LECTURE 13:GEOMETRICQUANTIZATIONCONTENTS11.Polarizations32.Geometric quantization63.QuantizingKahlermanifolds1.POLARIZATIONS Polarizations.As wehave explained,in a classical Hamiltonian mechanical system with configuration space a Riemannian manifold X, the classical phase space is taken to be thecotangent bundle T*X; while in the Schrodinger's formulation of the correspondingquantum mechanical system the quantum phase space is taken to be the Hilbertspace L?(X).The motivation for geometric quantization is trying to extend thecorrespondenceM = T*X L?(X)to more general symplectic manifolds. Last time we have introduced the geometricprequantization,M L?(M,L),where L?(M,L) is a twisted version of L?(M), which is obviously too large. Theidea of polarization is trying to"cut the variables in M to a half' canonically, sothat one can get an analogy of L?(X) instead of an analogy of L(T*X).Suppose (V,2)bea symplectic vector space of dimension 2n.One can extendthe symplectic form complex-linearly to the complexified vector space V C. Wewill call any complex Lagrangian subspace P of V C a polarization.Now let (M,w)be a symplectic manifold. Again we can complexify the tangentbundle of M to TMC = TM C and extend the symplectic form w to TMCcomplex-linearly.Definition 1.1. A polarization of (M,w) is a complex subbundle P of TMC satis-fying.Pis involutive:[P,P] CP. Each Pm is Lagrangian: w(P, P) = O, dimc Pm = n(1) If P is a polarization, so is P.Remark.1
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
2LECTURE13:GEOMETRICQUANTIZATION(2)Bythetheorem of Frobenius,the sub-bundlePis integrable.In otherwords,through each point m there is an integrable submanifold N of M whose(complexified)tangent spaceat m is Pm.These integrable submanifolds arecalled leaves of P.T Real polarization.Definition 1.2.A polarization is called real if P=PErample. Any cotangent bundle (T*X, wcan) admits a natural polarization. In fact,for each m = (r, ) e TX we setPm = Tm(T,X) C,i.e.Pm is the complexified tangent space of the fiber TX at m.In local coordinatesP is spanned (over C) by ,'s. As an immediate consequence, P is involutive andLagrangian. Moreover, by definition P=P. So P is a real polarization. We willcall P the vertical polarization of (T*X, wcan), since the integrable manifold of P arejust the vertical fibers. As a consequence, the space of all integral manifolds of Pcan be identified with X.Note that a vector field on T*X sits in P if and only ifL,(pr*f)= 0for any f e Co(X).Remark.If P is a real polarization,then PnTM is an involutive (real) subbundleof TM and each PmnTmM is a Lagrangian subspace of TmM.So one can define areal polarization without complexifying theTM.Erample.ConsiderthepuncturedplaneR?(o,0),equippedwiththesymplecticformdrAdy.Thenthecollectionof circlesCr= [(c,) / ? +y?=r2)forms a Lagrangian foliation of R2/{(0,0)). The tangent lined to these circles forma real polarization.Remark.Real polarizations, or more generally, polarizations, need not exist. Forexample, one can show that any real line bundle over S?must be trivial, and thusmust has a nowhere zero section. As a result, s2 has no real polarization, since anyvector field on s? must has a zero. M. Gotay constructed examples that admits nopolarizations even in the complex sense.Kahler polarization.Definition1.3.ApolarizationPiscalledKahlerif PnP=0.Erample. Let (M,w, J) be a Kahler manifold. Recall that this means. (M,w) is symplectic
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
3LECTURE13:GEOMETRICQUANTIZATION.(M, J) is complex, i.e.J is an almost complex structure so thatNr(u,u) :=[Ju, Ju] -J[Ju,u] -J[u, Ju] -[u,u] = 0. w(JX, JY) =w(X,Y), g(X,Y) := w(JX,Y) is a Riemannian metric on M.We letP=To1= [UETM&CI Ju=-iu].Then(1) Pis involutive: Suppose u, uEP, thenO=N(u, Jo) =-[Ju,l -J[Ju, Ju] + J[u,u]-[u, Jv=iu,u]+J[u,] +J[u,u]+i[u,]J[u,u] =-[u,u](2) P is Lagrangian: Suppose u, E P, thenw(u,v) =w(Ju, Jv) =-w(u,v) w(u,v) =0(3) PnP =0:By definitionP= Tio= uE TMCI Ju= iw].So Pis a Kahler polarization of (M,w).Similarly Pis aKahler polarization.We willcall Pthe holomorphic polarization and call P the anti-holomorphic polarization,since they are generated by 's and 's respectively.SoanyKahlermanifoldadmittwonaturalKahlerpolarizations.Conversely.ifsymplectic manifold (M,w) carries a Kahler polarization P, then there is a complexstructureJ onM whichiscompatiblewiththewandsuchthatPisitsholomorphicpolarization:Of course theonly way to defineJ is so thatJ=-i on Pand J=ion P.The integrability of this J follows from the integrability assumption on P2.GEOMETRIC QUANTIZATIONPolarized sections.Now let (M,w) be pre-quantizable symplectic manifold and (L,h, )be apre-quantum line bundle over M. Using polarization P one can reduce the pre-quantumspace L?(M,L) to amuch smaller one:Definition 2.1. A section s e Fo(M,L) is polarized with respect to a polarizationPif Vxs=O forall sections X in P.Roughly speaking, a section s is polarized if it is constant along integral mani-folds of P, i.e. only depends on “the other half variables". We will denote the spaceof all polarized sections with respect to P by Ip(M,L)
