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Geometric quantization
Geometric quantization
Geometric quantization
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In mathematical physics, geometric quantization is a mathematical approach to defining a quantum theory corresponding to a given classical theory. It attempts to carry out quantization, for which there is in general no exact recipe, in such a way that certain analogies between the classical theory and the quantum theory remain manifest. For example, the similarity between the Heisenberg equation in the Heisenberg picture of quantum mechanics and the Hamilton equation in classical physics should be built in.

Origins

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One of the earliest attempts at a natural quantization was Weyl quantization, proposed by Hermann Weyl in 1927. Here, an attempt is made to associate a quantum-mechanical observable (a self-adjoint operator on a Hilbert space) with a real-valued function on classical phase space. The position and momentum in this phase space are mapped to the generators of the Heisenberg group, and the Hilbert space appears as a group representation of the Heisenberg group. In 1946, H. J. Groenewold considered the product of a pair of such observables and asked what the corresponding function would be on the classical phase space.[1] This led him to discover the phase-space star-product of a pair of functions.

The modern theory of geometric quantization was developed by Bertram Kostant and Jean-Marie Souriau in the 1970s. One of the motivations of the theory was to understand and generalize Alexandre Kirillov's orbit method in representation theory.

Types

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The geometric quantization procedure falls into the following three steps: prequantization, polarization, and metaplectic correction. Prequantization produces a natural Hilbert space together with a quantization procedure for observables that exactly transforms Poisson brackets on the classical side into commutators on the quantum side. Nevertheless, the prequantum Hilbert space is generally understood to be "too big".[2] The idea is that one should then select a Poisson-commuting set of n variables on the 2n-dimensional phase space and consider functions (or, more properly, sections) that depend only on these n variables. The n variables can be either real-valued, resulting in a position-style Hilbert space, or complex analytic, producing something like the Segal–Bargmann space.[a] A polarization is a coordinate-independent description of such a choice of n Poisson-commuting functions. The metaplectic correction (also known as the half-form correction) is a technical modification of the above procedure that is necessary in the case of real polarizations and often convenient for complex polarizations.

Prequantization

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Suppose is a symplectic manifold with symplectic form . Suppose at first that is exact, meaning that there is a globally defined symplectic potential with . We can consider the "prequantum Hilbert space" of square-integrable functions on (with respect to the Liouville volume measure). For each smooth function on , we can define the Kostant–Souriau prequantum operator where is the Hamiltonian vector field associated to .

More generally, suppose has the property that the integral of over any closed surface is an integer. Then we can construct a line bundle with connection whose curvature 2-form is . In that case, the prequantum Hilbert space is the space of square-integrable sections of , and we replace the formula for above with with the connection. The prequantum operators satisfy for all smooth functions and .[3]

The construction of the preceding Hilbert space and the operators is known as prequantization.

Polarization

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The next step in the process of geometric quantization is the choice of a polarization. A polarization is a choice at each point in a Lagrangian subspace of the complexified tangent space of . The subspaces should form an integrable distribution, meaning that the commutator of two vector fields lying in the subspace at each point should also lie in the subspace at each point. The quantum (as opposed to prequantum) Hilbert space is the space of sections of that are covariantly constant in the direction of the polarization.[4][b] The idea is that in the quantum Hilbert space, the sections should be functions of only variables on the -dimensional classical phase space.

If is a function for which the associated Hamiltonian flow preserves the polarization, then will preserve the quantum Hilbert space.[5] The assumption that the flow of preserve the polarization is a strong one. Typically, not very many functions will satisfy this assumption.

Half-form correction

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The half-form correction—also known as the metaplectic correction—is a technical modification to the above procedure that is necessary in the case of real polarizations to obtain a nonzero quantum Hilbert space; it is also often useful in the complex case. The line bundle is replaced by the tensor product of with the square root of the canonical bundle of the polarization. In the case of the vertical polarization, for example, instead of considering functions of that are independent of , one considers objects of the form . The formula for must then be supplemented by an additional Lie derivative term.[6] In the case of a complex polarization on the plane, for example, the half-form correction allows the quantization of the harmonic oscillator to reproduce the standard quantum mechanical formula for the energies, , with the "" coming courtesy of the half-forms.[7]

Poisson manifolds

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Geometric quantization of Poisson manifolds and symplectic foliations also is developed. For instance, this is the case of partially integrable and superintegrable Hamiltonian systems and non-autonomous mechanics.

Example

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In the case that the symplectic manifold is the 2-sphere, it can be realized as a coadjoint orbit in . Assuming that the area of the sphere is an integer multiple of , we can perform geometric quantization and the resulting Hilbert space carries an irreducible representation of SU(2). In the case that the area of the sphere is , we obtain the two-dimensional spin-1/2 representation.

Generalization

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More generally, this technique leads to deformation quantization, where the ★-product is taken to be a deformation of the algebra of functions on a symplectic manifold or Poisson manifold. However, as a natural quantization scheme (a functor), Weyl's map is not satisfactory. For example, the Weyl map of the classical angular-momentum-squared is not just the quantum angular momentum squared operator, but it further contains a constant term 3ħ2/2. (This extra term is actually physically significant, since it accounts for the nonvanishing angular momentum of the ground-state Bohr orbit in the hydrogen atom.[8]) As a mere representation change, however, Weyl's map underlies the alternate phase-space formulation of conventional quantum mechanics.

