Polarisation, half-densities, and metaplectic correction

Intuition Beginner

The prequantum line bundle from the previous unit is too big to be the quantum Hilbert space. Its smooth sections form an infinite-dimensional space that knows about every direction in phase space at once, with no preferred separation of position from momentum. Quantum mechanics needs the wavefunction to depend on half the variables — position alone, in the Schrödinger picture — and the prequantum sections depend on all of them. The job of a polarisation is to cut the section space down by half.

A polarisation is a smooth choice, at every point of phase space, of a Lagrangian subspace through that point — a half-dimensional flat direction. Sections that are constant along these directions are the polarised sections, and they form the candidate quantum Hilbert space. The three flavours of polarisation produce the three standard quantum-mechanical pictures: a real polarisation by the vertical fibres of $T^{\ast }{\mathbb{R}}^n$ gives wavefunctions $\psi (q)$, the position representation; a Kähler polarisation by antiholomorphic directions on ${\mathbb{C}}^n$ gives holomorphic functions, the Bargmann-Fock representation; mixed polarisations interpolate between the two.

There is one more correction. Polarised sections are constant along leaves of a foliation, so any integral over the whole manifold of a polarised pair diverges. The fix is to tensor by the half-density bundle on the leaf space, so that pairs of polarised sections produce a finite number when paired transverse to the leaves. A second subtlety appears when you try to define half-densities globally: the square root of the determinant has a sign ambiguity that obstructs the construction. This is the metaplectic correction, named after the double cover $\mathrm{Mp}(2n)$ of the symplectic group $\mathrm{Sp}(2n)$. It lifts the sign ambiguity globally and produces a quantisation that is independent of the polarisation chosen.

Visual Beginner

Phase space drawn as a wavy surface across the bottom of the figure, with vertical dashed lines rising from each point — the leaves of a real polarisation. Above the surface, two stacked horizontal lines represent the symplectic group $\mathrm{Sp}(2n)$ and its double cover $\mathrm{Mp}(2n)$, with the metaplectic correction running through the upper line.

The two structures coupled in the picture — the polarisation foliating phase space, and the double cover lifting the sign ambiguity of the half-density — are the two ingredients that turn the prequantum bundle into a proper quantum Hilbert space.

Worked example Beginner

Take phase space to be the plane $M={\mathbb{R}}^2$ with coordinates $(q,p)$, where $q$ is position and $p$ is momentum. From the previous unit, the prequantum line bundle on this phase space is the product bundle $L=M\times \mathbb{C}$, equipped with a particular connection whose curvature reproduces the symplectic form.

Step 1. Choose the vertical polarisation: at every point $(q,p)$, take the line in the $p$-direction. This foliates phase space by vertical lines, each labelled by its $q$-value. A section of the prequantum bundle that is covariantly constant along these vertical lines is a function that does not depend on $p$ — call it $s(q)$. Working out the polarisation equation in coordinates uses the fact that the connection has no component in the vertical direction, so the polarised sections are functions of $q$ alone.

Step 2. The space of "vertical lines" — the leaf space — is just the $q$-line, ${\mathbb{R}}_q$. A half-density on this line is an expression of the form $f(q)|dq{|}^{1/2}$. Pairing two such half-densities along the $q$-axis gives the familiar inner product of $L^2({\mathbb{R}}_q)$.

Step 3. Polarised sections of $L$ tensored with half-densities on the leaf space form the space $\mathcal{H}=L^2({\mathbb{R}}_q,|dq|)$ — exactly the Schrödinger Hilbert space of one-dimensional quantum mechanics.

What this tells us: the polarisation choice and the half-density tensor produce a familiar quantum-mechanical Hilbert space directly from the prequantum data, without writing down $L^2$ by hand. A different polarisation on the same prequantum bundle — for example, the horizontal polarisation by lines of constant $p$ — would give $L^2({\mathbb{R}}_p,|dp|)$ instead, the momentum representation. The Blattner-Kostant-Sternberg pairing intertwines the two, and reproduces the Fourier transform as the canonical unitary equivalence.

Check your understanding Beginner

Formal definition Intermediate+

Let $(M,\omega )$ be a symplectic manifold of dimension $2n$ admitting a prequantum line bundle $(L,\nabla ,h)$ with curvature $F^{\nabla }=-i\omega /\hslash$, as constructed in 05.11.01. Denote by $T_{\mathbb{C}}M=TM{\otimes }_{\mathbb{R}}\mathbb{C}$ the complexified tangent bundle, and extend $\omega$ to a $\mathbb{C}$-bilinear $2$-form on $T_{\mathbb{C}}M$ by complex linearity.

A complex polarisation of $(M,\omega )$ is a smooth complex distribution $\mathcal{P}\subset T_{\mathbb{C}}M$ satisfying:

• (Lagrangian) ${\mathcal{P}}_x$ is a complex Lagrangian subspace of $(T_{\mathbb{C},x}M,{\omega }_x)$ of complex dimension $n$ at every $x\in M$;
• (Integrable) $[\Gamma (\mathcal{P}),\Gamma (\mathcal{P})]\subset \Gamma (\mathcal{P})$, so $\mathcal{P}$ is involutive as a sub-bundle of $T_{\mathbb{C}}M$;
• (Constant-rank intersection) $D=\mathcal{P}\cap \overline{\mathcal{P}}\cap TM$ has locally constant real rank.

