LECTURE 12: GEOMETRIC PREQUANTIZATION Contents 1. The idea of quantization 1 2. Complex line bundles 3 3. Geometric Prequantization 6 1. The idea of quantization ¶ Classical mechanics modelled on symplectic manifolds. Recall that in the Hamiltonian formulation of classical mechanics, the phase space of a mechanical system is a symplectic manifold (M, ω). The symplectic manifold could be a cotangent space T ∗X, or more generally an arbitrary symplectic manifold (e.g. a symplectic quotient). Any point in M represents a possible state of the system. A Hamiltonian function H is a (smooth) function on M which represents a conserved quantity of the system. The trajectory of the system is an integral curve of the Hamiltonian vector field ΞH. In local Darboux coordinates the integral curve is given by the system of Hamiltonian equations x˙(t) = ∂H ∂ξ , ˙ξ(t) = − ∂H ∂x . A classical observable of the system is just a smooth function a on M. We have seen in lecture 6 that the evolution of a satisfies the equation (1) ˙a = {a, H}, where {·, ·} is the Poisson bracket on C ∞(M) induced by the symplectic structure. ¶ Quantum mechanics modeled on Hilbert space. In the Schr¨odinger formulation of quantum mechanics, the state space of a quantum mechanical system is a Hilbert space (H,h·, ·i). (Or more precisely, the projectified Hilbert space PH.) A quantum state is a unit vector in H. A quantum Hamiltonian is a self-adjoint operator Hˆ acting on H, whose eigenvalues represents the quantum energy level of the system, and whose normalized eigenfunctions represents the corresponding quantum states. We shall denote the eigenvalues and 1
2 LECTURE 12: GEOMETRIC PREQUANTIZATION eigenfunctions of Hˆ by λj and ψj . The equation describing the evolution of the system is the Schr¨odinger equation i~ dψ(t) dt = Hψ. ˆ A quantum observable is a self-adjoint operator A acting on H. The expectation value of a quantum observable A in state ψ is given by hAiψ := hAψ, ψi. Proposition 1.1. The evolution of the quantum observable is d dthAiψ = 1 i~ h[A, Hˆ ]iψ. Proof. d dthAψ(t), ψ(t)i = h 1 i~ AHψ, ψ ˆ i + hAψ, 1 i~ Hψˆ i = 1 i~ h[A, Hˆ ]ψ, ψi. Comparing this with (1) we see that the quantum analogy of the Poisson bracket should be the Lie bracket of operators. ¶ The idea of quantization. The word “quantization” represents a procedure (a correspondence, a functor .) that convert a classical Hamiltonian system to its quantum analogue. More precisely, we want a “dictionary” (M, ω) (H,h·, ·i) H Hˆ a A {·, ·} [·, ·] At the very beginning of the whole story, Dirac proposed a set of axioms that a quantization procedure should satisfy. Dirac’s axiom: A quantization procedure assigns self-adjoint operators Q(a) on some Hilbert space H to classical observable a ∈ C ∞(M), so that (1) (Linearity) Q(λa + µb) = λQ(a) + µQ(b). (2) (Normalization) Q(1) = Id. (3) (Quantum condition) Q({a, b}) = 1 i~ [Q(a), Q(b)]. (4) (Minimality) A complete set of (Poisson commuting) functions is quantized to a complete set of (Lie commuting) operators.
LECTURE 12: GEOMETRIC PREQUANTIZATION 3 [A procedure that only satisfies (1)-(3) is called a prequantization.] Unfortunately it we shown by Groenewold and Van Hove that such a quantization procedure never exist. As a result, mathematicians have developed many different kinds of quantization procedure, each have a different emphasis. For example, the Weyl’s quantization works for nice symbols on R 2n , or more generally T ∗X, with quantum condition replaced by an asymptotic expansion. The deformation quantization concerns more on the affection on the Poisson algebra without indicating a Hilbert space. In the last two lectures of this course, we will discuss the so-called geometric quantization. It was proposed by Kostant and has the advantage that it is coordinate free and works for a very wide class of symplectic manifolds. The Hilbert space quantizing (M, ω) will be a space of sections of a complex line bundle over M. 2. Complex line bundles ¶ Complex line bundles via transition functions. Let M be a smooth manifold, L → M a smooth complex line bundle over M, and