See also

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Notes

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Citations

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  1. ^ Groenewold 1946, pp. 405–460.
  2. ^ Hall 2013, Section 22.3.
  3. ^ Hall 2013, Theorem 23.14.
  4. ^ Hall 2013, Section 23.4.
  5. ^ Hall 2013, Theorem 23.24.
  6. ^ Hall 2013, Sections 23.6 and 23.7.
  7. ^ Hall 2013, Example 23.53.
  8. ^ Dahl & Schleich 2002.

Sources

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  • Bates, S; Weinstein, A. (1996). Lectures on the Geometry of Quantization. American Mathematical Society. ISBN 978-0-8218-0798-9.
  • Dahl, J.; Schleich, W. (2002). "Concepts of radial and angular kinetic energies". Physical Review A. 65 (2) 022109. arXiv:quant-ph/0110134. Bibcode:2002PhRvA..65b2109D. doi:10.1103/PhysRevA.65.022109. S2CID 39409789.
  • Giachetta, G.; Mangiarotti, L.; Sardanashvily, G. (2005). Geometric and Algebraic Topological Methods in Quantum Mechanics. World Scientific. ISBN 981-256-129-3.
  • Groenewold, H. J. (1946). "On the Principles of elementary quantum mechanics". Physica. 12 (7): 405–460. Bibcode:1946Phy....12..405G. doi:10.1016/S0031-8914(46)80059-4.
  • Hall, B.C. (2013). Quantum Theory for Mathematicians. Graduate Texts in Mathematics. Vol. 267. Springer. ISBN 978-1-4614-7115-8.
  • Kong, K. (2006). From Micro to Macro Quantum Systems, (A Unified Formalism with Superselection Rules and Its Applications). World Scientific. ISBN 978-1-86094-625-7.
  • Śniatycki, J. (1980). Geometric Quantization and Quantum Mechanics. Springer. ISBN 0-387-90469-7.
  • Vaisman, I. (1991). Lectures on the Geometry of Poisson Manifolds. Birkhauser. ISBN 978-3-7643-5016-1.
  • Woodhouse, N.M.J. (1991). Geometric Quantization. Clarendon Press. ISBN 0-19-853673-9.
[edit]
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Geometric quantization

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Geometric quantization is a mathematical framework in physics and that associates a of quantum states and a representation of the algebra of classical observables as self-adjoint operators to a classical modeled as a , employing tools from such as line bundles and polarizations to ensure compatibility with Dirac's conditions. The theory emerged in the mid-20th century as an effort to geometrize the transition from classical to , building on and ; it was independently formulated by Jean-Marie Souriau in his 1966 paper "Quantification géométrique," which introduced prequantization via circle bundles over phase spaces, and by in his 1970 work "Quantization and Unitary Representations," which connected quantization to unitary representations of Lie groups on coadjoint orbits. At its core, geometric quantization proceeds in stages: first, prequantization constructs a Hermitian over the (M, ω) equipped with a connection whose curvature form equals ω/iℏ, yielding a prequantum of global sections and operator representations that satisfy [Â_f, Â_g] = iℏ Â_{{f,g}} for {f,g}; this step, however, produces an infinite-dimensional space unsuitable for physical wave functions. To refine this, a polarization is selected—a maximal positive Lagrangian subbundle of the complexified tangent bundle, such as the holomorphic or real polarization—to restrict the quantum states to sections that are covariantly constant along the polarization directions, effectively reducing the dependence on half the phase space coordinates and yielding a finite-dimensional Hilbert space for compact manifolds. An important correction, half-form quantization, addresses issues with the inner product by incorporating half-density bundles, ensuring unitarity and the correct transformation properties under coordinate changes, particularly for Kähler polarizations where the quantum Hilbert space consists of holomorphic sections of the bundle. Beyond basic mechanics, geometric quantization has profound applications in linking irreducible unitary representations of Lie groups to quantizations of coadjoint orbits via the Kirillov-Kostant-Souriau (KKS) correspondence, influencing areas like integrable systems, , and modern approaches to on curved spacetimes.

