construction to satisfactorily quantize a very simple system, the n-dimensional harmonic oscillator, and only using a Kähler polarization. Certainly the formalism developed here is applicable to other finite-dimensional systems and other polarizations, especially if one employs the pairing mechanism alluded to earlier. However one clear limitation of the program to date is that the final step in the program, the half-form construction or metaplectic correction, cannot yet be carried out for systems with an infinite number of degrees of freedom [3]. This is perhaps the most interesting class of systems as it includes field theories, some of which (such as quantum electrodynamics) are very well understood, and some of which (such as quantum general relativity) are not very well understood at all. Of course geometric quantization is not necessarily the only or even always the best way to quantize a classical theory. Algebraic procedures such as deformation quantization and Weyl quantization also serve to elucidate the relationship between quantum and classical mechanics. This may seem then like a lot of sophisticated mathematical machinery to develop only to accomplish a relatively simple task, but in doing so we have set out on a path which we hope leads to a deeper understanding of quantum mechanics and its geometric underpinnings. Canonical quantization (which is contained in geometric quantization), often applied in physical applications, is quite effective at producing a meaningful and practically applicable quantum theory. However, it 25

26 is based on particular position and momentum coordinates (i.e. a single polarization) without giving insight into how quantizations based on other polarizations are related; it is not clear from that perspective how quantizing could be a symplectically invariant procedure. In contrast, the tools developed by geometric quantization such as the pairing mechanism give a very clear picture of how these different quantization procedures based on different polarizations are related. Furthermore, both the pairing mechanism, which involves projective representations, and the natural emergence of the metaplectic group appear to be geometric manifestations of a deep connection between quantum mechanics and central extensions of the symplectic group. This connection remains to be fully explored. 26

27 A Holonomies of a Complex Line Bundle We first begin by choosing a line bundle π : B M with connection and choose a local trivialization (U, τ) for some neighborhood U M and τ : M C π 1 (U). The trivialization yields a unit section s = τ(, 1) from which we can define the potential one-form Θ Ω 1 (U) via s = iθ s. Now any section s over U can be written as s = ψ s for some complex-valued function ψ ω 0 (U); for any such general section then, X s = (X(ψ) iι X Θ) s = (dψ(x) iι X Θ) s =ι X (d iθ) ψ s To construct the holonomy operator, we first consider a closed curve γ : [0, 1] U parameterized by t [0, 1], s = ψ s a section over the image of γ, and γ the tangent vector field defined over the image of γ. Then s is parallel to γ if 0 = γ s ( ) dψ = dt iι γθψ s. This is an ODE in ψ with (unique) solution ( ) ψ(t) = exp it Θ ψ(0); γ 27

28 thus, the holonomy operator around a closed curve γ is given by multiplication by the complex number ( ) ξ = exp i Θ. γ B Operators Here we construct all self-adjoint operators that preserve a Kähler polarization. Using the metric to raise or lower indices, e.g. p a = g ab p b, introduce the complex coordinates z a =p a + iq a z a =p a iq a p a = 1(z + z) 2 qa = 1 i(z z) 2 then dz a = dp a + idq a d z a = dp a idq a z a = 1 ( 2 p a i ) q a z a = 1 ( 2 p a + i ) q a and the Hamiltonian vector field of a function f (eq. 1.7) is ( f X f = 2i z a z a f z a ) z a. (B.1) We have a Kähler polarization P whose leaves are surfaces of constant z a so that vectors tangent to leaves lie in the span of z a. For the flow of a Hamiltonian vector field X f to preserve the polarization it must Lie drag vectors tangent to P into vectors tangent to P (eq. 5.2) ( ) [ L Xf z a = X f, ] z a = ( 2 ) ( z a (X f 2 ) f ) = 2i z a z b z b 2i f z a z b z b 2 f z a z b = 0 (B.2) which, together with the requirement that f is an observable and hence real, restricts f to be of the form (eq. 5.3) f = 1 2 i w az a 1 2 iwa z a U abz a z b + c (B.3) where w and c are complex constants, and U ab = Ūba. 28

29 The explicit forms of wavefunctions and operators depend on the choice of gauge, a particular symplectic potential θ (app. C). In the gauge adapted to P, θ = p a dq a = 1 2 i z adz a, (B.4) the wave functions ψ = φν are the product of φ(z) a holomorphic function on M (a section of the prequantum bundle B) and a half-form ν (a section of the half-form bundle). Note that if we choose a standard volume form µ of P, any other volume form is a complex function of z times µ. Similarly any half-form is a complex function of z times µ, so in our wavefunctions we may absorb the extra function into φ and always choose ν = µ without loss of generality. Explicitly, ψ = φ (z)ν = φ (z)g(z) µ = φ(z) µ (B.5) Denote by f an operator acting on ψ and by ˆf an operator acting on φ(z) only. A function f determines an operator f via ( f = i L Xf i ) ι X f θ + f (B.6) and fψ = ( ˆfφ) µ i φl Xf µ (B.7) where ˆf is defined in eq First we compute the operator ˆf corresponding to an f whose X f preserves P (eq. B.3), ˆf = i X f + f ι Xf θ ( = i 2i f ) ( z a z a + f ( 1i z 2 a) 2i f ) z a = 2 f ( ) z a z a + f f z a z a = i w a z a + U ab z a z b + 1i w 2 az a + c. The choice w = c = 0 and U ab = δ ab yields the Hamiltonian h of the n-dimensional harmonic 29

30 oscillator, so ĥ is as given in eq. 5.4, ĥ = z a z a. Next we compute the extra Lie derivative term in f. (B.8) We take as our standard volume form µ = (4π ) n/2 µ 0 where µ 0 = d n z = dz 1 dz n, and our standard half-form ν = µ, then compute L Xf µ 0 = d ( ι Xf d n z ) ( f = 2i d ( = i div U a b L Xf µ = i (tr U) µ zb) µ 0 = i = L Xf ν 2 = 2νL Xf ν L Xf ν = 1 i (tr U) ν 2 i φl Xf µ = 1 2 (tr U) µ ) dn z z a ( = w a d ι ι z a ( ) z a U b a zb µ 0 = i ( z a dn z Ub a δb a ) ( + i d Ub a zb ι dn z z a ) µ 0 = i (tr U) µ 0 ) so that the operator f is fψ = (( ˆf tr U ) φ) µ (B.9) or, with the understanding that derivatives act only on φ(z), f = i w a z a + U ab z a z b + 1i w 2 az a + c + 1 tr U. (B.10) 2 The choice w = c = 0 and U ab = δ ab (the Hamiltonian h of the n-dimensional harmonic oscillator), using tr δ = n, now yields the operator h ( h = z a ) z a + 1n 2 (B.11) which has the correct eigenvalue spectrum. For example the one-dimensional harmonic oscillator (n = 1) has eigenvalues (N + 1 ) corresponding to eigenfunctions φ(z) that are homogeneous 2 polynomials in z of degree N. 30