Construction of H (k) and H (k)

Define a family of∂-operators onL^k at σ∈ T by

∇^0,1_σ =1

2(1 +iIσ)∇.

For every σ∈ T consider the subspace ofC^∞(M,L^k)given by H_σ^(k)=H⁰(Mσ,L^k) =

s∈C^∞(M,L^k)

∇^0,1_σ s= 0 .

We will assume these subspaces of holomorphic sections form a smooth finite rank sub-bundle H^(k)of the trivial bundleH^(k)=T ×C^∞(M,L^k).

Let∇ˆ^t denote the trivial connection in the trivial bundleH^(k). LetD(M,L^k)denote the vector space of differential operators acting onC^∞(M,L^k). For any smooth one formuinT with values inD(M,L^k)we have a connection∇ˆ in H^(k)given by

∇ˆV = ˆ∇^t_V −u(V) for any vector fieldV onT.

Chapter 3

Geometric quantization

The possible states of a quantum system are vectors in a Hilbert space called the state-space.

Each observable is represented by a self-adjoint linear operator acting on the state space. Most quantum systems have a classical limit, and a way of relating the observables of the quantum mechanical system and the classical system. However often the construction is the other way around, one starts with a classical system, and wants to obtain a corresponding quantum mechanical system, by some way ofquantization.

From physics we know canonical quantization, which for examples as the hydrogen atom matches observations. In mathematics, the general quest is to make a well defined quantization scheme, that can be used on any phase space, and that reproduce canonical quantization on (R²ⁿ, ω).

Geometric quantization is an attempt at such a quantization scheme, which in its most complete form involves metaplectic quantization. This quantization schemes however depends on the choice of a so-calledpolarization, which in the case we will consider will simply be a complex structure compatible with the given symplectic form. This quantization scheme, in its current state of development, fails to establish the independence of the polarization in general.

3.1 Prequantization

Definition 3.1(Prequantum line bundle). Aprequantum line bundle(L, h,∇)over a symplec- tic manifold(M, ω)is a complex line bundleLwith a Hermitian structurehand a compatible connection∇ whose curvature satisfies

F_∇(X, Y) =−iω(X, Y).

A Hermitian structurehis called compatible with∇if for any vector fieldX and any two sectionss1, s2ofL we have

X(s₁, s₂) =h(∇_X(s₁), s₂) +h(s₁,∇_X(s₂))

Definition 3.2 (Prequantizable). We call a symplectic manifold(M, ω)prequantizableif there exists a prequantum line bundle over it.

Note that the curvature of a connection∇onL →M is a2-formF_∇∈Ω²(End(L)), with values in the Endomorphism bundleEnd(L) =L ⊗ L^∗. SinceL →M is a line bundle, we can create a global section of End(L)→M by choosing the identity, henceEnd(L)is trivial.

End(L)'M×C 27

which means the curvature can be seen as a2-form onM with values inC. A symplectic manifold(M, ω)is prequantizable if and only if

hω 2π

i∈Im(H²(M;Z)→H²(M;R)),

see [48].

Hence for a symplectic manifold (M, ω), prequantization assigns a line bundle L with curvatureω. There is a prequantum operatorf 7→fˆgiven by

f ψˆ = (−i~∇X_f +f)s for alls∈C^∞(M,L).

Example 3.3. Let us take a look at the exampleR²ⁿ with coordinates(pⁱ, qⁱ), i= 1, . . . , n.

Thenω=dpⁱ∧dqⁱ is a symplectic form, andω=d(P

ip_idq_i). Letα=P

ip_idq_i. We can take the trivial line bundleL=R²ⁿ×Cwith connection∇v=v+ ⁱ

~v·α, as the prequantum line bundle. Then

Xq_i = ∂

∂p_i and Xp_i =− ∂

∂q_i. We calculate∇X_qi and∇X_pi

∇X_qi = ∂

∂pi

+ i

~

∂

∂pi

·X

j

p_jdq_j= ∂

∂pi

and

∇X_pi =− ∂

∂qi

− i

~

∂

∂qi

·X

i

pidqi=− ∂

∂qi

− i

~ pi. Hence

ˆ

q_is= (−i~

∂

∂p_i +q_i)s ˆ

pis= (−i~(− ∂

∂qi

− i

~

pi) +pi)ψ=i~ ∂

∂qi

s for alls∈C^∞(M,L).

Recall that in canonical quantization we consider wave functionsψdepending only on say the q_i-variables and the quantization of the coordinate functions are then given bypˆ_iψ=i~_∂q^∂_iψ and qˆ_iψ = q_iψ. But notice that if we restrict to the subspace of s’s in prequantization setup which are covariant constant along∂/∂p_i’s, we do actually perfectly recreate canonical quantization, since such sections will be determined by their restriction to say the subspace wherep_i vanish and thus the resulting functions only depend on the p_i’s and the prequantum operator reproduce the operators from canonical quantization perfectly. Requiring that the sections are covariant constant along the∂/∂pi’s can be generalized to the notion of a polarization, which makes sense on any symplectic manifold. We will now briefly recall the definition of a general complex polarization, referring the reader to [48] for further details.

Complex Polarizations and geometric quantization

Let(M, ω)be a symplectic manifold. Assume that we have a prequantum line bundle(L, h,∇) on(M, ω).

A complex polarization of a symplectic manifold(M, ω)is a complex distributionP on M such that for eachm∈M,Pm⊂(TmM)_Cis Lagrangian, the dimension of D=P∩P∩T M is constant and P is integrable.

Given a polarizationP we can considers the vector space ofP-polarized sections H_P^(k)=

s∈C^∞(M,L^k)| ∇_Zs= 0, ∀Z∈C^∞(M, P) .

This is the quantum vector space which geometric quantization associated to(M, ω)equipped with the polarizationP atlevelk∈Z.

Example 3.4. For R²ⁿ with the standard symplectic structure we can in the coordinates introduced in the previous section letP =span{∂/∂p₁, . . . ∂/∂p_n}. Then

H_P ={s∈C^∞(M,L)| ∇ ∂

∂pis= 0}

is precisely the subspace we considered to get canonical quantization.

As another example, let us first consider almost complex structuresJ which are compatible withω, that isgJ(X, Y) :=ω(X, J Y)defines a Riemannian metric onM.

The almost complex structureJ induces a splitting of the complexified tangent space T M_C=T⁰MJ⊕T⁰⁰MJ

into eigenspaces of J corresponding to the eigenvalues iand−i. Let π_J^1,0= 1

2(Id−iJ), π_J^0,1= 1

2(Id +iJ),

denote the projections. ThenT⁰MJ= Im(π_J^1,0)andT⁰⁰MJ= Im(π^0,1_J ).

Hence we can defineP =T⁰MJ and thenP will be integrable if an only ifJ is integrable.

We observe that the condition on the dimension ofD is trivially true sinceD= 0in this case.

In fact, whenD= 0the sub-bundleP is always thei-eigenspace of some uniquely determined J.

However, geometric quantization does actually not reproduce the correct quantization of the harmonic oscillator, since the spectrum of the quantization of the Hamiltonian differs from the correct one by a shift. Metaplectic quantization is modification of geometric quantization, which does reproduce the canonical quantization of the harmonic oscillator.