Mathematical Background

Symplectic Manifolds

A symplectic manifold is a pair (M,ω)(M, \omega), where MM is a smooth manifold of even dimension 2n2n and ω\omega is a closed, non-degenerate 2-form on MM. The closedness condition means that the exterior derivative satisfies dω=0d\omega = 0, ensuring the form is locally exact in a manner compatible with the manifold's topology. Non-degeneracy implies that for every point p∈Mp \in M and every nonzero tangent vector v∈TpMv \in T_p M, there exists a vector w∈TpMw \in T_p M such that ω(v,w)≠0\omega(v, w) \neq 0, which induces a non-singular pairing on the tangent space. This structure captures the phase space of classical mechanics, where positions and momenta are treated on equal footing. The Darboux theorem asserts that around any point in a , there exist local coordinates (q1,…,qn,p1,…,pn)(q^1, \dots, q^n, p_1, \dots, p_n) in which the symplectic form takes the standard expression ω=∑i=1ndqi∧dpi\omega = \sum_{i=1}^n dq^i \wedge dp_i. These coordinates, often called canonical or Darboux coordinates, highlight the intrinsic geometric uniformity of , as the local form is independent of the specific choice of point. This theorem underscores that lacks local invariants beyond the dimension, unlike where plays a role. The powers of the symplectic form yield a natural volume element on the manifold. Specifically, for a 2n2n-dimensional , the symplectic volume form is given by ωnn!\frac{\omega^n}{n!}, which provides a orientation and measure. This volume form derives the Liouville measure on the , which is invariant under the canonical transformations preserving ω\omega and plays a central role in for computing phase space volumes. A prototypical example of a symplectic manifold is the cotangent bundle T∗QT^*Q of a smooth configuration manifold QQ, equipped with its canonical symplectic form. Here, points in T∗QT^*Q represent positions in QQ paired with momenta in the cotangent spaces, and the symplectic structure arises naturally from the geometry of differentials on QQ. For instance, when Q=RnQ = \mathbb{R}^n, T∗Q=R2nT^*Q = \mathbb{R}^{2n} with the standard symplectic form ∑dqi∧dpi\sum dq^i \wedge dp_i, serving as the phase space for nn free particles.

Hamiltonian Mechanics

In Hamiltonian mechanics, the phase space of a classical system is modeled as a (M,ω)(M, \omega), where the Hamiltonian function H:M→RH: M \to \mathbb{R} serves as the representing the total energy of the system. This function encodes the dynamics, with observables corresponding to smooth functions on MM. The dynamics are governed by the XHX_H, defined through the relation ιXHω=−dH\iota_{X_H} \omega = -dH, where ι\iota denotes the interior product with the symplectic form ω\omega. In (qi,pi)(q^i, p_i) on MM, the components of XHX_H take the explicit form XH=∑i(∂H∂pi∂∂qi−∂H∂qi∂∂pi).X_H = \sum_i \left( \frac{\partial H}{\partial p_i} \frac{\partial}{\partial q^i} - \frac{\partial H}{\partial q^i} \frac{\partial}{\partial p_i} \right). The integral curves of XHX_H generate the of the system, yielding : dqidt=∂H∂pi,dpidt=−∂H∂qi.\frac{dq^i}{dt} = \frac{\partial H}{\partial p_i}, \quad \frac{dp_i}{dt} = -\frac{\partial H}{\partial q^i}. These equations describe how position and coordinates evolve along the flow. A key algebraic structure arises from the Poisson bracket, defined for smooth functions f,g∈C∞(M)f, g \in C^\infty(M) as {f,g}=ω(Xf,Xg)=Xfg=−Xgf\{f, g\} = \omega(X_f, X_g) = X_f g = -X_g f. This bilinear operation satisfies bilinearity, antisymmetry, the Leibniz rule, and the Jacobi identity {f,{g,h}}+{g,{h,f}}+{h,{f,g}}=0\{f, \{g, h\}\} + \{g, \{h, f\}\} + \{h, \{f, g\}\} = 0, endowing C∞(M)C^\infty(M) with the structure of a Lie algebra. The Poisson bracket quantifies the time evolution of observables, with dfdt={f,H}\frac{d f}{dt} = \{f, H\} for any f∈C∞(M)f \in C^\infty(M). The flow ϕt\phi_t generated by XHX_H is a symplectomorphism, preserving the symplectic form ω\omega via ϕt∗ω=ω\phi_t^* \omega = \omega for all tt. This preservation implies conservation laws, such as the invariance of phase space volumes under the flow, as captured by Liouville's theorem, which follows from the fact that Hamiltonian vector fields are divergence-free with respect to the Liouville measure induced by ωn/n!\omega^n / n!.

Historical Development

Early Motivations

Canonical quantization, the standard procedure for transitioning from classical Hamiltonian mechanics to quantum mechanics by promoting coordinates and momenta to operators satisfying commutation relations, encountered significant challenges that motivated a more geometric approach. A primary issue was the operator ordering ambiguity, where classical products of non-commuting phase space variables, such as qpqp, could be mapped to multiple quantum operators like q^p^\hat{q}\hat{p} or p^q^\hat{p}\hat{q}, leading to non-unique Hermitian Hamiltonians and inconsistent results for observables. This ambiguity became particularly acute in systems with nonlinear terms or constraints, as highlighted by the Groenewold-van Hove no-go theorem, which demonstrated that no linear map from classical Poisson algebras to quantum operator algebras could preserve both products and brackets for all functions on phase space. Another limitation arose in curved phase spaces, where canonical quantization failed to uphold the Ehrenfest theorem in its classical form, preventing expectation values from evolving according to classical Hamilton's equations due to the lack of a covariant operator framework. In such spaces, the theorem's derivation assumes flat geometry, and deviations lead to discrepancies between quantum dynamics and classical trajectories, underscoring the need for a quantization method intrinsic to the symplectic structure. Paul Dirac addressed related algebraic foundations by proposing quantization as the passage from the classical Poisson algebra—where {f,g}\{f, g\} denotes the Poisson bracket—to irreducible representations of the corresponding Lie algebra of operators, with commutators [f^,g^]=iℏ{f^,g^}[\hat{f}, \hat{g}] = i\hbar \{\hat{f}, \hat{g}\}, aiming to preserve the structure of classical observables. However, this approach struggled with global consistency on non-trivial manifolds. Hermann Weyl's 1927 quantization scheme attempted to resolve ordering issues through symmetric (Weyl) ordering, mapping classical functions via Fourier transforms to operators on L2(Rn)L^2(\mathbb{R}^n), but it exhibited limitations for systems with non-linear symmetries, failing to produce operators for certain observables like the square of and not fully accommodating curved geometries or groups. In the 1950s, these problems spurred geometric efforts to quantize , with researchers like Peter Bergmann, , John Wheeler, and developing , treating spatial metrics as dynamical variables on curved s to avoid flat-space assumptions, though constraints from invariance complicated operator definitions. Similar geometric insights were applied to motion, where phase space on coadjoint orbits of rotation groups required symplectic structures to capture rotational symmetries, prefiguring later quantization techniques. Representation theory played a crucial role in these early motivations, providing tools to classify irreducible representations of symmetry groups like SU(2) for , ensuring that quantum states transformed correctly under group actions and revealing how classical Poisson structures on orbits correspond to unitary representations. This perspective, rooted in works linking representations to geometry, highlighted the inadequacy of coordinate-based canonical methods for capturing irreducible sectors of symmetry-reduced systems, paving the way for a fully geometric framework.