π : L → M the projection map. Recall that this means • At each m ∈ M, Lm = π −1 (m) is a complex vector space of dimension 1 • there exists an open covering {Ui} of M and diffeomorphisms ϕi : π −1 (Ui) → Ui × C so that for m ∈ Ui , the restriction ϕi |Lm : Lm → {m} × C is a linear isomorphism. Consider the transition maps gij : Ui ∩ Uj → C ∗ = GL(C) defined by gij (m) = ϕi |Lm ◦ (ϕj |Lm) −1 . Obviously the transition functions {gij} satisfy the relations gii = 1, gijgji = 1, gijgjkgki = 1. Conversely, it is well known that given any open covering {Ui} of M and functions {gij : Ui ∩ Uj → C ∗} satisfying relations above, there exists a complex line bundle L over M so that {gij} are the transition functions of L. So line bundles are completely determined by their transition functions. Using transition functions one can characterize whether two line bundles are isomorphic: two line bundles L and Le are isomorphic if and only if on a common refinement {Ui} of the defining coverings, there exists smooth functions λi → C ∗ such that on Ui ∩ Uj , λigijλ −1 j = ˜gij
4 LECTURE 12: GEOMETRIC PREQUANTIZATION Now suppose L is an Hermitian line bundle over M. That means, we have an Hermitian inner product h·, ·im on Lm, which varies smoothly on m and is preserved by ϕi |Lm. As a consequence, each transition map becomes gij : Ui ∩ Uj → S 1 . ¶ Connections and curvature forms. Now suppose L is a complex line bundle over smooth manifold M. The space of smooth sections of L is denoted by Γ∞(M, L). The space of smooth k-forms on M with coefficients in L is Ω k (M, L) := Γ∞(M, ∧ kT ∗M ⊗ L). Note that Ω0 (M, L) = Γ∞(M, L). In the case L = M × C is a trivial bundle, one has the identification Ωk (M, L) ' Ω k (M). Definition 2.1. A connection ∇ on L is a linear map ∇ : Γ∞(M, L) → Ω 1 (M, L) so that for any f ∈ C ∞(M) and any s ∈ Γ ∞(M, L), one has the Leibniz rule ∇(fs) = df ⊗ s + f∇s. For any smooth vector field X ∈ Vect(M), one can contract X with ∇ to get the covariant derivative in the direction of X, ∇X : Γ∞(M, L) → Γ ∞(M, L), ∇Xs := ιX∇s. A connection ∇ on L can be extended uniquely to a linear map ∇ : Ωk (M, L) → Ω k+1(M, L) so that for any α ∈ Ω k (M) and β ∈ Ω • (M, L), ∇(α ∧ β) = dα ∧ β + (−1)kα ∧ ∇β. Using this generalized Leibniz rule, it is easy to see that for any f ∈ C ∞(M) and any β ∈ Ω • (M, L), the map ∇2 : Ω• (M, L) → Ω •+2(M, L) satisfies ∇2 (fβ) = ∇(f∇β + df ∧ β) = f∇2β + df ∧ ∇β − df ∧ ∇β = f∇2β. As a consequence, the map ∇2 is given by “multiplication by a 2-form”, i.e. there exists a 2-form Ω ∈ Ω 2 (M) so that for any s ∈ Γ ∞(M, L), ∇2 s = Ωs. Definition 2.2. The 2-form Ω is called the curvature of ∇. Exercise 1. As a map from Γ∞(M, L) to Γ∞(M, L), Ω(X, Y ) = ∇X∇Y − ∇Y ∇X − ∇[X,Y ]
LECTURE 12: GEOMETRIC PREQUANTIZATION 5 ¶ Differential geometry of complex line bundles. Now suppose L be an Hermitian line bundle over M. Definition 2.3. We say a connection ∇ on L is unitary, or is compatible with the Hermitian structure, if for any s, t ∈ Γ ∞(M, L), dhs, ti = h∇s, ti + hs, ∇ti. Now let {ei} be a unitary frame of the Hermitian line bundle L. Then there exists locally defined connection 1-form θi so that ∇ei = θiei . Note that if ˜ei is another local frame with ˜ei = g(x)ei , then ∇e˜i − ∇(gei) = (dg + gθi)ei = (d log g + θi)˜ei , i.e. ˜θi = d log g + θi . As a consequence, ˜θi − θi is a closed 1-form. Moreover, since ei is unitary, 0 = d(ei , ei) = θi + ¯θi , i.e. θi is pure imaginary. Exercise 2. Using connection 1-forms, one can write the curvature 2-form as Ω = dθ − θ ∧ θ = dθ. (It is not exact since θ is not globally defined.) As a consequence, Ω is globally defined, closed, purely imaginary 2-form. Definition 2.4. The first Chern class of the line bundle L is c1(L) := [ 1 2πi Ω] ∈ H 2 deRham(M, R). Remark. According to the famous Chern-Weil theorem, c1(L) is independent of the choice of the connection and the Hermitian metric on L, and thus is a topological invariant of L. ¶ Weil’s theorem. Recall how the de Rham isomorphism sends an element [a] ∈ H2 deRham(M, R) to an element [c] ∈ H2 Cech(M, R): • First take a good cover {Ui} so that all intersections are contractible. • For [a] ∈ H2 deRham(M), one can find a 1-form bi on Ui so that a = dbi on Ui . • On the contractible set Ui ∩ Uj , dbi = dbj . So one can find a function cij on Ui ∩ Uj so that bi − bj = dcij
