# An Invitation to Geometric Quantization

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1 An Invitation to Geometric Quantization Alex Fok Department of Mathematics, Cornell University April 2012

2 What is quantization? Quantization is a process of associating a classical mechanical system to a Hilbert space. Through this process, classical observables are sent to linear operators on the Hilbert space. Particle moving on R 1. The configuration space is the space of all possible positions of the particle, which is R 1. The phase space is T R = {(q, p) q R 1, p T q R 1 } where q is the position and p is the momentum. Alex Fok (Cornell University) Geometric Quantization April / 29

3 What is quantization? Suppose the particle is subject to a potential energy which depends on q(an example is the simple harmonic oscillator). Then + V (q) = constant 2m Turning the crank of quantization, H = L 2 (R 1 ) p 2 q M x = Multiplication by x p i d dx p V (q) + M 2m 2m V (Schrödinger operator) Alex Fok (Cornell University) Geometric Quantization April / 29

4 Symplectic manifolds Definition (X, ω) is a symplectic manifold if X is a manifold ω is a closed, non-degenerate 2-form A compact symplectic manifold (X, ω) plays the rôle of a classical phase space. Alex Fok (Cornell University) Geometric Quantization April / 29

5 Symplectic manifolds (X, ω) = (S 2, area form). ω p (ξ, η) = ξ η, n p Two questions: Can we always quantize any (X, ω)? What is the corresponding Hilbert space? Alex Fok (Cornell University) Geometric Quantization April / 29

6 First attempt We want the Hilbert space to be a certain space of sections of a complex line bundle L on X equipped with a connection and a covariant inner product, such that curv( ) = ω So this imposes a condition on ω already. Proposition [ω] H 2 (X, Z) iff there exists (L,,, ) such that curv( ) = ω. Alex Fok (Cornell University) Geometric Quantization April / 29

7 First attempt Definition (X, ω) is prequantizable if [ω] H 2 (X, Z). Remark [ω] = c 1 (L). The class of ω is topological in nature and does not depend on. Alex Fok (Cornell University) Geometric Quantization April / 29

8 First attempt Definition (L,,, ) are prequantum data of (X, ω) if curv( ) = ω, is covariant under The Hilbert space is { H = s Γ(L) X } s, s ωn n! < + Alex Fok (Cornell University) Geometric Quantization April / 29

9 First attempt Given a function f C (X ), what is the associated operator Q f? Definition X f is the symplectic vector field such that ι Xf ω = df Definition Q f = Xf + if Alex Fok (Cornell University) Geometric Quantization April / 29

10 First attempt Proposition Q f is skew-adjoint with respect to the inner product, on Γ(L) defined by s, s = s, s ωn n! X Alex Fok (Cornell University) Geometric Quantization April / 29

11 A classical example Let X = S 2 the unit sphere in R 3 centered at the origin, ω = Area form. If L = TS 2 = Riemannian connection induced from that of T R 3, = Riemannian metric Then (L,,, ) are prequantum data. Alex Fok (Cornell University) Geometric Quantization April / 29

12 A classical example We have X f = Tangent vectors of latitudes More precisely, if p = (ϕ, θ) in spherical coordinates, (X f ) p = sin ϕ(cos θ i + sin θ j) Q f X f = 0 Alex Fok (Cornell University) Geometric Quantization April / 29

13 Disadvantage of first attempt H obtained from this quantization scheme is too large to handle. One way to get around this is to introduce polarization and holomorphic sections to cut down the dimensions of H. Then we need to impose more structures on the compact symplectic manifold. A natural candidate: Kähler manifold. Alex Fok (Cornell University) Geometric Quantization April / 29

14 Kähler manifolds Definition (X, ω, J) is a Kähler manifold if ω is a symplectic 2-form. J is an integrable almost complex structure, i.e. X is a complex manifold and J corresponds to multiplication by i on each fiber of T U where U is a holomorphic chart. ω and J are compatible in the sense that ω(, J ) is positive definite. Alex Fok (Cornell University) Geometric Quantization April / 29

15 s of Kähler manifolds X = C n = {(x 1 + 1y 1,, x n + 1y n ) x i, y i R, 1 i n}. Identifying T p C n with C n, letting e i be the i-th standard basis vector and f i = 1e i, we define J by Je i = f i, Jf i = e i and ω = n dx i dy i i=1 Then (C n, ω, J) is a Kähler manifold. Actually any Kähler manifold locally looks like (C n, ω, J). Alex Fok (Cornell University) Geometric Quantization April / 29

16 s of Kähler manifolds (S 2, ω, J), where ω = area form and J is the rotation by π 2 counterclockwise on tangent spaces when viewing S 2 from outside, is Kähler. CP n and any smooth projective subvarieties of CP n are Kähler. Alex Fok (Cornell University) Geometric Quantization April / 29

17 s of Kähler manifolds The coadjoint action of a compact Lie group G on g is defined by Ad gγ, ξ = γ, Ad g 1ξ Let O γ be the orbit of the coadjoint action of G passing through γ Int(Λ +). Then O γ = G/T, T being a maximal torus of G. T β O γ = g/t. Using the above identifications, we define a 2-form ω β (ξ, η) = β([ξ, η]) ω, called the Kostant-Kirillov-Souriau form, is symplectic and integral. Alex Fok (Cornell University) Geometric Quantization April / 29

