4 · geometric quantization

the quantization problem

Let $(M,\omega)$ be a symplectic manifold of dimension $2n$, regarded as the classical phase space. The Poisson bracket of two functions $f,g\in C^\infty(M)$ is $$ \{f,g\}=\omega(X_f,X_g), $$ where $X_f$ is the Hamiltonian vector field defined by $$ \iota_{X_f}\omega=-df. $$ A quantization map is expected to send (a suitable subclass of) classical observables to self-adjoint operators on a Hilbert space $\mathcal{H}$ so that $$ \widehat{\{f,g\}}=\frac{1}{i\hbar}[\hat f,\hat g] $$ and constants act by multiplication. Dirac's early hope that this correspondence could hold for all smooth functions is obstructed; geometric quantization isolates a maximal subclass for which a consistent construction exists.

remark (relation to chapter 2). the Heisenberg algebra of Chapter 2 is the special case $M=T^*\mathbb{R}^n$ with the linear observables $q_i,p_i,1$. geometric quantization extends that kinematics to curved phase spaces and explains which larger algebras of functions remain quantizable.

prequantization

Prequantization begins by seeking a complex Hermitian line bundle $L\to M$ equipped with a Hermitian connection $\nabla$ whose curvature satisfies $$ F_\nabla=-\frac{i}{\hbar}\omega. $$ Such a bundle exists if and only if the cohomology class $[\omega/(2\pi\hbar)]$ is integral: $$ \Bigl[\frac{\omega}{2\pi\hbar}\Bigr]\in H^2(M,\mathbb{Z}). $$ This is the prequantum integrality condition. When it is satisfied, the connection is determined up to tensoring by flat Hermitian line bundles (characters of $\pi_1(M)$ when $M$ is connected).

Sections of $L$ form a complex vector space $\Gamma(L)$. For each classical observable $f$ one defines a prequantum operator $$ \hat f=-i\hbar\nabla_{X_f}+f $$ acting on (sufficiently regular) sections. These operators satisfy $$ [\hat f,\hat g]=i\hbar\widehat{\{f,g\}} $$ identically on an appropriate domain. Thus prequantization realizes the Poisson algebra as a Lie algebra of differential operators. The difficulty is that the representation is highly reducible: the space $\Gamma(L)$ is too large and the operators do not yet act irreducibly.

example (prequantum $S^2$). for $(S^2,\omega)$ with $\int_{S^2}\omega=2\pi n\hbar$ and $n\in\mathbb{Z}_{>0}$, the prequantum line bundle is the Hopf line bundle of Chern class $n$. holomorphic sections later recover the spin-$j=n/2$ representation of $\operatorname{SU}(2)$.

polarization

A polarization is a Lagrangian distribution that selects which classical observables will act by multiplication (or by first-order operators of a controlled type). A real polarization is an integrable Lagrangian sub-bundle $P\subset TM$ (or its complexification) invariant under the Hamiltonian flows of a maximal Poisson-commuting family of functions. A complex polarization is a complex Lagrangian distribution $P\subset TM\otimes\mathbb{C}$ satisfying $P\cap\overline{P}=0$ together with involutivity.

The most important examples are: the vertical polarization on $T^*Q$, whose leaves are the cotangent fibres, yielding essentially $L^2(Q)$; the holomorphic polarization on a Kahler manifold, whose leaves are the anti-holomorphic tangent spaces, yielding holomorphic sections of $L$ (Kahler quantization); and mixed polarizations that appear for systems with both configuration and momentum constraints.

Once a polarization $P$ is chosen, one restricts attention to the space of polarized sections $$ \Gamma_P(L)=\{s\in\Gamma(L)\mid\nabla_Xs=0\text{ for all }X\in\Gamma(P)\}. $$ This space is still not a Hilbert space; an inner product must be supplied by additional geometric data.

example (vertical polarization). on $T^*\mathbb{R}^n$ with coordinates $(q,p)$ and $\omega=dq\wedge dp$, the vertical polarization is spanned by $\partial/\partial p$. polarized sections are independent of $p$ after a standard trivialization, recovering wave functions $\psi(q)$.

half-forms and the quantum hilbert space

Polarized sections behave as densities of weight $1/2$ with respect to the leaf space of the polarization. To obtain a well-defined $L^2$ inner product one tensors $L$ with the half-form bundle $\delta_P^{1/2}$ associated with the canonical bundle of the polarization. The resulting space of square-integrable polarized half-form sections, $$ \mathcal{H}=\bigl\{s\in\Gamma_P(L\otimes\delta_P^{1/2})\bigm|\|s\|^2<\infty\bigr\}, $$ is the quantum Hilbert space. The half-form correction supplies the metaplectic correction and ensures that the quantization of the harmonic oscillator yields the spectrum $\hbar\omega(n+\tfrac12)$ rather than $\hbar\omega n$.

remark (metaplectic structure). existence of half-forms requires a metalinear or metaplectic structure on the polarization. this is the geometric origin of the double cover already encountered for $\operatorname{Mp}(2n,\mathbb{R})$ in Chapter 2.

quantizable observables and the pairing

Not every classical function preserves the polarization. A function $f$ is quantizable with respect to $P$ when its Hamiltonian vector field maps polarized sections to polarized sections. The corresponding quantum operator is the restriction of the prequantum operator to $\mathcal{H}$, corrected by a Lie-derivative term arising from the half-form bundle: $$ \hat f=-i\hbar\nabla_{X_f}+f-\frac{i\hbar}{2}\operatorname{div}_P X_f. $$ The resulting map is a Lie-algebra homomorphism from the Poisson algebra of quantizable functions into operators on $\mathcal{H}$.