6 LECTURE 12: GEOMETRIC PREQUANTIZATION • On Ui ∩ Uj ∩ Uk, dcij + dcjk + dcki = 0. So the function cijk = cij + cjk + cki is a constant function. They define a Cech cohomology class in H2 (M, R). Back to c1(L). On each Ui we have Ω = dθi , and on Ui ∩ Uj we have θi − θj = d log gij . So the de Rham isomorphism sends c1(L) = [ 1 2πiΩ] to [c] ∈ H2 (M, R) with cijk = 1 2πi (log gij + log gjk + log gki). Since the gij ’s are the transition functions and satisfy the cocycle condition gijgjkgki = 1, we get e 2πicijk = 1. This means cijk ∈ Z, i.e. [c] ∈ H2 (M, Z). In conclusion, we c1(L) is an integral cohomology class: Proposition 2.5. c1(L) ∈ H2 (M, Z). Conversely, we have Theorem 2.6 (Weil). Let M be a smooth manifold and ω a real, closed 2-from whose cohomology class [c] is integral. Then there is a unique Hermitian line bundle L over M with unitary connection ∇ so that c1(L) = [c]. Sketch of Proof. Existence: Reverse the arguments above. Since [c] is integral, e 2πcijk = 1. As a consequence, gijgjkgkl = 1. So gij ’s are the transition function for some line bundle whose first Chern class is [c]. Uniqueness: Suppose c1(L) = c1(Le). Let hij = 1 2πigij and define h˜ ij similarly. Then the functions hˆ ij = hij − h˜ ij satisfies the relation hˆ ij + hˆ jk + hˆ ki = 0. We take a partition of unity ρk and let λi = e 2πiPhˆ kiρk . Then λigijλ −1 j = e 2πi(hij+ P(hˆ ki−hˆ kj )ρk) = e 2πi(hij−hˆ ij ) = ˜gij . So as line bundles L ' Le. 3. Geometric Prequantization Recall that a prequantization is a process that assign to each a ∈ C ∞(M) a self-adjoint operator Q(a) on some Hilbert space H so that the conditions (1), (2) and (3) of Dirac’s axiom holds. Definition 3.1. A symplectic manifold (M, ω) is called pre-quantizable if [ ω 2π ] ∈ H 2 (M, Z). An Hermitian line bundle (L, h, ∇) over (M, ω) with Ω = ω i~ is called a pre-quantum line bundle.
LECTURE 12: GEOMETRIC PREQUANTIZATION 7 Now let (M, ω) be a pre-quantizable symplectic manifold, and (L, h, ∇) a prequantum line bundle over M. The Hilbert space H that we are going to use is H = L 2 (M, L), where the inner product between two sections is given by hs1, s2i = 1 (2π~) n Z M h(s1, s2) ω n n! . For any a ∈ C ∞(M) we let Ξa be the Hamiltonian vector field associated to a, and let ma be the operator “multiplication by a” on H. Definition 3.2. For any a ∈ C ∞(M, R) we define Q(a) = −i~∇Ξa + ma. Proposition 3.3. The operator Q(a) is self-adjoint on H. Proof. It is enough to check that the operator i∇Ξa is self-adjoint: hi∇Ξa s1, s2i = Z M h(i∇Ξa s1, s2) ω n n! = i Z M Ξa(h(s1, s2))ω n n! + Z M h(s1, i∇Ξa s2))ω n n! = i Z M Ξa(h(s1, s2))ω n n! + hs1, i∇Ξa s2i. Recall from PSet 2 that for any smooth functions a and b on M, R M {a, b}ω n = 0. So Z M Ξa(h(s1, s2))ω n = 0. The conclusion follows. Now we are ready to prove Theorem 3.4 (Kostant-Souriau). The assignment a Q(a) is a prequantization, i.e. satisfies conditions (1), (2) and (3) in Dirac’s axioms. Proof. The conditions (1) and (2) are obvious. To prove (3), we calculate 1 i~ [Q(a), Q(b)] = 1 i~ (Q(a)Q(b) − Q(b)Q(a)) = 1 i~ [(−i~∇Ξa + ma)(−i~∇Ξb + mb) − (−i~∇Ξb + mb)(−i~∇Ξa + ma)] = 1 i~ [(−i~) 2 (∇Ξa∇Ξb − ∇Ξb∇Ξa ) − i~(∇Ξamb + ma∇Ξb − ∇Ξbma − mb∇Ξa )] = i~[∇Ξa , ∇Ξb ] − (db(Ξa) − da(Ξb)) = i~[∇Ξa , ∇Ξb ] + 2{a, b}.
8 LECTURE 12: GEOMETRIC PREQUANTIZATION On the other hand, since Ω(X, Y ) = [∇X, ∇Y ] − ∇[X,Y ] we get [∇Ξa , ∇Ξb ] = Ω(Ξa, Ξb) + ∇[Ξa,Ξb] = 1 i~ ω(Ξa, Ξb) + ∇[Ξa,Ξb] = − 1 i~ {a, b} + ∇Ξ{a,b} , so the conclusion follows. Remark. The problem for this geometric prequantization is that the Hilbert space H = L 2 (X, L) is too large and thus is very far from satisfying the minimality condition (4) of Dirac’s axiom. In fact, one can regard L 2 (M, L) as a “twisted version” of L 2 (M). Comparing to the Weyl’s quantization which quantize (T ∗X, ωcan) to L 2 (X), we see that in some sense one should try to “cut L 2 (M, L) into a half”.