18 s of Kähler manifolds Consider the complexified Lie algebra g C and its root space decmposition g C = t C α R Let {H α, X α } α R be the Chevalley basis of g C which satisfies 2 α, β [H α, X β ] = β, β X β [X α, X α ] = H α for α R + Then g α g/t = α R + span R {e α, f α } where e α = 1(X α + X α ), f α = X α X α. Alex Fok (Cornell University) Geometric Quantization April / 29

19 s of Kähler manifolds Define J by Then (O γ, ω, J) is Kähler. Je α = f α, Jf α = e α Alex Fok (Cornell University) Geometric Quantization April / 29

20 Kähler polarization Suppose dim C X = n. Consider the complexified tangent bundle TX R C Definition A complex rank n subbundle F TX R C is a positive-definite polarization if It is integrable, i.e. closed under Lie bracket. For all X, Y F, ω C (X, Y ) = 0 1ωC (, ) is positive-definite. Alex Fok (Cornell University) Geometric Quantization April / 29

21 Kähler polarization X = C n. Then Then T p X R C = span C {e 1, f 1,, e n, f n } F = span C {e 1 + 1f 1,, e n + 1f n } is a positive-definite polarization. F should be thought of as the holomorphic direction in TX R C. Alex Fok (Cornell University) Geometric Quantization April / 29

22 Second attempt: Kähler quantization Theorem Let (X, ω, J) be a compact Kähler manifold with positive-definite polarization F, and (L,,, ) be prequantum data. Let X quantum = {s Γ(L) Θ s = 0 for all Θ F } Then X quantum is finite-dimensional. We define the quantization of (X, ω, J) to be X quantum, which is the space of holomorphic sections of the prequantum line bundle L. Alex Fok (Cornell University) Geometric Quantization April / 29

23 Second attempt: Kähler quantization X = (O γ, ω, J) as in a previous example. A positive-definite polarization is F = span C {e α + 1f α } α R + = span C {X α } α R + So F = span C {X α } α R +. Note that L = G T C γ is a prequantum line bundle. By Borel-Weil Theorem, X quantum = space of holomorphic sections of L = Irreducible representation of G with highest weight γ Alex Fok (Cornell University) Geometric Quantization April / 29

24 Second attempt: Kähler quantization X = (S 2, ω, J) with L = TS 2. Then X quantum is a 3-dimensional complex vector space. Identifying S 2 with C through stereographic projection, we can describe three holomorphic vector fields of S 2 which form a basis of X quantum as follows the vector field s 0 generated by the infinitesimal action of 1 Lie(S 1 ) of the S 1 -action on C given by rotation z e iθ z, the vector field s 2 generated by the infinitesimal action of 1 Lie(R 1 ) of the R 1 -action on C given by translation z z + a, the vector field s 2 generated by the infinitesimal action of 1 Lie(R 1 ) of the R 1 -action on C given by z 1 z + a Alex Fok (Cornell University) Geometric Quantization April / 29

25 Quantization of G-Kähler manifolds One may further consider a compact Kähler manifold with prequantum data and a nice G-action where G is a compact Lie group. By nice we mean G preserves both ω and J. There exists a map called moment map µ : X g which is G-equivariant(here G acts on g by coadjoint action) and ι ξ ω = d µ, ξ for all ξ g We say G acts on X in a Hamiltonian fashion. Alex Fok (Cornell University) Geometric Quantization April / 29

26 Quantization of G-Kähler manifolds Let Q ξ := Q µ,ξ = ξ + i µ, ξ This gives a g-action on Γ(L). In nice cases, e.g. G is simply-connected, this action can be integrated to a G-action. It turns out that G acts on X quantum, which makes it a G-representation. Question: What can we say about the multiplicities of weights of this representation? Alex Fok (Cornell University) Geometric Quantization April / 29

27 Quantization commutes with reduction Definition Let G act on (X, ω, J) in a Hamiltonian fashion with moment map µ. Assume that 0 is a regular value of µ and G acts on µ 1 (0) freely. The symplectic reduction of X by G is defined to be X G := µ 1 (0)/G X G can be thought of as the fixed points of the phase space X. One can construct prequantum data (L G, G,, G ) and positive-definite polarization of X G from those of X by restriction and quotienting. Alex Fok (Cornell University) Geometric Quantization April / 29

28 Quantization commutes with reduction Theorem (Guillemin-Sternberg, 82) dim(x G quantum) = dim((x G ) quantum ) By virtue of this theorem, one can compute the multiplicity of the trivial representation in X quantum by looking at the quantization of X G. Alex Fok (Cornell University) Geometric Quantization April / 29

29 Quantization commutes with reduction X = (S 2, ω, J), G = S 1 acts on X by rotation around the z-axis, with µ being the height function. Note that e iθ s 0 = s 0, e iθ s 2 = e 2iθ s 2, e iθ s 2 = e 2iθ s 2 So as S 1 -representations, X quantum = C0 C 2 C 2 It follows that dim(x S1 quantum) = 1. On the other hand, X S 1 = a point A line bundle over a point is simply a 1-dimensional vector space. So dim((x S 1) quantum ) = 1. Alex Fok (Cornell University) Geometric Quantization April / 29

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