Formalization by Kostant and Souriau

The formalization of geometric quantization as a rigorous mathematical framework for associating quantum Hilbert spaces to classical symplectic manifolds emerged independently in the late through the contributions of Jean-Marie Souriau and . Souriau first introduced the core concepts in his paper "Quantification géométrique," which formulated prequantization via circle bundles over s. These ideas were detailed and expanded in his book Structure des Systèmes Dynamiques (published in French by Dunod and later translated into English as Structure of Dynamical Systems: A Symplectic View of Physics), which presented prequantum bundles as a geometric tool to encode the structure of dynamical systems into a over the , equipped with a connection whose matches the symplectic form up to a factor related to Planck's constant. This approach emphasized the underlying and provided a pathway to quantization by associating observables to vector fields and sections of the bundle. Concurrently, Kostant developed a parallel geometric perspective in his 1970 paper "Quantization and Unitary Representations," published in the Lecture Notes in Mathematics series by Springer, where he formalized prequantization and extended it to construct unitary representations of groups. Kostant's method integrated Kirillov's orbit method, which classifies unitary representations via coadjoint orbits in the dual of a , by applying geometric quantization directly to these orbits as symplectic manifolds. This framework highlighted the role of symplectic reduction and coadjoint actions in bridging classical Poisson structures to quantum operators, particularly for semisimple groups. By the early 1970s, parallel developments by Kostant and Souriau converged on the crucial notion of polarization, a choice of complex subbundle of the that reduces the dimensionality of the prequantum to yield a true quantum of half the dimension. This step addressed limitations in prequantization by selecting Lagrangian submanifolds compatible with the symplectic structure, enabling the construction of square-integrable sections as wave functions. Initial applications focused on quantizing coadjoint orbits of compact Lie groups, where the resulting realized irreducible unitary representations, thus linking geometric quantization to and verifying its consistency with known quantum mechanical examples like the .

Prequantization

Line Bundles and Connections

In geometric quantization, the prequantum structure begins with the construction of a L→ML \to M over the (M,ω)(M, \omega), equipped with a connection ∇\nabla whose satisfies F∇=−iωF_\nabla = -i \omega (with ℏ=1\hbar = 1). This condition ensures that the connection encodes the symplectic geometry, as the 2-form F∇F_\nabla is a global section of Λ2T∗M⊗End⁡(L)\Lambda^2 T^*M \otimes \operatorname{End}(L) representing the infinitesimal holonomy, directly tying the bundle's geometry to ω\omega. Such a prequantum line bundle exists if and only if the cohomology class [ω]/2π[\omega]/2\pi lies in the integral second cohomology group H2(M,Z)H^2(M, \mathbb{Z}), known as the integrality or prequantizability condition. This topological obstruction guarantees that ω\omega can be represented by the curvature of a connection on a , allowing the classical to be "quantized" at the prequantum level. Locally, over an open cover {Uα}\{U_\alpha\} of MM, the bundle LL admits trivializations with transition functions gαβ:Uα∩Uβ→S1g_{\alpha\beta}: U_\alpha \cap U_\beta \to S^1 given by eiθαβe^{i \theta_{\alpha\beta}}, where the θαβ\theta_{\alpha\beta} are real-valued smooth functions satisfying the cocycle condition θαβ+θβγ−θαγ=2πnαβγ\theta_{\alpha\beta} + \theta_{\beta\gamma} - \theta_{\alpha\gamma} = 2\pi n_{\alpha\beta\gamma} for some integer-valued function nαβγn_{\alpha\beta\gamma} on triple intersections. These transition functions ensure the bundle is well-defined globally, with the Hermitian metric preserving unitarity (|g_{\alpha\beta}| = 1), and the connection ∇\nabla is specified locally by 1-forms θα\theta_\alpha on each UαU_\alpha such that the curvature matches dθα=−iωd\theta_\alpha = -i \omega on UαU_\alpha. The connection arises from a canonical 1-form θ\theta on the frame bundle L+L^+ (the principal U(1)U(1)-bundle associated to LL), which is pulled back to define the covariant derivative; specifically, for a vector field XX and section ss, ∇Xs=Xs+i(s∗θ)(X)s\nabla_X s = X s + i (s^* \theta)(X) s, where s∗θs^* \theta is the pullback. This θ\theta is the tautological form on L+L^+ satisfying θ(ξ)=1\theta(\xi) = 1 for fundamental vector fields ξ\xi generating the U(1)U(1)-action. A example occurs on the R2n\mathbb{R}^{2n} with coordinates (qi,pi)(q^i, p_i) and standard symplectic form ω=∑dpi∧dqi\omega = \sum dp_i \wedge dq^i; here, LL is trivialized globally, and the connection is given by the Liouville 1-form θ=∑pidqi\theta = \sum p_i dq^i, whose yields dθ=ωd\theta = \omega, satisfying the curvature condition up to the factor −i-i.