A polarisation is real if $\mathcal{P}=\overline{\mathcal{P}}$, equivalently $\mathcal{P}=D_{\mathbb{C}}$ for a real Lagrangian foliation $D\subset TM$; it is Kähler if $\mathcal{P}\cap \overline{\mathcal{P}}=0$ at every point and the Hermitian form $h(v,w)=i\omega (v,\bar{w})$ is positive on $\mathcal{P}$. Mixed polarisations interpolate between these extremes ^{[Kirillov EMS Vol 4 Part II Ch. 2; Woodhouse 1992 Ch. 5]}.

Notation conventions

Throughout this unit:

• $(M,\omega )$ is a symplectic manifold of real dimension $2n$.
• $(L,\nabla ,h)$ is the prequantum line bundle from 05.11.01, with $F^{\nabla }=-i\omega /\hslash$.
• $\mathcal{P}\subset T_{\mathbb{C}}M$ is a polarisation; its complex conjugate is $\overline{\mathcal{P}}$.
• $D=\mathcal{P}\cap \overline{\mathcal{P}}\cap TM$ is the real distribution underlying the polarisation; $E=(\mathcal{P}+\overline{\mathcal{P}})\cap TM$ is its real envelope.
• ${\Gamma }_{\mathcal{P}}(L)=\{s\in \Gamma (L):{\nabla }_{X}s=0\text{ for all }X\in \Gamma (\mathcal{P})\}$ is the space of polarised sections.
• $|\Lambda {|}^s(V)\to V$ denotes the bundle of $s$-densities of a real vector bundle $V$ for $s\in \mathbb{R}$; sections transform as $|\det g{|}^s$ under frame changes.
• $\mathrm{Sp}(2n,\mathbb{R})$ is the real symplectic group; $\mathrm{Mp}(2n)$ is its connected double cover.
• $\hslash >0$ is fixed; the integrality condition on $[\omega /(2\pi \hslash )]$ is in force.

Polarised sections

A section $s\in \Gamma (L)$ is polarised with respect to $\mathcal{P}$ when ${\nabla }_{X}s=0$ for every $X\in \Gamma (\mathcal{P})$. The set ${\Gamma }_{\mathcal{P}}(L)$ of polarised sections is a vector subspace of $\Gamma (L)$ that is much smaller — generically infinite-dimensional in the right way, with the natural Hilbert structure coming from the half-density correction below.

For a real polarisation by leaves of a fibration $\pi :M\to B$ with Lagrangian fibres, ${\Gamma }_{\mathcal{P}}(L)$ consists of sections covariantly constant along the fibres; pulling back via $\pi$, these correspond to sections of a bundle over $B$ associated to the holonomy of $\nabla$ along the fibres.

For a Kähler polarisation given by a compatible complex structure $J$ on $M$, $\mathcal{P}=T^{0,1}M$ is the antiholomorphic tangent bundle, and ${\Gamma }_{\mathcal{P}}(L)$ is the space of holomorphic sections $H^0(M,L)$ of $L$ as a holomorphic line bundle. The integrability condition on $\mathcal{P}$ is the Newlander-Nirenberg condition for $J$.

Half-density bundle on the leaf space

The naive inner product ${\int }_{M}h(s_1,s_2){\omega }^n/n!$ diverges on polarised sections: such sections are constant along the leaves of $\mathcal{P}\cap \overline{\mathcal{P}}$, so the integrand is fibre-wise constant times the symplectic volume form, which is unbounded transversally.

The fix is to replace $\Gamma (L)$ by sections of $L\otimes |\Lambda {|}^{1/2}(M/D)$, where $|\Lambda {|}^{1/2}(M/D)$ is the bundle of half-densities on the leaf space — the quotient of the half-density bundle of $M$ by the half-density bundle of the leaves of the real distribution $D=\mathcal{P}\cap \overline{\mathcal{P}}\cap TM$. For two such polarised sections $s_1\otimes {\nu }_1$ and $s_2\otimes {\nu }_2$, the inner product

$$⟨s_1\otimes {\nu }_1,s_2\otimes {\nu }_2⟩={\int }_{M/D}h(s_1,s_2)\overline{{\nu }_1}{\nu }_2$$

is well-defined and finite. The quantum Hilbert space is the completion

$${\mathcal{H}}_{\mathcal{P}}=\overline{{\Gamma }_{\mathcal{P}}(L\otimes |\Lambda {|}^{1/2}(M/D))}.$$

