Geometric quantization. A classical symplectic manifold supplies observables as real functions, their Poisson bracket, and Hamiltonian flows. Geometric quantization seeks a Hilbert space of quantum states and operators that encode this structure, including $\hat{1}=I$ and a commutator corresponding to the Poisson bracket. A direct prescription cannot make every classical product become an operator product without qualification; operator ordering and the choice of physically admissible observables matter. The aim is to build the state space and operators from the geometry rather than from a preferred coordinate chart.

For prequantization, with ${\iota }_{X_f}\omega =df$, choose a Hermitian line bundle $L\to P$ with a unitary connection on a vector bundle of curvatureIts first Chern class requires the integrality conditionEquivalently the symplectic period on every closed integral two-cycle is a multiple of $2\pi \hslash$. Conversely an integral class supplies a line bundle, and a unitary connection can be adjusted by a one-form to have the required curvature. Locally choose $\theta$ with $\omega =-d\theta$; then $\nabla =d-i\theta /\hslash$. Potentials differing by an exact one-form change the local phase of the section. Integrality is precisely what makes these local descriptions glue consistently.

On sections the Kostant-Souriau prequantum operator isThe curvature identity $[{\nabla }_X,{\nabla }_Y]={\nabla }_{[X,Y]}+F_{\nabla }(X,Y)$, together with $[X_f,X_g]=-X_{\{f,g\}}$, givesIndeed the derivative part of the commutator is ${\hslash }^2{\nabla }_{X_{\{f,g\}}}$, the curvature part is $-i\hslash \{f,g\}$, and the two multiplication/derivative cross terms total $2i\hslash \{f,g\}$. These combine to the displayed operator identity. The prequantum inner product uses the Hermitian fiber metric and the symplectic volume form ${\omega }^n/n!$, with appropriate square-integrability and operator-domain conditions.

This is not yet the desired quantum state space: a prequantum section depends on all $2n$ phase-space variables, whereas a wavefunction normally depends on only $n$ configuration variables. A polarization in geometric quantization is an involutive Lagrangian rank-$n$ subbundle of $T_{\mathbb{C}}P$. Polarized sections obey ${\nabla }_{X}s=0$ along that distribution. The symplectic form vanishes on it, so its restricted curvature vanishes and these differential conditions are locally compatible.

For $T^{\ast }Q$, the vertical polarization is spanned locally by ${\partial }_{p_i}$. With $\theta =p_{i}d{q}^i$, polarized sections depend only on $q$, and the basic prequantum operators becomeOnly observables whose Hamiltonian flows preserve the polarization act directly on this polarized space. A generic quadratic kinetic energy does not preserve the vertical polarization, so obtaining its quantum operator needs an additional construction rather than simply restricting every prequantum operator. A compatible complex structure can instead provide a Kähler polarization, yielding holomorphic-type states, as for oscillator variables. The Blattner-Kostant-Sternberg pairing compares suitable state spaces of different polarizations and can implement some transported observables.

Finding a useful global polarization is a genuine problem: it may not exist as a nonsingular real foliation; leaf holonomy can obstruct globally nonzero flat sections; singular leaves and compact leaves impose further conditions. A metaplectic correction uses half-forms to improve the inner product and coordinate transformation law and account for familiar zero-point shifts, when its additional square-root data exist. Thus prequantum integrality, polarization, Hilbert-space completion and observable domains are separate steps, each with geometric constraints.

Applications of Stokes's theorem. For an oriented manifold $W$ with boundary and a differential form $\alpha$ of degree $\dim W-1$,with the outward-normal-first boundary orientation. Locally the identity is integration by parts in a half-space; a partition of unity reduces the general manifold to that local calculation, and contributions along internal patch boundaries cancel. The Generalized Stokes theorem unifies the fundamental theorem of calculus, circulation/flux formulas and the relation between local conservation laws and conserved integrated charges.

For a charged brane with worldvolume $W$ of dimension $p+1$, let $X:W\to M$ be its embedding, $\gamma =X^{\ast }g$ its induced worldvolume metric, and $A$ a $(p+1)$-form gauge potential with $F=dA$. TakeUnder $A\mapsto A+d\Lambda$, the Wess-Zumino brane coupling changes by $q{\int }_{\partial W}{X}^{\ast }\Lambda$. It is invariant for a closed worldvolume, or for suitable boundary conditions; open branes can require compensating boundary terms.