Prequantum Operators

In geometric quantization, the prequantum operators act on the space of sections of the prequantum L→ML \to M introduced in the preceding section on line bundles and connections. The prequantum is the space L2(M,L)L^2(M, L) of square-integrable sections of LL, formed by completing the pre-Hilbert space of compactly supported smooth sections Γc(M,L)\Gamma_c(M, L) with respect to the inner product ⟨ψ1,ψ2⟩=∫M⟨ψ1,ψ2⟩L ωnn!\langle \psi_1, \psi_2 \rangle = \int_M \langle \psi_1, \psi_2 \rangle_L \, \frac{\omega^n}{n!}, where ⟨⋅,⋅⟩L\langle \cdot, \cdot \rangle_L is the Hermitian metric on LL and ω\omega is the symplectic form on the manifold MM. The prequantum quantization map Q:C∞(M)→End(L2(M,L))Q: C^\infty(M) \to \mathrm{End}(L^2(M, L)) associates to each smooth classical observable f∈C∞(M)f \in C^\infty(M) a Q(f)Q(f) defined by Q(f)ψ=−i∇Xfψ+fψQ(f) \psi = -i \nabla_{X_f} \psi + f \psi for ψ∈Γ(M,L)\psi \in \Gamma(M, L), where XfX_f is the of ff satisfying df=−ιXfωdf = -\iota_{X_f} \omega, and ∇\nabla is the connection on LL with curvature form curv(∇)=−iω\mathrm{curv}(\nabla) = -i \omega (in units where ℏ=1\hbar = 1). The ∇X\nabla_X along a XX acts on sections as ∇Xψ=Xψ+(X⌟A)ψ\nabla_X \psi = X \psi + (X \lrcorner A) \psi, where AA is the local connection 1-form representing ∇\nabla, ensuring that Q(f)Q(f) is well-defined globally and independent of local trivializations of LL. These operators preserve the classical in the quantum setting: for smooth functions f,g∈C∞(M)f, g \in C^\infty(M), [Q(f),Q(g)]=i{f,g},[Q(f), Q(g)] = i \{f, g\}, where {f,g}\{f, g\} is the on MM, reproducing the Dirac quantization condition [Q(f),Q(g)]=i{f,g}Q(1)[Q(f), Q(g)] = i \{f, g\} Q(1) with Q(1)Q(1) the identity operator. Additionally, the prequantum structure provides a unitary representation of the Hamiltonian flows on L2(M,L)L^2(M, L). The flow ϕtf\phi_t^f generated by XfX_f acts unitarily via with respect to ∇\nabla: if Utfψ=(ϕtf)∗ψU_t^f \psi = (\phi_t^f)^* \psi denotes the pullback of sections, then UtfU_t^f is unitary and satisfies ddtUtf=−iQ(f)Utf\frac{d}{dt} U_t^f = -i Q(f) U_t^f, intertwining the classical dynamics with quantum .

Polarization and Quantization

Choice of Polarization

In geometric quantization, the choice of polarization reduces the prequantum by selecting a subspace of sections that correspond to quantum wave functions, resolving the overcompleteness inherent in prequantization. A polarization PP is defined as an integrable Lagrangian subbundle of the complexified TM⊗CTM \otimes \mathbb{C}, which is maximally with respect to the complexified symplectic form ωC\omega_\mathbb{C}. This integrability ensures that PP is locally spanned by complete vector fields, allowing for a consistent of the , while the maximal isotropy condition implies that PP has half the rank of TM⊗CTM \otimes \mathbb{C} and that ωC\omega_\mathbb{C} vanishes on PP. The concept originates in the foundational works of Kostant and Souriau. Two prominent types of polarizations are Kähler and real polarizations, each suited to different geometric settings. A Kähler polarization arises from a compatible almost complex structure JJ on the , such that ω(⋅,J⋅)\omega(\cdot, J\cdot) defines a Hermitian metric; here, PP is the (0,1)(0,1)-bundle T0,1MT^{0,1}M, and polarized sections are holomorphic sections of the prequantum . This choice is particularly natural on Kähler manifolds, like the , where it aligns with standard holomorphic quantization. In contrast, a real polarization consists of a real Lagrangian subbundle P∩P‾P \cap \overline{P} that foliates the manifold into Lagrangian leaves, with polarized sections required to be constant along these leaves; examples include the vertical polarization on cotangent bundles, where leaves are the fibers themselves. For the resulting to be well-defined and physically meaningful, polarizations must satisfy specific criteria, including positivity and ellipticity. A positive polarization admits a positive definite Hermitian metric compatible with the symplectic and almost complex , ensuring the inner product on sections is non-degenerate. Ellipticity requires that the of the associated ∂ˉ\bar{\partial}-operator (or its real analog) is non-degenerate, guaranteeing that the polarized sections form a space of solutions to an complex, which facilitates global analysis and square-integrability. These properties, emphasized in refinements to the original Kostant-Souriau framework, ensure the polarization yields a finite-dimensional representation in compact cases or a suitable otherwise. With a chosen polarization PP, the geometric quantization map QPQ_P acts on the space of polarized sections of the prequantum bundle, refining the prequantum operators to produce operators on this reduced space and ensuring the quantization respects the to Lie bracket correspondence up to the polarization. This map typically takes the form of a Kostant-Souriau operator restricted to sections covariantly constant along PP, preserving the of observables.