Counterexamples to common slips

• Polarisation is in $T_{\mathbb{C}}M$, not in $TM$. The Kähler polarisation $T^{0,1}M$ is genuinely complex and has no real version; insisting on a real Lagrangian distribution rules out the Bargmann-Fock construction entirely.
• Integrability is essential. A non-integrable Lagrangian distribution gives a section space that fails to have a workable transverse structure: the polarisation equation ${\nabla }_{X}s=0$ admits only the zero solution when extended to the Lie subalgebra generated by $\mathcal{P}$, because the obstruction $F^{\nabla }(X,Y)$ for $X,Y\in \mathcal{P}$ vanishes precisely when $\omega {|}_{\mathcal{P}}=0$, which is the Lagrangian condition.
• Half-densities are not densities. A half-density transforms by $|\det g{|}^{1/2}$ under change of frame; squaring gives a full density that integrates over the leaf space. The sign of the square root is the source of the metaplectic obstruction.
• Polarisations need not exist globally. On a compact symplectic manifold without a compatible complex structure, no Kähler polarisation exists; on a non-fibred symplectic manifold, no global real polarisation exists. Existence of some polarisation is itself a genuine condition with topological content.

Key theorem with proof Intermediate+

Theorem (polarised section space and the metaplectic correction). Let $(M,\omega )$ be a symplectic manifold with prequantum line bundle $(L,\nabla ,h)$, and let $\mathcal{P}\subset T_{\mathbb{C}}M$ be a polarisation with real distribution $D=\mathcal{P}\cap \overline{\mathcal{P}}\cap TM$ of locally constant rank $k$. Then:

(i) The half-density bundle $|\Lambda {|}^{1/2}(M/D)\to M$ is a real line bundle whose transition functions are absolute squared roots of determinants of leaf-transverse frame changes.

(ii) The bundle $L\otimes |\Lambda {|}^{1/2}(M/D)$ supports a Hermitian inner product on its polarised sections.

(iii) A consistent global half-density structure — equivalently, an unambiguous square root of the leaf-transverse determinant bundle — exists if and only if the second Stiefel-Whitney class $w_2(M/D)$ vanishes, equivalently if $M$ admits a metaplectic structure: a lift of the structure group of the symplectic frame bundle of $TM$ from $\mathrm{Sp}(2n,\mathbb{R})$ to its double cover $\mathrm{Mp}(2n)$.

(iv) The Blattner-Kostant-Sternberg pairing between two polarisations ${\mathcal{P}}_1$ and ${\mathcal{P}}_2$ in general position is the integral kernel $⟨s_1,s_2{⟩}_{BKS}$ obtained by integrating along the leaves of ${\mathcal{P}}_1\cap \overline{{\mathcal{P}}_2}$. When transverse, the BKS pairing is a unitary isomorphism ${\mathcal{H}}_{{\mathcal{P}}_1}\to {\mathcal{H}}_{{\mathcal{P}}_2}$. ^{[Kostant 1970; Blattner 1973; Woodhouse 1992 Ch. 5-6]}.

Proof. (i) The leaf-transverse tangent bundle $T(M/D)=TM/D$ has rank $2n-k$; its frame bundle is a principal $\mathrm{GL}(2n-k,\mathbb{R})$-bundle. Half-densities are sections of the associated line bundle for the one-dimensional representation $g\mapsto |\det g{|}^{1/2}$. The transition functions on overlaps $U_{\alpha \beta }$ are $|\det g_{\alpha \beta }{|}^{1/2}$ where $g_{\alpha \beta }$ are the original $\mathrm{GL}$-cocycle data. Since $|\det g{|}^{1/2}$ is a continuous positive function, the cocycle property is preserved and $|\Lambda {|}^{1/2}(M/D)$ is a smooth real line bundle.

(ii) For polarised sections $s_1\otimes {\nu }_1,s_2\otimes {\nu }_2$ of $L\otimes |\Lambda {|}^{1/2}(M/D)$, define $$ \langle s_1 \otimes \nu_1, s_2 \otimes \nu_2 \rangle = \int_{M/D} h_L(s_1, s_2),\overline{\nu_1},\nu_2. $$ The integrand $h_L(s_1,s_2)\overline{{\nu }_1}{\nu }_2$ is a full density on $M/D$: the half-density factors $\overline{{\nu }_1}{\nu }_2$ multiply to give a density, the Hermitian metric $h_L$ on $L$ produces a scalar, and the integration is over the leaf space $M/D$ (well-defined since polarised sections are constant along leaves of $D$). Polarisation of $s_i$ and the connection compatibility $\nabla h_L=0$ guarantee that $h_L(s_1,s_2)$ is constant along $D$-leaves, so the integral collapses to a finite leaf-space integral.

(iii) The obstruction to a global square root of the determinant line bundle is the second Stiefel-Whitney class $w_2$ of the relevant rank-$2n-k$ vector bundle ^{[Woodhouse 1992 Ch. 6]}. For a Kähler polarisation, this reduces to the existence of a square root of the canonical bundle $K_M=\det T^{\ast 1,0}M$, equivalently a holomorphic ${\mathrm{Spin}}^c$-structure or a half-canonical bundle $K_M^{1/2}$. The general statement (Kostant 1970) is the lift of the symplectic frame bundle structure group from $\mathrm{Sp}(2n,\mathbb{R})$ to its connected double cover $\mathrm{Mp}(2n)$. Such a lift exists iff the obstruction class — the pull-back of the generator of $H^2(B\mathrm{Sp}(2n);\mathbb{Z}/2)$ along the classifying map — vanishes, which for symplectic vector bundles is the second Stiefel-Whitney class of the underlying real bundle.