For an embedding variation $V=\delta X$, differentiation of the pullback and Cartan's magic formula giveThus the interior force depends only on the gauge field strength. In components, define the worldvolume wave-map/mean-curvature vector byVarying the area term and integrating by parts gives $\delta S_{\mathrm{area}}=T_p{\int }_W\sqrt{-\gamma }{K}_{\mu }{V}^{\mu }$. Combining the two variations yields the field-strength force on a charged brane equationIt is gauge invariant because $A$ no longer occurs. Its tangential projection vanishes identically by antisymmetry, consistently with worldvolume reparametrization invariance. At $p=0$, the chosen Lorentzian worldline convention has $K=-\ddot{X}$ in proper time and recovers the usual Lorentz-force sign.

For topological conservation, let the spatial manifold compactify to a closed oriented $n$-manifold $\Sigma$, and let a field $U:\Sigma \to N$ take values in a closed oriented $n$-dimensional target with normalized volume form $\Omega$, ${\int }_N\Omega =1$. ThenThis is the degree of a map between oriented manifolds: for a regular value it is the signed sum of inverse images. To relate the integral to that count, use a normalized target form supported near a regular value and change variables on its inverse-image neighborhoods. Any two normalized top forms on a connected closed oriented target differ by an exact form, whose pulled-back integral vanishes by the Generalized Stokes theorem. Hence the value is independent of this localization and equals the integer signed count.

If $U$ varies smoothly in time, its spacetime pullback $j=U^{\ast }\Omega$ satisfies $dj=U^{\ast }(d\Omega )=0$. Apply Stokes theorem to the spacetime slab between two slices: the difference of their charges is minus the flux through the spatial boundary. With no flux at infinity, $Q$ is conserved independently of the field equations. Equivalently a smooth homotopy cannot change the degree. Singularities, altered boundary conditions or nonzero boundary flux are the ways the argument can fail.

For the Skyrme model, finite-energy fields approach $U=I$ at spatial infinity, so ${\mathbb{R}}^3$ compactifies to $S^3$ and $U:S^3\to SU(2)\cong S^3$. With the conventional target orientation chooseThis is a normalized closed target volume form. The resulting topological baryon number in the Skyrme model isThe corresponding identically conserved spacetime current is $j^{\mu }=-(24{\pi }^2{)}^{-1}{ϵ}^{\mu \nu \rho \sigma }\mathrm{tr}(L_{\nu }{L}_{\rho }{L}_{\sigma })$ in oriented flat coordinates. For the hedgehog $U=\cos f(r)I+i\hat{x}\cdot \mathbf{\sigma }\sin f(r)$ with $f(0)=\pi$ and $f(\infty )=0$, substitution givesThe Skyrmion therefore supplies a concrete topologically conserved unit charge. Topological conservation alone does not guarantee a stable finite-size static solution: the positive Skyrme term, with four derivatives, opposes collapse of the two-derivative energy. Under size rescaling, these contributions behave as $E_2\propto \lambda$ and $E_4\propto {\lambda }^{-1}$, allowing a nonzero equilibrium size while preserving the degree.

Use ${\iota }_{X_f}\omega =df$ and $\{f,g\}=\omega (X_f,X_g)$. For an actual infinitesimal left Lie group action, define ${\xi }_P(p)={\frac{d}{dt}|}_0\exp (t\xi )\cdot p$. Its fundamental vector fields obey $[{\xi }_P,{\eta }_P]=-[\xi ,\eta {]}_P$ with this convention.

Existence and its global obstruction. Preservation of the symplectic form gives ${\mathcal{L}}_{{\xi }_P}\omega =0$. By Cartan's magic formula, ${\iota }_{{\xi }_P}\omega$ is closed. Locally it equals $d{\mu }_{\xi }$; globally this requires its class in $H_{\mathrm{dR}}^1(P)$ to vanish. If it does for every $\xi$, choose potentials linearly in $\xi$ and defineThis gives the component Hamiltonians of a weakly Hamiltonian action. The assumption in the question does not alone guarantee a global moment map: on $T^2$ with $\omega =dx\wedge dy$, translation in $x$ preserves $\omega$, but its contraction is $dy$, whose integral around the $y$ circle is nonzero. It is not an exact one-form. Thus no global real-valued component Hamiltonian exists for that circle action. The intended construction is valid locally, or globally with this exactness condition, for example if $H_{\mathrm{dR}}^1(P)=0$.