When two different polarizations $P$ and $P'$ are used, the Blattner-Kostant-Sternberg (BKS) pairing provides a sesquilinear pairing between the corresponding spaces of polarized sections. Under favorable conditions the pairing induces a unitary intertwiner between the two quantizations, implementing the quantum analogue of a canonical transformation.

example (position versus momentum). on $T^*\mathbb{R}$, the vertical and horizontal polarizations yield the position and momentum representations. the BKS pairing reduces to the Fourier transform, recovering the unitary equivalence guaranteed by Stone-von Neumann.

moment maps and symmetry

Let $G$ be a Lie group acting on $M$ by symplectomorphisms, with equivariant moment map $$ \mu:M\to\mathfrak{g}^*. $$ If the action lifts to an action on the prequantum bundle $L$ that preserves the connection, the components of $\mu$ become quantizable observables. Their quantum operators furnish a projective representation of $G$ on $\mathcal{H}$. Equivariance of the moment map guarantees that the Lie-algebra relations are preserved (up to the central cocycles already visible classically).

This mechanism recovers the representation-theoretic constructions of Chapter 2 (Heisenberg group, $\operatorname{SU}(2)$, metaplectic representation) from the geometry of the classical phase space.

example (rotations on $T^*\mathbb{R}^3$). the moment map for the cotangent-lifted $\operatorname{SO}(3)$ action is angular momentum $\boldsymbol{\ell}=\mathbf{q}\times\mathbf{p}$. quantization with the vertical polarization yields the orbital operators $\mathbf{L}$ of Chapter 1, realizing integer spins on $L^2(\mathbb{R}^3)$.

the groenewold-van hove obstruction

One may ask whether the prequantum operators can be restricted to an irreducible representation of the full Poisson algebra $C^\infty(M)$. The Groenewold-van Hove theorem answers in the negative for $M=T^*\mathbb{R}^n$: there is no irreducible representation of the full Poisson algebra by (essentially) self-adjoint operators on a Hilbert space that extends the Schrodinger representation of the Heisenberg algebra and sends the constant function $1$ to the identity.

Consequently any consistent quantization map must be defined only on a proper subalgebra of observables: the quantizable functions selected by a polarization. Geometric quantization makes this restriction geometrically natural rather than ad hoc.

example (polynomials of degree $\ge 3$). already for $T^*\mathbb{R}$, attempting to quantize all polynomials while preserving the Poisson bracket and the Heisenberg representation leads to a contradiction at cubic order. the vertical polarization retains position multiplication and momentum derivatives, but not arbitrary higher symbols as a Lie homomorphism.

prototype examples

cotangent bundle $T^*Q$. vertical polarization yields the Schrodinger representation on $L^2(Q)$ (with half-form densities on $Q$). position functions act by multiplication; momenta act by covariant differentiation.

cotangent bundle of a lie group. left- or right-invariant polarizations recover the representation theory of the group and its coadjoint orbits (Kirillov-Kostant-Souriau orbit method).

kahler manifolds. holomorphic polarization produces the Bargmann-Fock space when $M=\mathbb{C}^n$, and more generally the spaces of holomorphic sections that appear in geometric quantization of coadjoint orbits of compact groups.

harmonic oscillator. the metaplectic correction supplied by half-forms shifts the zero-point energy by $\tfrac12\hbar\omega$, matching the spectrum obtained algebraically in Chapter 2.

summary

Geometric quantization converts a symplectic manifold $(M,\omega)$ into a quantum theory in three geometric steps. First one constructs a prequantum Hermitian line bundle $(L,\nabla)$ with curvature $F_\nabla=-i\omega/\hbar$, so that the prequantum operators $\hat f=-i\hbar\nabla_{X_f}+f$ realise the Poisson algebra on $\Gamma(L)$. Next a polarization $P$ cuts $\Gamma(L)$ down to polarized sections $\Gamma_P(L)$, on which a maximal Poisson-commutative subalgebra acts by multiplication. Finally half-forms produce the Hilbert space $\mathcal{H}=L^2\bigl(\Gamma_P(L\otimes\delta_P^{1/2})\bigr)$ and supply the metaplectic correction. Moment maps lift classical symmetries to (projective) unitary representations on $\mathcal{H}$.

The Groenewold-van Hove obstruction confirms that the procedure cannot be an isomorphism on the whole Poisson algebra; the geometric choices of bundle and polarization select the largest consistent subclass of quantizable observables. The construction recovers the Schrodinger representation, the Bargmann-Fock space, and the representation theory of compact and nilpotent groups as special cases, and prepares the ground for the orbit method and the geometric interpretation of induced representations in the next chapter.