(iv) Let ${\mathcal{P}}_1,{\mathcal{P}}_2$ be two polarisations with ${\mathcal{P}}_1+\overline{{\mathcal{P}}_2}$ of constant rank. The leaf space of ${\mathcal{P}}_1\cap \overline{{\mathcal{P}}_2}$ carries an induced symplectic structure (by symplectic reduction), and there is a canonical half-form pairing $$ \nu_1 \otimes \overline{\nu_2}\mapsto \int_{\text{leaves of } \mathcal{P}1 \cap \overline{\mathcal{P}2}} \nu_1,\overline{\nu_2} $$ where the integrand is a density along each leaf. For $s_1 \in \mathcal{H}{\mathcal{P}1}$and$s_2 \in \mathcal{H}{\mathcal{P}2}$, the BKS pairing is $$ \langle s_1, s_2\rangle{BKS} = \int{\text{leaves}} h_L(s_1, s_2),\nu_1,\overline{\nu_2}. $$ When the polarisations are transverse (${\mathcal{P}}_1\cap \overline{{\mathcal{P}}_2}=0$), the leaves degenerate to points and the pairing is the ordinary inner product on the common Hilbert space. When the polarisations are mutually generic, the BKS kernel is a finite-rank operator with adjoint property making it a unitary intertwiner ^{[Blattner 1973; Sternberg 1975]}. $\square$

Bridge. The metaplectic correction is the natural endpoint of the prequantum-polarisation construction: it ties the half-form bundle to the global topology of the symplectic frame bundle, reproducing the universal Maslov class 05.12.01 as the obstruction class. The structure that organises these pieces is the Shale-Weil representation of $\mathrm{Mp}(2n)$ on $L^2({\mathbb{R}}^n)$, the projective representation of $\mathrm{Sp}(2n)$ that the Stone-von Neumann theorem produces; the BKS pairing realises this representation explicitly via the polarisation-change kernel. On the finite-dimensional symplectic vector space ${\mathbb{R}}^{2n}$, the metaplectic correction enters as the canonical $1/2$-shift in the harmonic oscillator spectrum and as the integer-versus-half-integer character of the Maslov-corrected Lagrangian intersection number. The construction generalises in three directions explored downstream. For the orbit method 05.11.04, the metaplectic correction realises the canonical Liouville measure on a coadjoint orbit as a half-form; for Floer cohomology 05.08.02, the Maslov class controls the grading and the metaplectic shift produces the $\mathbb{Z}/2$-grading anomaly in non-orientable contexts; for semiclassical asymptotics, the metaplectic correction is the $1/4$-Maslov-phase appearing in the WKB expansion at every caustic.

Exercises Intermediate+

Advanced results Master

Existence of polarisations and the Newlander-Nirenberg theorem. For a symplectic manifold $(M,\omega )$, the existence of a Kähler polarisation is equivalent to the existence of a compatible complex structure $J$ with $\omega (J\cdot ,J\cdot )=\omega (\cdot ,\cdot )$ and $g(\cdot ,\cdot )=\omega (\cdot ,J\cdot )$ a Riemannian metric. The Newlander-Nirenberg theorem then ensures that the almost-complex distribution $T^{0,1}M$ is integrable iff the Nijenhuis tensor of $J$ vanishes — automatically true for $J$ from a Kähler structure, where the integrability comes from the $\mathrm{d}\omega =0$ condition combined with compatibility. Real polarisations exist on cotangent bundles $T^{\ast }X$ and on integrable systems via action-angle coordinates 05.02.04; mixed polarisations interpolate, and a unified existence theorem on compact $M$ is the Frobenius-Hess theorem: every symplectic manifold admits some polarisation locally, but global existence is obstructed by the Maslov class.

The universal Shale-Weil representation. The metaplectic group $\mathrm{Mp}(2n)$ acts on $L^2({\mathbb{R}}^n)$ via the Shale-Weil representation, the unique faithful unitary representation of $\mathrm{Mp}(2n)$ lifting the projective representation of $\mathrm{Sp}(2n)$ produced by the Stone-von Neumann theorem ^{[Weil 1964]}. Concretely, for a symplectic matrix $g=\left(\begin{array}{cc}A& B\\ C& D\end{array}\right)\in \mathrm{Sp}(2n,\mathbb{R})$ acting on phase space coordinates, the metaplectic operator $U(g):L^2({\mathbb{R}}^n)\to L^2({\mathbb{R}}^n)$ is given (up to the metaplectic sign ambiguity) by a Fourier integral operator with explicit quadratic generating function. The Shale-Weil representation realises every metaplectic operator as a chained polarisation-change kernel, the BKS pairing on ${\mathbb{R}}^{2n}$ for the family of linear polarisations parametrised by $\Lambda (n)$. The metaplectic sign — the choice of branch in $\sqrt{\det (A+iB)}$ for the quadratic-form normal-form of $g$ — is precisely the Maslov-index correction.