Matching the Lie algebra. On each connected component, the moment-map equivariance obstruction is the constantIndeed, the identities $[X_f,X_g]=-X_{\{f,g\}}$ and $[{\xi }_P,{\eta }_P]=-[\xi ,\eta {]}_P$ show that its differential vanishes. The Jacobi identity for the Poisson bracket saysChanging ${\mu }_{\xi }$ by a linear constant $b(\xi )$ changes $c$ to $c-b([\xi ,\eta ])$. Consequently the necessary and sufficient infinitesimal condition is the cocycle must be a coboundary, so that the constants can be chosen withEquivalently its class in $H^2(\mathfrak{g},\mathbb{R})$ vanishes. For a connected group this infinitesimal equivariance integrates to equivariance under its coadjoint action. A disconnected group can impose additional conditions. Here “coincides” means representing the specified Lie brackets; it does not guarantee injectivity when the group action has ineffective generators.

Proof using the Killing form. A nondegenerate Killing form $B$ makes $\mathfrak{g}$ semisimple. First, it is perfect: the $B$-orthogonal complement of $[\mathfrak{g},\mathfrak{g}]$ is the center by invariance, and the center is zero by nondegeneracy. For symplectic fields,by Cartan's magic formula and the antihomomorphism sign. Every commutator therefore has an exact contraction. Perfectness supplies component Hamiltonians for every generator, even when $P$ is not simply connected: this is Hamiltonian existence for a semisimple symplectic action.

It remains to eliminate the cocycle, not merely to name an existence theorem. Define a linear operator $D$ on $\mathfrak{g}$ by $B(D\xi ,\eta )=c(\xi ,\eta )$. Antisymmetry of $c$ makes $D$ skew for $B$. Combining its cocycle identity with invariance of $B$ givesChoose $Z\in \mathfrak{g}$ by $B(Z,\xi )=\mathrm{tr}(D{\mathrm{ad}}_{\xi })$, possible because $B$ is nondegenerate. Since $D$ is a derivation, $[D,{\mathrm{ad}}_{\xi }]={\mathrm{ad}}_{D\xi }$. Cyclicity of the matrix trace then givesThus $D={\mathrm{ad}}_Z$, and $c(\xi ,\eta )=B(Z,[\xi ,\eta ])$. Taking $b(\xi )=B(Z,\xi )$ removes $c$, proving the required condition. This applies to semisimple Lie groups, including $SU(n)$ for $n\ge 2$, $SO(3)$ and $SL(n,\mathbb{R})$ for $n\ge 2$. It is nondegeneracy, not positive definiteness, that matters. Groups with a nonzero central Lie algebra, such as $U(3)$, do not satisfy this Killing-form criterion, although particular actions can still have equivariant moment maps.

The isotropic oscillator. On $T^{\ast }{\mathbb{R}}^3$, putThe action $z\mapsto Uz$, $U\in U(3)$, preserves the Hermitian norm and the symplectic form, and therefore preserves $H$. A Hermitian matrix $A$ parametrizes the Lie algebra generator $\xi =-iA$. Its unitary moment map of an isotropic oscillator isSince $\{z_j,{\bar{z}}_k\}=-i{\delta }_{jk}$, direct differentiation yieldsThe matrix $-i[A,B]$ is Hermitian and corresponds precisely to $[{\xi }_A,{\xi }_B]$; no central defect remains. The full moment map can be identified, using the pairing with Hermitian matrices, with the rank-one matrix $z{z}^{†}$. The central generator $A=I$ gives ${\mu }_I=H$, whose flow is $z(t)=e^{-it}z(0)$. Globally $U(3)=(SU(3)\times U(1))/{\mathbb{Z}}_3$, so the “$U(1)$ factor” refers to this central circle, rather than a literal direct-product decomposition.

The real orthogonal subgroup acts geometrically by $(x,p)\mapsto (Rx,Rp)$, the cotangent lift of a diffeomorphism of ${\mathbb{R}}^3$. If $K$ is real antisymmetric and $A=iK$, then ${\mu }_A=-x^{T}Kp$. For $Kv=\Omega \times v$, this is $\Omega \cdot (x\times p)$, so its three components are the usual angular momentum $L_i={ϵ}_{ijk}{x}_j{p}_k$.

The remaining six independent Hermitian directions can be taken as real symmetric matrices. Their moment functions spanThe diagonal ones are the individual oscillator energies; their sum is $H$. The off-diagonal ones correlate two oscillator modes. Their flows mix positions and momenta and are not cotangent lifts of the obvious spatial rotation action: for a real symmetric $A$, $\dot{x}=Ap$ and $\dot{p}=-Ax$, rather than $\dot{x}=Kx$, $\dot{p}=Kp$. All nine quadratic functions commute with $H$, since the entire unitary group action preserves the oscillator Hamiltonian.