LECTURE 13: GEOMETRIC QUANTIZATION Contents 1. Polarizations 1 2. Geometric quantization 3 3. Quantizing K¨ahler manifolds 6 1. Polarizations ¶ Polarizations. As we have explained, in a classical Hamiltonian mechanical system with configuration space a Riemannian manifold X, the classical phase space is taken to be the cotangent bundle T ∗X; while in the Schr¨odinger’s formulation of the corresponding quantum mechanical system the quantum phase space is taken to be the Hilbert space L 2 (X). The motivation for geometric quantization is trying to extend the correspondence M = T ∗X L 2 (X) to more general symplectic manifolds. Last time we have introduced the geometric prequantization, M L 2 (M, L), where L 2 (M, L) is a twisted version of L 2 (M), which is obviously too large. The idea of polarization is trying to “cut the variables in M to a half” canonically, so that one can get an analogy of L 2 (X) instead of an analogy of L 2 (T ∗X). Suppose (V, Ω) be a symplectic vector space of dimension 2n. One can extend the symplectic form Ω complex-linearly to the complexified vector space V ⊗C. We will call any complex Lagrangian subspace P of V ⊗ C a polarization. Now let (M, ω) be a symplectic manifold. Again we can complexify the tangent bundle of M to TMC = TM ⊗ C and extend the symplectic form ω to TMC complex-linearly. Definition 1.1. A polarization of (M, ω) is a complex subbundle P of TMC satisfying • P is involutive: [P, P] ⊂ P. • Each Pm is Lagrangian: ω(P, P) = 0, dimC Pm = n. Remark. (1) If P is a polarization, so is P. 1
2 LECTURE 13: GEOMETRIC QUANTIZATION (2) By the theorem of Frobenius, the sub-bundle P is integrable. In other words, through each point m there is an integrable submanifold N of M whose (complexified) tangent space at m is Pm. These integrable submanifolds are called leaves of P. ¶ Real polarization. Definition 1.2. A polarization is called real if P = P. Example. Any cotangent bundle (T ∗X, ωcan) admits a natural polarization. In fact, for each m = (x, ξ) ∈ T X we set Pm = Tm(TxX) ⊗ C, i.e. Pm is the complexified tangent space of the fiber TxX at m. In local coordinates P is spanned (over C) by ∂ ∂ξj ’s. As an immediate consequence, P is involutive and Lagrangian. Moreover, by definition P = P. So P is a real polarization. We will call P the vertical polarization of (T ∗X, ωcan), since the integrable manifold of P are just the vertical fibers. As a consequence, the space of all integral manifolds of P can be identified with X. Note that a vector field v on T ∗X sits in P if and only if Lv(pr∗ f) = 0 for any f ∈ C ∞(X). Remark. If P is a real polarization, then P ∩ TM is an involutive (real) subbundle of TM and each Pm ∩ TmM is a Lagrangian subspace of TmM. So one can define a real polarization without complexifying the TM. Example. Consider the punctured plane R 2 \ {(0, 0}, equipped with the symplectic form dx ∧ dy. Then the collection of circles Cr = {(x, y) | x 2 + y 2 = r 2 } forms a Lagrangian foliation of R 2 \ {(0, 0)}. The tangent lined to these circles form a real polarization. Remark. Real polarizations, or more generally, polarizations, need not exist. For example, one can show that any real line bundle over S 2 must be trivial, and thus must has a nowhere zero section. As a result, S 2 has no real polarization, since any vector field on S 2 must has a zero. M. Gotay constructed examples that admits no polarizations even in the complex sense. ¶ K¨ahler polarization. Definition 1.3. A polarization P is called K¨ahler if P ∩ P = 0. Example. Let (M, ω, J) be a K¨ahler manifold. Recall that this means • (M, ω) is symplectic