Quantum-classical correspondence and BKS as Fourier integral operator. The Blattner-Kostant-Sternberg pairing for general polarisations on a symplectic manifold realises the geometric quantisation as a category-theoretic structure: the category of polarised symplectic manifolds has symplectic manifolds with polarisations as objects, and BKS-type kernels as morphisms. The composition of two BKS kernels is a third kernel, computed by stationary-phase integration over the intermediate leaf space — this is the geometric origin of the Hörmander-Maslov calculus of Fourier integral operators. The metaplectic correction enters as the $1/4$-Maslov phase at every caustic of the composition. The functor from polarised symplectic manifolds to unitary Hilbert spaces is well-defined up to the BKS-kernel equivalence, and the obstruction to functoriality is the universal Maslov class of 05.12.01. This category-theoretic packaging is the modern view of geometric quantisation, due to Weinstein 1981 (Symplectic groupoids and Poisson manifolds) and later refinements; the BKS pairing is the bridge.

The harmonic oscillator and the metaplectic ground-state shift. For the harmonic oscillator on ${\mathbb{R}}^{2n}$ with Hamiltonian $H=\frac{1}{2}{\sum }_i(q_i^2+p_i^2)$, the prequantum operator $\hat{H}=-i\hslash {\nabla }_{X_H}+H$ acts on holomorphic polynomials in $z=(q+ip)/\sqrt{2}$ via the Euler operator $\hslash {\sum }_i{z}_i\partial /\partial z_i$, with spectrum $\{{\sum }_i{n}_i\hslash :n_i\in {\mathbb{Z}}_{\ge 0}\}$. The metaplectic correction tensoring with $K_{{\mathbb{C}}^n}^{1/2}$ produces the corrected operator $\hat{H}+\frac{n}{2}\hslash$, with spectrum $\{{\sum }_i(n_i+1/2)\hslash \}$ — the physical harmonic-oscillator spectrum with ground-state energy $\frac{n}{2}\hslash$. The shift $\frac{n}{2}\hslash$ is the zero-point energy, the most direct experimental signature of the metaplectic correction. The same shift appears in the Casimir effect, the Lamb shift, and the vacuum-fluctuation energy of every free quantum field; geometric quantisation accounts for all of these as instances of the half-form correction.

Synthesis. Geometric quantisation produces a quantum Hilbert space from a symplectic manifold in three layers, each addressing one of the three obstructions identified in the previous units. The prequantum line bundle 05.11.01 addresses the integrality obstruction — the symplectic class must lift to integral cohomology — and produces a section space that is too big. The polarisation cuts this space down by selecting a half-dimensional integrable Lagrangian distribution; the resulting polarised section space is the candidate quantum Hilbert space. The half-density tensor produces a well-defined inner product transverse to the leaves, completing the polarised sections into a separable Hilbert space. The metaplectic correction addresses the Maslov-class obstruction to the global square root of the determinant, ensuring that the half-form bundle is unambiguously defined and that the resulting quantisation is independent of the polarisation chosen — the Blattner-Kostant-Sternberg pairing realises this independence as a unitary intertwiner. The three constructions together produce a functor from a category of polarised symplectic manifolds to a category of Hilbert spaces, the geometric quantisation functor, and identify three classical-mechanical obstructions — integrality of the symplectic form, existence of a polarisation, vanishing of $w_2$ — as the topological conditions for the functor to be well-defined.

Full proof set Master

Proposition (consistency of the polarisation equation). Let $(L,\nabla )$ be the prequantum line bundle on $(M,\omega )$ with $F^{\nabla }=-i\omega /\hslash$. For an integrable Lagrangian distribution $\mathcal{P}\subset T_{\mathbb{C}}M$, the polarised section space ${\Gamma }_{\mathcal{P}}(L)=\{s:{\nabla }_{X}s=0\text{ for all }X\in \mathcal{P}\}$ is locally non-empty.

Proof. Pick a point $x\in M$ and a foliation chart in which $\mathcal{P}$ is the span of vector fields $\partial /\partial y_1,\dots ,\partial /\partial y_n$ on a neighbourhood $U$ of $x$ (complemented by $\partial /\partial z_1,\dots ,\partial /\partial z_n$). In the trivialisation of $L$ over $U$, the connection $\nabla$ has connection $1$-form $A={\sum }_i{A}_{i}d{y}_i+{\sum }_j{B}_{j}d{z}_j$ with $F^{\nabla }=dA=-i\omega /\hslash$ Lagrangian (i.e., $\omega {|}_{\mathcal{P}}=0$). The polarised condition ${\nabla }_{\partial /\partial y_i}s=(\partial /\partial y_i-i{A}_i)s=0$ is a system of $n$ first-order PDE on the $2n$ variables $(y,z)$, with consistency $$ (\partial/\partial y_i - iA_i)(\partial/\partial y_j - iA_j),s = (\partial/\partial y_j - iA_j)(\partial/\partial y_i - iA_i),s, $$ which expands to $F^{\nabla }(\partial /\partial y_i,\partial /\partial y_j)=-i\omega (\partial /\partial y_i,\partial /\partial y_j)/\hslash =0$ (by Lagrangian). So the PDE system is integrable, with local solutions $s=e^{i\Phi (y)}f(z)$ for any smooth $f$ and any local antiderivative $\Phi$ of ${\sum }_i{A}_{i}d{y}_i$ along the $y$-leaf. Hence ${\Gamma }_{\mathcal{P}}(L){|}_U$ is non-empty and parametrised by smooth functions on the local leaf space. $\square$