Hilbert Space Construction

In geometric quantization, the choice of polarization PP on the prequantum line bundle L→ML \to M defines the space of polarized sections, which consist of smooth sections ψ∈Γ(M,L)\psi \in \Gamma(M, L) that are covariantly constant along the distribution PP, meaning ∇Xψ=0\nabla_X \psi = 0 for all vector fields X∈Γ(M,P)X \in \Gamma(M, P). For complex polarizations, such as Kähler polarizations, these reduce to holomorphic sections of LL. This selection ensures that the sections encode the "physical states" by restricting to a half-dimensional subspace transverse to the classical directions associated with PP. The physical HPH_P is then the completion of the space of square-integrable polarized sections, denoted L2(M,L,P)L^2(M, L, P), equipped with an inner product given by ⟨ψ,ϕ⟩=∫MH(ψ,ϕ‾) ωnn!\langle \psi, \phi \rangle = \int_M H(\psi, \overline{\phi}) \, \frac{\omega^n}{n!}, where HH is the Hermitian metric on LL and ωnn!\frac{\omega^n}{n!} is the induced by the symplectic . For real polarizations, the may be taken over the M/DM/D by the leaves of the distribution D=P∩P‾D = P \cap \overline{P}, using a compatible measure to account for the . This construction yields a whose dimension reflects the reduced phase space, with the inner product ensuring unitarity of the representation. The quantization map QP:C∞(M)→End(HP)Q_P: C^\infty(M) \to \mathrm{End}(H_P) assigns to each classical observable f∈C∞(M)f \in C^\infty(M) a on HPH_P via QP(f)ψ=projP(−i∇Xfψ)+fψQ_P(f) \psi = \mathrm{proj}_P(-i \nabla_{X_f} \psi) + f \psi, where XfX_f is the of ff, ∇\nabla is the connection on LL, and projP\mathrm{proj}_P denotes orthogonal projection onto the polarized sections. Here, the term −i∇Xf-i \nabla_{X_f} arises from the prequantum operator, adjusted to preserve the polarization. This operator satisfies the Dirac quantization condition [QP(f),QP(g)]=i{f,g}+O(ℏ)[Q_P(f), Q_P(g)] = i \{f, g\} + O(\hbar) up to lower-order terms, with exactness holding for basic functions—those constant along the leaves of PP—while non-basic functions lead to anomalies requiring corrective structures like half-densities. A canonical example occurs with the vertical (position) polarization on the M=T∗QM = T^*Q, where PP is spanned by ∂/∂pj\partial/\partial p_j. The polarized sections are independent of the momenta pjp_j, yielding HP≅L2(Q,dq)H_P \cong L^2(Q, dq) with the standard , and the quantization recovers the Schrödinger representation: position operators act by Q(qj)ψ=qjψQ(q_j) \psi = q_j \psi, while momentum operators act as Q(pj)ψ=−i∂qjψQ(p_j) \psi = -i \partial_{q_j} \psi (in units where ℏ=1\hbar = 1). This aligns with standard on configuration space, illustrating how geometric quantization generalizes familiar representations.