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)
4LECTURE13:GEOMETRICQUANTIZATIONErample. Consider the vertical polarization P of (T*X,wcan) described above. If wetake L=T*X x C to be the trivial line bundle and take to be theusual exteriordifferential, then by fixing aglobal trivializing section one can identify any sectionof this line bundle withfunctions on T*X, and polarized sections becomes functionsindependent of 's, i.e. pull-back through π of functions on X:TP(T*X,T*X × C) = π*C~(M)Erample. For the Kahler polarization P of a compact Kahler manifold (M,w), if onetakeL to bea holomorphic line bundle and chooseto betheChern connection(which we will explain later in this lecture), then polarized sections are exactlyholomorphic sections of L.[Reducing classical observables.Unfortunately, after cutting L?(M,L) to L(M,L), new problems appears.Re-call that the prequantization procedure sends any classical observable,i.e.any realvalued smooth function a,tothe self adjoint operatorQadefinedbyQ(a) =-iV=a+ma.Now suppose s E Lp(M,L) is a polarized section.We would like Q(a)s to bepolarized also, but this does not always happen.Proposition 2.2. For any X,a, s one has Vx(Q(a)s) = Q(a)Vxs - ihV(x,=a)s.Proof. We calculateVx(Q(a)s)-Q(a)Vxs=Vx(-inV=as+as)-(-inV=.Vxs+aVxs)= (-in)[Vx, V=a]s + (Vxa)s.As in lasttime, we usetheformula2(X,Y) = [Vx, Vy] - V(x,)and the assumption =Iw to get(-ih)[Vx, V=a]s =w(X,三a)s +(-ih)V(x,=a]s口and the conclusionfollows.As a result, if we want Q(a)s is also polarized if s is polarized, one can onlyconsider those classical observables a such that for any section X in P, [X, Ea] alsosits in P.Definition 2.3. Given any polarization P, the space of polarization preservingfunctions isthe subspace Cp(M)defined byCp(M) := (a E C(M) I [Xa, X] Er(P) for all X E T(P)).Erercise 1.If a is a polarization preserving function, then Ea preserves the leavesof the foliation defined by the distributionP
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.
LECTURE13:GEOMETRICQUANTIZATION5One mustshowProposition 2.4. The subspace Cp(M) of Co(M) is closed under the Poissonbracket.Proof. Suppose a,b E Cp(M) and X e T(P). Then[X,三(a,b] = [X,[Ea, 三]] = [X,Ea], 三b] + [三b, X],三a] e I(P),口Sowe reduce the Hilbert space to a much smaller space which consists of po-larized sections, and reduce the space of observables to the space of polarizationpreserving 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 distri-butional polarized sections. c.f. J.Sniatycki, Geometric quantization and quantummechanics.Another problem arisen in defining an inner product in the space of polarizedsections. In the case where the integral manifolds of P are compact, one can use theinduced inner product from M, i.e. integrate with respect to the Liouville measurewm. However, if the integral manifolds of P are noncompact, like the case of verticalbundles,this does not work.In fact as we have seen, in the case of trivial linebundle and trivial connection, polarized sections are just the pull back of functionsonM.TheyarenolongersquareintegrableoverMwithrespecttowm,duethenon-compactness of e direction. What people really used in this example is push-forwarding the functions (si,s2) as a function on M to a function on X, the spaceof integral manifolds of P, and then useameasure on X to integrate.In general one can use the same idea, i.e. push-forwarding the functions (si, s2)whichare constant along thedirection of integral manifolds of P,tofunctions on themanifold M/P of integral manifolds (we need toassume that the space of integralmanifolds is a manifold), and integrate via a measure on M/P. The problem isthat there is no God-giving measure on M/P.One way to solve this problem is tointroduce half densities on M/P. So instead of consider polarized sections, whichare sections of L over M but can beidentified with sections over someHermitianline bundle L/P over M/P,one can consider sections of the linebundleL/P[T(M/P)/2 over M/P, where [T(M/P)}/2 is the half density bundle over M/PThe sections of the later space can be paired intrinsically,which gives us a Hilbertspace structure.Problems still exist if one compare quantization constructed as above with realexamplesfromphysics.Andpeopleinventedsocalledhalf-formcorrectiontoelim-inate this problem. I will not discuss details here
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.