Proposition (well-definedness of the half-density inner product). Let $\mathcal{P}$ be a real polarisation with leaf-space projection $\pi :M\to B$ and prequantum line bundle $(L,\nabla ,h)$. For polarised sections $s_1\otimes {\nu }_1,s_2\otimes {\nu }_2$ of $L\otimes |\Lambda {|}^{1/2}(M/D)$, the integrand $h(s_1,s_2)\overline{{\nu }_1}{\nu }_2$ is the pull-back of a smooth density on $B=M/D$, and its integral is finite when $B$ is compact (or finite for sections with appropriate decay when $B$ is non-compact).

Proof. Since $s_1,s_2$ are polarised, ${\nabla }_X{s}_i=0$ for all $X\in \Gamma (\mathcal{P})$. The connection compatibility $\nabla h=0$ gives $X(h(s_1,s_2))=h({\nabla }_X{s}_1,s_2)+h(s_1,{\nabla }_X{s}_2)=0$, so $h(s_1,s_2)$ is constant along $\mathcal{P}$-leaves. Hence $h(s_1,s_2)$ descends to a smooth function on $B=M/D$.

The half-density product $\overline{{\nu }_1}{\nu }_2$ is a full density on $M/D$ — sections of $|\Lambda {|}^{1/2}(M/D)\otimes |\Lambda {|}^{1/2}(M/D)=|\Lambda {|}^1(M/D)$. The integrand $h(s_1,s_2)\overline{{\nu }_1}{\nu }_2$ is a smooth density on $B$, hence integrable over compact $B$ by ordinary measure theory. For non-compact $B$, an $L^2$-decay condition on the half-densities ensures finite norm, and the completed space ${\mathcal{H}}_{\mathcal{P}}=\overline{{\Gamma }_{\mathcal{P}}(L\otimes |\Lambda {|}^{1/2}(M/D))}$ is the desired Hilbert space ^{[Bates-Weinstein 1997 Ch. 7]}. $\square$

Proposition (existence and uniqueness of $\mathrm{Mp}(2n)$). The symplectic group $\mathrm{Sp}(2n,\mathbb{R})$ has ${\pi }_1(\mathrm{Sp}(2n))=\mathbb{Z}$, and the metaplectic group $\mathrm{Mp}(2n)$ is the unique connected double cover, corresponding to the index-$2$ subgroup $2\mathbb{Z}\subset \mathbb{Z}$.

Proof. The maximal compact subgroup of $\mathrm{Sp}(2n,\mathbb{R})$ is $U(n)$, embedded as the stabiliser of the standard compatible complex structure $J_0$. The inclusion $U(n)\hookrightarrow \mathrm{Sp}(2n)$ is a homotopy equivalence (by Iwasawa decomposition $\mathrm{Sp}(2n)=U(n)\cdot AN$ with $AN$ contractible). Hence ${\pi }_1(\mathrm{Sp}(2n))={\pi }_1(U(n))$, which by $\det :U(n)\to U(1)$ giving a deformation-retraction-compatible isomorphism on ${\pi }_1$ equals ${\pi }_1(U(1))=\mathbb{Z}$. Connected covers of $\mathrm{Sp}(2n)$ correspond to subgroups of ${\pi }_1(\mathrm{Sp}(2n))=\mathbb{Z}$ via covering-space theory; the index-$2$ subgroup $2\mathbb{Z}$ is unique, defining a unique connected double cover $\mathrm{Mp}(2n)\to \mathrm{Sp}(2n)$. The metaplectic group inherits a Lie-group structure from this covering and acts on $L^2({\mathbb{R}}^n)$ via the Shale-Weil representation ^{[Weil 1964; Sternberg 1975]}. $\square$

Proposition (BKS pairing is unitary for transverse polarisations). Let ${\mathcal{P}}_1,{\mathcal{P}}_2$ be two transverse polarisations on $(M,\omega )$ with ${\mathcal{P}}_1\cap \overline{{\mathcal{P}}_2}=0$ everywhere, and let ${\mathcal{H}}_{{\mathcal{P}}_i}$ be the corresponding quantum Hilbert spaces with metaplectic correction. The BKS pairing ${\mathcal{H}}_{{\mathcal{P}}_1}\to {\mathcal{H}}_{{\mathcal{P}}_2}$ is a unitary isomorphism.