LECTURE 13: GEOMETRIC QUANTIZATION 3 • (M, J) is complex, i.e. J is an almost complex structure so that NJ (u, v) := [Ju, Jv] − J[Ju, v] − J[u, Jv] − [u, v] = 0. • ω(JX, JY ) = ω(X, Y ), • g(X, Y ) := ω(JX, Y ) is a Riemannian metric on M. We let P = T0,1 = {v ∈ TM ⊗ C | Jv = −iv}. Then (1) P is involutive: Suppose u, v ∈ P, then 0 = NJ (u, Jv) = −[Ju, v] − J[Ju, Jv] + J[u, v] − [u, Jv] = i[u, v] + J[u, v] + J[u, v] + i[u, v] =⇒J[u, v] = −i[u, v]. (2) P is Lagrangian: Suppose u, v ∈ P, then ω(u, v) = ω(Ju, Jv) = −ω(u, v) =⇒ ω(u, v) = 0. (3) P ∩ P = 0: By definition P = T1,0 = {v ∈ TM ⊗ C | Jv = iv}. So P is a K¨ahler polarization of (M, ω). Similarly P is a K¨ahler polarization. We will call P the holomorphic polarization and call P the anti-holomorphic polarization, since they are generated by ∂ ∂z¯j ’s and ∂ ∂zj ’s respectively. So any K¨ahler manifold admit two natural K¨ahler polarizations. Conversely, if a symplectic manifold (M, ω) carries a K¨ahler polarization P, then there is a complex structure J on M which is compatible with the ω and such that P is its holomorphic polarization: Of course the only way to define J is so that J = −i on P and J = i on P. The integrability of this J follows from the integrability assumption on P. 2. Geometric quantization ¶ Polarized sections. Now let (M, ω) be pre-quantizable symplectic manifold and (L, h, ∇) be a prequantum line bundle over M. Using polarization P one can reduce the pre-quantum space L 2 (M, L) to a much smaller one: Definition 2.1. A section s ∈ Γ ∞(M, L) is polarized with respect to a polarization P if ∇Xs = 0 for all sections X in P. Roughly speaking, a section s is polarized if it is constant along integral manifolds of P, i.e. only depends on “the other half variables”. We will denote the space of all polarized sections with respect to P by ΓP (M, L)
4 LECTURE 13: GEOMETRIC QUANTIZATION Example. Consider the vertical polarization P of (T ∗X, ωcan) described above. If we take L = T ∗X × C to be the trivial line bundle and take ∇ to be the usual exterior differential, then by fixing a global trivializing section one can identify any section of this line bundle with functions on T ∗X, and polarized sections becomes functions independent of ξ’s, i.e. pull-back through π of functions on X: Γ ∞ P (T ∗X, T∗X × C) = π ∗C ∞(M). Example. For the K¨ahler polarization P of a compact K¨ahler manifold (M, ω), if one take L to be a holomorphic line bundle and choose ∇ to be the Chern connection (which we will explain later in this lecture), then polarized sections are exactly holomorphic sections of L. ¶Reducing classical observables. Unfortunately, after cutting L 2 (M, L) to L 2 P (M, L), new problems appears. Recall that the prequantization procedure sends any classical observable, i.e. any real valued smooth function a, to the self adjoint operator Qa defined by Q(a) = −i~∇Ξa + ma. Now suppose s ∈ L 2 P (M, L) is a polarized section. We would like Q(a)s to be polarized also, but this does not always happen. Proposition 2.2. For any X, a, s one has ∇X(Q(a)s) = Q(a)∇Xs − i~∇[X,Ξa]s. Proof. We calculate ∇X(Q(a)s) − Q(a)∇Xs = ∇X(−i~∇Ξa s + as) − (−i~∇Ξa∇Xs + a∇Xs) = (−i~)[∇X, ∇Ξa ]s + (∇Xa)s. As in last time, we use the formula Ω(X, Y ) = [∇X, ∇Y ] − ∇[X,Y ] and the assumption Ω = 1 ~ ω to get (−i~)[∇X, ∇Ξa ]s = ω(X, Ξa)s + (−i~)∇[X,Ξa]s and the conclusion follows. As a result, if we want Q(a)s is also polarized if s is polarized, one can only consider those classical observables a such that for any section X in P, [X, Ξa] also sits in P. Definition 2.3. Given any polarization P, the space of polarization preserving functions is the subspace C ∞ P (M) defined by C ∞ P (M) := {a ∈ C ∞(M) | [Xa, X] ∈ Γ(P) for all X ∈ Γ(P)}. Exercise 1. If a is a polarization preserving function, then Ξa preserves the leaves of the foliation defined by the distribution P.