Corrections and Refinements

Half-Form Bundles

In geometric quantization, when constructing the HΠH_\Pi associated to a general polarization Π\Pi, the prequantum operators QΠ(f)Q_\Pi(f) for classical observables ff fail to yield a unitary representation of the of observables. Instead, they produce a , where the operators differ by phase factors that depend on the choice of polarization. To address this issue, half-form bundles are introduced as a correction mechanism. For complex polarizations, the half-density bundle, denoted K1/2K^{1/2}, is defined as the square root of the bundle of (n,0)(n,0)-densities on the MM, specifically K1/2=∣Λn,0T∗M∣1/2K^{1/2} = |\Lambda^{n,0} T^* M|^{1/2}. The transition functions for K1/2K^{1/2} are the square roots of the transition functions for the full density bundle Λn,0T∗M\Lambda^{n,0} T^* M, requiring the of such a square root, which holds under suitable topological conditions like the vanishing of the first Stiefel-Whitney class of the bundle. For real polarizations, the half-form bundle is constructed using half-densities on the quotient space M/ΠM / \Pi. The corrected quantum line bundle is then the tensor product of the prequantum line bundle LL with K1/2K^{1/2}, yielding L⊗K1/2L \otimes K^{1/2}. Sections of this bundle are of the form ψ⊗κ1/2\psi \otimes \kappa^{1/2}, where ψ\psi is a polarized section of LL and κ\kappa is a section of the density bundle with ∣κ∣=1|\kappa| = 1. The inner product on these sections is defined by integrating the Hermitian metric on LL over the quotient space M/ΠM / \Pi, using the half-densities to induce a finite volume measure on the leaves of the polarization. The quantization operators are adjusted to act covariantly on these sections. For a smooth function ff on MM, with XfX_f, the operator is given by Q(f)(ψ⊗κ1/2)=[−i∇Xfψ+12Xf(log⁡h)ψ]⊗κ1/2+fψ⊗κ1/2,Q(f) (\psi \otimes \kappa^{1/2}) = \left[ -i \nabla_{X_f} \psi + \frac{1}{2} X_f (\log h) \psi \right] \otimes \kappa^{1/2} + f \psi \otimes \kappa^{1/2}, where ∇\nabla is the connection on LL and hh is the Hermitian metric on the bundle. This modification includes a term accounting for the action of XfX_f on the half-density part, ensuring compatibility with the polarization. With the half-form correction, the quantized operators satisfy the commutation relation [Q(f),Q(g)]=iQ({f,g})[Q(f), Q(g)] = i Q(\{f, g\}) up to a , yielding a of the of classical observables.

Metaplectic Correction

The , denoted Mp(2n,R)\mathrm{Mp}(2n, \mathbb{R}), is a central extension of the Sp(2n,R)\mathrm{Sp}(2n, \mathbb{R}) by Z2≅{±1}\mathbb{Z}_2 \cong \{\pm 1\}, specifically a double cover given by the short 1→Z2→Mp(2n,R)→Sp(2n,R)→11 \to \mathbb{Z}_2 \to \mathrm{Mp}(2n, \mathbb{R}) \to \mathrm{Sp}(2n, \mathbb{R}) \to 1. This structure arises naturally in the context of quantization, where symplectic transformations on the must lift to unitary operators on the quantum , but generically yield projective representations due to phase ambiguities. The of Mp(2n,R)\mathrm{Mp}(2n, \mathbb{R}) is the same as that of Sp(2n,R)\mathrm{Sp}(2n, \mathbb{R}), but the group incorporates additional topological features, such as its non-simply connected nature, which resolves inconsistencies in representing certain loops in the symplectic group. In geometric quantization, the process yields a projective unitary representation of Mp(2n,R)\mathrm{Mp}(2n, \mathbb{R}) on the Hilbert space HΠH_\Pi associated to a polarization Π\Pi, rather than a true representation of Sp(2n,R)\mathrm{Sp}(2n, \mathbb{R}). This projective nature accounts for the cocycle character of the representation, where the central Z2\mathbb{Z}_2 factor introduces phases that ensure the quantization map commutes with symplectic diffeomorphisms up to unitarity. For instance, the metaplectic representation on spaces like the or L2(Rn)L^2(\mathbb{R}^n) provides explicit unitary operators for quadratic Hamiltonians, linking classical symplectic flows to quantum evolution. The half-form correction, building on the density adjustments from half-form bundles, integrates the metaplectic structure by incorporating the square root of the determinant bundle δΠ=det⁡(T∗M/Π)1/2\delta_\Pi = \det(T^*M / \Pi)^{1/2}, which is a line bundle of half-forms. Sections of the corrected prequantum bundle are then pairs s⊗αs \otimes \alpha, where ss is a section of the prequantum line bundle and α\alpha is a half-form, ensuring the inner product ⟨s⊗α,s′⊗α′⟩=∫M(s⊗α)(s′⊗α′)‾\langle s \otimes \alpha, s' \otimes \alpha' \rangle = \int_M (s \otimes \alpha) \overline{(s' \otimes \alpha')} is well-defined and finite even for non-square-integrable polarized sections. This metaplectic half-form bundle requires a choice of metalinear structure, equivalent to a metaplectic structure on the manifold, and corrects the prequantum connection to ∇corr=∇+12∇δΠ\nabla_{\mathrm{corr}} = \nabla + \frac{1}{2} \nabla^{\delta_\Pi}, preserving polarization while achieving unitarity. The Blattner-Kostant-Sternberg (BKS) pairing provides a mechanism to compare quantizations across different polarizations Π\Pi and Π′\Pi', defined as a sesquilinear map ⟨⟨⋅,⋅⟩⟩:HΠ×HΠ′→C\langle \langle \cdot, \cdot \rangle \rangle : H_\Pi \times H_{\Pi'} \to \mathbb{C} via an integral kernel that projects sections from one polarization to the other. For strongly admissible pairs of polarizations, this pairing identifies HΠH_\Pi and HΠ′H_{\Pi'} up to unitary isomorphism, with the kernel constructed from the half-form corrected operators to handle non-polarization-preserving functions. In examples like the transition from vertical to horizontal polarization on cotangent bundles, the BKS map coincides with the Bargmann transform, ensuring equivalence. For nice symplectic manifolds—those admitting a and compatible polarizations—the metaplectic correction via the BKS pairing guarantees that the resulting quantizations are of the choice of polarization up to . This holds when the second Stiefel-Whitney class of the anti- bundle vanishes, allowing global square roots and consistent lifts to the , thus providing a quantum theory from the classical data.