Proof sketch. For transverse polarisations, the leaf space of ${\mathcal{P}}_1\cap \overline{{\mathcal{P}}_2}$ degenerates to $M$ itself (no integration over leaves needed). The pairing kernel is the half-form intertwiner constructed from the symplectic structure: at each $x\in M$, the symplectic pairing ${\omega }_x:{\mathcal{P}}_{1,x}\otimes \overline{{\mathcal{P}}_{2,x}}\to \mathbb{C}$ is non-degenerate (transversality), and induces a canonical isomorphism $|\det {\omega }_x{|}^{1/2}:|\Lambda {|}^{1/2}({\mathcal{P}}_{1,x})\otimes |\Lambda {|}^{1/2}(\overline{{\mathcal{P}}_{2,x}})\to \mathbb{C}$. Globally, this assembles into a pairing kernel.

To show unitarity, compute the operator product $K^{\ast }K$ where $K$ is the BKS kernel. The product is a kernel on ${\mathcal{H}}_{{\mathcal{P}}_1}$ obtained by integrating the kernel against its conjugate over the ${\mathcal{P}}_2$-fibration. By Parseval-Plancherel for the local model of the transverse pairing (which is a Fourier transform in the Heisenberg-group setting), the product kernel is the identity. Hence $K$ is unitary ^{[Blattner 1973; Sternberg 1975; Bates-Weinstein 1997 Ch. 8]}. For the position-momentum pair on $T^{\ast }{\mathbb{R}}^n$, the BKS kernel is the Fourier transform, and unitarity is Plancherel's theorem. $\square$

Connections Master

• Prequantum line bundle 05.11.01. The polarisation construction takes as input the prequantum bundle and produces the quantum Hilbert space as polarised sections tensored with half-densities. The integrality condition from the prequantum unit is necessary for the present construction; the polarisation choice is the additional data that turns the prequantum data into a quantum theory.

• Lagrangian Grassmannian and the universal Maslov class 05.12.01. The Maslov class $\mu \in H^1(\Lambda (n);\mathbb{Z})$ is the obstruction to a global square root of the determinant on the Lagrangian Grassmannian, equivalently the obstruction to a metaplectic structure on a symplectic manifold. The metaplectic correction in geometric quantisation is the global resolution of this obstruction at the level of polarised sections; the BKS pairing inherits a Maslov phase factor at every caustic in the leaf space.

• Prequantisation of the spin coadjoint orbit 05.11.02. The Kähler polarisation on ${\mathcal{O}}_j\cong {\mathbb{CP}}^1$ produces the spin-$j$ representation as polarised sections of $\mathcal{O}(2j)$; with the metaplectic correction $K_{{\mathbb{CP}}^1}^{1/2}=\mathcal{O}(-1)$, the corrected bundle is $\mathcal{O}(2j-1)$ with $2j$-dimensional section space — the spin-$(j-1/2)$ representation. The metaplectic shift converts integer spin to half-integer spin, recovering the experimentally observed spin-$1/2$ spectrum from a geometric construction.

• Hamiltonian vector field and Poisson bracket 05.02.01. The polarisation projector takes the prequantum representation $\hat{f}=-i\hslash {\nabla }_{X_f}+f$ on the full prequantum section space to its restriction on the polarised subspace ${\mathcal{H}}_{\mathcal{P}}$. The commutator $[\hat{f},\hat{g}]=-i\hslash \widehat{\{f,g\}}$ from the prequantum unit descends to the polarised representation when $\hat{f},\hat{g}$ preserve the polarisation — a condition the quantisation map imposes selectively, producing the Kostant-Souriau quantisation of observables that are polynomially adapted to the polarisation.

• CCR algebra and Stone-von Neumann 12.14.01. The BKS pairing between the position and momentum polarisations on $T^{\ast }{\mathbb{R}}^n$ is the Fourier transform, the canonical Stone-von Neumann unitary intertwiner between the Schrödinger and momentum representations of the Heisenberg algebra. Geometric quantisation provides a geometric proof of Stone-von Neumann: the BKS pairing between any two transverse polarisations is unitary, and the parameter space of polarisations is connected, so all Hilbert spaces obtained are canonically isomorphic up to the metaplectic sign.

• Hermitian metric and Chern connection 03.05.20. The Kähler polarisation on a Kähler manifold $M$ produces polarised sections as holomorphic sections of $L\otimes K_M^{1/2}$, where $K_M^{1/2}$ is the half-canonical bundle supplying the metaplectic correction. The Chern connection on $L$ restricts to the $\bar{\partial }$-operator on holomorphic sections; the half-canonical structure on $K_M^{1/2}$ requires $w_2(M)=0$ in $H^2(M;\mathbb{Z}/2)$, equivalent to a spin${}^c$ structure compatible with the symplectic form.

Historical & philosophical context Master

The polarisation construction was introduced by Jean-Marie Souriau in the 1970 monograph Structure des Systèmes Dynamiques (Dunod, Ch. V-VI) and independently by Bertram Kostant in 1970 in Quantization and Unitary Representations, Lecture Notes in Mathematics 170 §3-§5 ^{[Souriau 1970; Kostant 1970]}. Both authors recognised that the prequantum line bundle alone produces a section space too large to serve as the quantum Hilbert space and that a polarisation — an integrable Lagrangian distribution — provides the required half-dimensional reduction. Kostant's framing emphasised the representation-theoretic content: polarised sections of the prequantum bundle on an integral coadjoint orbit produce irreducible representations of the Lie group, completing the Borel-Weil construction to non-compact and non-Kähler cases.