LECTURE 13: GEOMETRIC QUANTIZATION 5 One must show Proposition 2.4. The subspace C ∞ P (M) of C ∞(M) is closed under the Poisson bracket. Proof. Suppose a, b ∈ C ∞ P (M) and X ∈ Γ(P). Then [X, Ξ{a,b}] = [X, [Ξa, Ξb]] = [[X, Ξa], Ξb] + [[Ξb, X], Ξa] ∈ Γ(P). So we reduce the Hilbert space to a much smaller space which consists of polarized sections, and reduce the space of observables to the space of polarization preserving functions. ¶ Still more problems and more subtle modifications. There are still more problems. For example, it is possible that the space of polarized smooth sections is empty. The solution to this problem is to consider distributional polarized sections. c.f. J. Sniatycki, Geometric quantization and quantum mechanics. Another problem arisen in defining an inner product in the space of polarized sections. In the case where the integral manifolds of P are compact, one can use the induced inner product from M, i.e. integrate with respect to the Liouville measure ω m. However, if the integral manifolds of P are noncompact, like the case of vertical bundles, this does not work. In fact as we have seen, in the case of trivial line bundle and trivial connection, polarized sections are just the pull back of functions on M. They are no longer square integrable over M with respect to ω m, due the non-compactness of ξ direction. What people really used in this example is pushforwarding the functions hs1, s2i as a function on M to a function on X, the space of integral manifolds of P, and then use a measure on X to integrate. In general one can use the same idea, i.e. push-forwarding the functions hs1, s2i which are constant along the direction of integral manifolds of P, to functions on the manifold M/P of integral manifolds (we need to assume that the space of integral manifolds is a manifold), and integrate via a measure on M/P. The problem is that there is no God-giving measure on M/P. One way to solve this problem is to introduce half densities on M/P. So instead of consider polarized sections, which are sections of L over M but can be identified with sections over some Hermitian line bundle L/P over M/P, one can consider sections of the line bundle L/P ⊗ |T(M/P)| 1/2 over M/P, where |T(M/P)| 1/2 is the half density bundle over M/P. The sections of the later space can be paired intrinsically, which gives us a Hilbert space structure. Problems still exist if one compare quantization constructed as above with real examples from physics. And people invented so called half-form correction to eliminate this problem. I will not discuss details here.
6 LECTURE 13: GEOMETRIC QUANTIZATION 3. Quantizing Kahler manifolds ¨ ¶ Holomorphic line bundles over compact K¨ahler manifolds. Now suppose (L, h) is a Hermitian line bundle over a compact K¨ahler manifold (M, ω, J). We will assume moreover that L is a holomorphic line bundle, in other words, (1) L is itself a complex manifold, (2) The projection π : L → M is holomorphic map, (3) The transition maps gαβ : Uα ∩ Uβ → C ∗ are holomorphic. From the decomposition Ωk (M, C) = ⊕p+q=kΩ p,q(M) one gets a natural decomposition Ω k (M, L) = M p+q=k Ω p,q(M, L), where Ω p,q(M, L) = Γ∞(M,((∧ pT 1,0 ) ∧ (∧ qT 0,1 )) ⊗ L). Since M is a complex manifold, the exterior differential d decompose as d = ∂ + ¯∂, where ¯∂ : Ωp,q(M) → Ω p,q+1(M) is called the Dolbeault operator. We can extend ¯∂ to a linear map ¯∂ : Ωp,q(M, L) → Ω p,q+1(M, L), locally by the formula ¯∂α ⊗ e = ( ¯∂α) ⊗ e, where e is a local frame. Lemma 3.1. The Dolbeault operator ¯∂ : Ωp,q(M, L) → Ω p,q+1(M, L) defined above is well-defined. Proof. Suppose e 0 is another local holomorphic frame. Then there is a holomorphic function g so that e = ge0 . Now suppose α ⊗ e = β ⊗ e 0 , where α, β ∈ Ω p,q(M). Then gα = β. It follows ( ¯∂β) ⊗ e 0 = ( ¯∂αg) ⊗ e 0 = ( ¯∂α)g ⊗ e 0 = ( ¯∂α) ⊗ e. Definition 3.2. A section s ∈ Γ ∞(M, L) is holomorphic if ¯∂s = 0. The space of all holomorphic sections of a holomorphic line bundle L is denoted by Γhol(M, L). It is well-known that this is always a finite dimensional vector space, whose dimension is calculated by Riemann-Rock. ¶ Chern connection. Now suppose ∇ : Γ∞(M, L) → Ω 1 (M, L) = Ω1,0 (M, L) ⊕ Ω 0,1 (M, L) be a connection on L. We will denote by ∇0.1 : Γ∞(M, L) → Ω 0,1 (M, L)
LECTURE 13: GEOMETRIC QUANTIZATION 7 the (0, 1)-component of ∇. The following theorem is the analogue of the Levi-Civita connection in this complex geometry setting: Theorem 3.3. There is a unique connection ∇ on L which is (1) compatible with the hermitian structure: dhs1, s2i = h∇s1, s2i + hs1, ∇s2i. (2) compatible with the holomorphic structure: ∇0,1 = ¯∂. This connection is called the Chern connection. Proof. Exercise. (Hint: By choosing a local holomorphic frame e, all connections compatible with the holomorphic structure have the form ∇ = d + A, where A is a locally defined (1, 0)-form. Using the metric compatibility condition one can show A = d log h = ∂ log h, where h(x) = he, ei.) The curvature is defined just as last time. Exercise 2. Let e be a local holomorphic frame and let h(x, x) = he, ei. Then the curvature 2-form is Ω = ¯∂∂h. As a consequence, Ω is a purely imaginary (1, 1)-form. In particular, the first Chern class c1(L) = [ 1 2πi Ω] ∈ H 1,1 (M, R) ∩ H 2 (M, Z). There is a holomorphic version of Weil’s theorem, due to Lefschetz, saying that the “first Chern class map” c1, c1 : {holomorphic line bundle L over compact Kahler M} → H 1,1 (M, R)∩H 2 (M, Z), is surjective. ¶ Geometric quantization of K¨ahler manifolds. Now suppose (M, ω, J) is a compact K¨ahler manifold and [ ω 2π ] ∈ H 1,1 (M, R) ∩ H 2 (M, Z). Then one can find a holomorphic Hermitian line bundle (L, h, ∇) over M with Chern connection ∇, whose curvature Ω = ω i~ . This is our pre-quantization data. Remark. Usually one will take ~ = 1 N , where N is a large integer. This is the same as replacing (L, ~) by (L ⊗N , h⊗N ). We can further impose the K¨ahler polarization P on TMC . P is locally spanned by vector fields ∂ ∂z¯j ’s. So a section s ∈ Γ ∞(M, L) is polarized by P if and only if ¯∂s = 0, i.e. if and only if s ∈ Γhol(M, L) is a holomorphic section of L. (One may replace L by L ⊗n of course.)
8 LECTURE 13: GEOMETRIC QUANTIZATION So in this setting, the quantum phase space is H = Γhol(M, L), and the inner product is the L 2 inner product with respect to ω n n! . Example. Consider S 2 = CP1 . This is not quite satisfied since the space Γhol(M, L) is a finite dimensional space. Seems not large enough. ¶ Berezin-Toeplitz quantization. Again let (M, ω) be a pre-quantizable compact K¨ahler manifold, and let (L, ~, ∇) be a holomorphic Hermitian line bundle over M with Chern connection, whose curvature Ω = ω. For each positive integer N we let πN : L 2 (M, L ⊗N ) → Γhol(M, L ⊗N ) be the orthogonal projection. In Berezin-Teoplitz quantization, the quantum space is defined to be H = M N Γhol(M, L ⊗N ). The inner product is defined in the usual way. For any smooth function f ∈ C ∞(M), we associate to it the operator Tf = M N T (N) f , where T (N) f is defined as T (N) f = πNmfπN : Γhol(M, L ⊗N ) → Γhol(M, L ⊗N ), where mf is again the map “multiplication by f”. Remark. Unlike the operator Q(a), the operators T (N) f ’s are no longer Lie algebra homomorphisms. Theorem 3.4. One has (1) kfk∞ + C N ≤ kT (N) f k ≤ kfk∞. (2) kT (N) f T (N) g − T (N) f g k = O( 1 N ). (3) N i[T (N) f , T(N) g ] − T (N) {f,g} k = O( 1 N ). The proof depends on the symbolic calculus of generalized Toeplitz operators
LECTURE 13: GEOMETRIC QUANTIZATION 9 ¶ Generalized Toeplitz operators. In general, let Ω be a strictly pseudoconvex domain, and X = ∂Ω be its boundary. Let H be H = {f ∈ C∞(X) | f can be extended to a holomorphic function in Ω}, called the Hardy space, and let π : L 2 (X) → H be the orthogonal projection, called the Szeg¨o projector. A Toeplitz operator T : C ∞(X) → C ∞(X) of order k is an operator of the form T = πP π, where P is a classical pseudodifferential operator of order k. As in the case of PsDOs, Teoplitz operators form a graded ring under composition and there is a nice symbolic calculus. To relate this to our setting, one consider the dual bundle (L ∗ , h∗ ) of (L, h), and the dual disk bundle D ∗ = {(z, v) ∈ L ∗ | |v| 2 h∗ ≤ 1}. It was proven by Grauert that D∗ is a pseudoconvex domain in L ∗ . Let X = ∂D∗ be the circle bundle. Then the Hardy space H is preserved by the natural S 1 -action, and thus decomposes as H = M N H (N) , where H (N) = {f ∈ H | f(e iθz) = e iNθf(z). Fact: One can identify holomorphic sections Γhol(M, L N ) with H(N) . And Tf defined above can be identified with πmfπ