The half-density and metaplectic correction were developed by Robert Blattner, Kostant, and Shlomo Sternberg in a sequence of papers from 1973 to 1975. The signature reference is Blattner's 1973 Quantization in representation theory (Proc. Symp. Pure Math. 26) ^{[Blattner 1973]}, introducing the pairing between Hilbert spaces of distinct polarisations now called the Blattner-Kostant-Sternberg pairing. Sternberg's 1975 On the role of field theories in our physical conception of geometry (Lecture Notes in Physics 50) refined the half-form intertwiner and provided the connection to the Shale-Weil representation of the metaplectic group ^{[Sternberg 1975]}. The metaplectic group itself was introduced earlier by André Weil in 1964 (Sur certains groupes d'opérateurs unitaires, Acta Math. 111) in the context of theta-functions and automorphic forms — Weil's construction of the metaplectic group as the unique connected double cover of $\mathrm{Sp}(2n,\mathbb{R})$ is what makes the geometric-quantisation metaplectic correction unambiguous.

Victor Guillemin and Sternberg's 1977 monograph Geometric Asymptotics (AMS Math Surveys 14) collected the polarisation-half-density-metaplectic apparatus into a unified treatment connecting geometric quantisation to semiclassical asymptotics and Fourier integral operator theory, identifying the Maslov phase as the WKB correction at every caustic ^{[Guillemin-Sternberg 1977]}. Nicholas Woodhouse's 1980 monograph Geometric Quantization (Oxford University Press; 2nd ed. 1992) provided the canonical textbook account, and Sean Bates and Alan Weinstein's 1997 Lectures on the Geometry of Quantization (Berkeley Math. Lecture Notes 8) offered the modern functorial perspective grounded in symplectic groupoids. Kirillov's EMS Volume 4 Part II (2001) integrated the polarisation machinery into the orbit method, completing the pedagogical arc from Souriau-Kostant 1970 to the modern research literature.

Bibliography Master

@incollection{Kostant1970,
  author    = {Kostant, Bertram},
  title     = {Quantization and unitary representations},
  booktitle = {Lectures in Modern Analysis and Applications III},
  series    = {Lecture Notes in Mathematics},
  volume    = {170},
  publisher = {Springer},
  year      = {1970},
  pages     = {87--208},
}

@book{Souriau1970pol,
  author    = {Souriau, Jean-Marie},
  title     = {Structure des Syst\`emes Dynamiques},
  publisher = {Dunod},
  address   = {Paris},
  year      = {1970},
}

@inproceedings{Blattner1973,
  author    = {Blattner, Robert J.},
  title     = {Quantization in representation theory},
  booktitle = {Harmonic Analysis on Homogeneous Spaces (Williamstown 1972)},
  series    = {Proceedings of Symposia in Pure Mathematics},
  volume    = {26},
  publisher = {American Mathematical Society},
  year      = {1973},
  pages     = {147--165},
}

@incollection{Sternberg1975,
  author    = {Sternberg, Shlomo},
  title     = {On the role of field theories in our physical conception of geometry},
  booktitle = {Differential Geometric Methods in Mathematical Physics II},
  series    = {Lecture Notes in Physics},
  volume    = {50},
  publisher = {Springer},
  year      = {1975},
  pages     = {1--80},
}

@book{Woodhouse1992pol,
  author    = {Woodhouse, Nicholas M. J.},
  title     = {Geometric Quantization},
  publisher = {Oxford University Press},
  year      = {1992},
  edition   = {2nd},
}

@book{BatesWeinstein1997pol,
  author    = {Bates, Sean and Weinstein, Alan},
  title     = {Lectures on the Geometry of Quantization},
  series    = {Berkeley Mathematics Lecture Notes},
  volume    = {8},
  publisher = {American Mathematical Society},
  year      = {1997},
}

@book{GuilleminSternberg1977pol,
  author    = {Guillemin, Victor and Sternberg, Shlomo},
  title     = {Geometric Asymptotics},
  series    = {Mathematical Surveys},
  volume    = {14},
  publisher = {American Mathematical Society},
  year      = {1977},
}

@article{Weil1964,
  author    = {Weil, Andr\'e},
  title     = {Sur certains groupes d'op\'erateurs unitaires},
  journal   = {Acta Mathematica},
  volume    = {111},
  year      = {1964},
  pages     = {143--211},
}

@incollection{KirillovEMSpol,
  author    = {Kirillov, Alexandre A.},
  title     = {Geometric Quantization},
  booktitle = {Dynamical Systems IV: Symplectic Geometry and Its Applications},
  editor    = {Arnol'd, V. I. and Novikov, S. P.},
  series    = {Encyclopaedia of Mathematical Sciences},
  volume    = {4},
  publisher = {Springer},
  year      = {2001},
  edition   = {2nd},
}

@article{Maslov1965pol,
  author    = {Maslov, V. P.},
  title     = {Theory of Perturbations and Asymptotic Methods},
  publisher = {Izdat. Moskov. Univ.},
  address   = {Moscow},
  year      = {1965},
}