Examples

Quantization of the 2-Sphere

The 2-sphere S2S^2 serves as a prototypical compact in geometric quantization, realizable as the coadjoint of the SU(2) at level j∈R≥0j \in \mathbb{R}_{\geq 0}. In this setting, the symplectic form is given by ω=jr2sin⁡θ dθ∧dϕ\omega = \frac{j}{r^2} \sin \theta \, d\theta \wedge d\phi, where rr denotes the radius of the sphere and coordinates (θ,ϕ)(\theta, \phi) are the standard spherical ones; this form satisfies the integrality condition [ω]/2π=2j∈Z[\omega]/2\pi = 2j \in \mathbb{Z} when jj is a non-negative , ensuring prequantizability. The total symplectic volume is then 4πj4\pi j, reflecting the scaling by the parameter jj. The prequantum line bundle over S2≅CP1S^2 \cong \mathbb{CP}^1 is the Hopf O(j)\mathcal{O}(j) (more precisely, O(2j)\mathcal{O}(2j) to match the ), a complex with first c1(O(2j))=2jc_1(\mathcal{O}(2j)) = 2j. Equipped with a compatible connection whose curvature is ω\omega, this bundle realizes the prequantization, where the Kostant-Souriau operators act as covariant derivatives on sections, preserving the classical up to central extension. Choosing the Kähler polarization associated to the Fubini-Study metric on CP1\mathbb{CP}^1, the of quantum states consists of the space of holomorphic sections of O(2j)\mathcal{O}(2j). These sections form an orthonormal basis given by the Yml(θ,ϕ)Y_m^l(\theta, \phi) with fixed l=jl = j and magnetic quantum numbers m=−j,−j+1,…,jm = -j, -j+1, \dots, j, providing a complete L2L^2-orthonormal set under the inner product induced by the Liouville measure. The dimension of this is 2j+12j + 1, precisely matching the dimension of the irreducible spin-jj representation of SU(2). A illustration arises in quantizing the h(θ)=rcos⁡θh(\theta) = r \cos \theta, the moment map for the U(1) action generating rotations about the z-axis. The corresponding quantum operator JzJ_z acts diagonally on the basis {Ymj}\{Y_m^j\} with eigenvalues mm (in units ℏ=1\hbar = 1), reproducing the −j,−j+1,…,j-j, -j+1, \dots, j and thereby realizing the spin-jj representation of SU(2) on the . This example demonstrates how geometric quantization recovers the of compact Lie groups from coadjoint orbit data.

Cotangent Bundle Quantization

The cotangent bundle T∗RnT^*\mathbb{R}^n serves as the phase space for a free particle in nn-dimensional Euclidean space, equipped with the canonical symplectic structure ω=∑i=1ndqi∧dpi\omega = \sum_{i=1}^n dq_i \wedge dp_i, where q=(q1,…,qn)q = (q_1, \dots, q_n) are position coordinates and p=(p1,…,pn)p = (p_1, \dots, p_n) are momentum coordinates. This symplectic form arises as the exterior derivative of the tautological one-form θ=∑i=1npi dqi\theta = \sum_{i=1}^n p_i \, dq_i, ensuring the integrality condition for prequantization when scaled by Planck's constant ℏ\hbar. In geometric quantization, the vertical polarization is selected, consisting of the distribution spanned by the vector fields {∂/∂pi}i=1n\{\partial/\partial p_i\}_{i=1}^n, whose integral manifolds are the fibers Tq∗Rn≅RnT^*_q \mathbb{R}^n \cong \mathbb{R}^n over each point q∈Rnq \in \mathbb{R}^n. The prequantum over T∗RnT^*\mathbb{R}^n is trivial, L=T∗Rn×CL = T^*\mathbb{R}^n \times \mathbb{C}, with a Hermitian connection whose is ω/ℏ\omega / \hbar and whose connection one-form is −(i/ℏ)θ-(i/\hbar) \theta. Polarized sections of this bundle, covariant constant along the vertical polarization, are functions s(q,p)=ψ(q)s(q,p) = \psi(q) independent of pp, yielding wave functions ψ(q)\psi(q) on the configuration space Rn\mathbb{R}^n. The resulting Hilbert space is the standard L2(Rn,dq)L^2(\mathbb{R}^n, dq), where dq=dq1⋯dqndq = dq_1 \cdots dq_n is the Lebesgue measure, recovering the position representation of quantum mechanics. The quantization map applied to the classical kinetic energy Hamiltonian H=12m∑i=1npi2H = \frac{1}{2m} \sum_{i=1}^n p_i^2 yields the Schrödinger operator −ℏ22mΔq-\frac{\hbar^2}{2m} \Delta_q, where Δq=∑i=1n∂2/∂qi2\Delta_q = \sum_{i=1}^n \partial^2 / \partial q_i^2 is the Laplacian on Rn\mathbb{R}^n, via the prequantum operators p^i=−iℏ∂/∂qi\hat{p}_i = -i\hbar \partial / \partial q_i acting on ψ(q)\psi(q). In this case, the half-form correction, which typically adjusts for the lack of a natural density on the leaves of the polarization, is trivial due to the flat Euclidean structure of the base, preserving the standard L2L^2 inner product without modification.

Generalizations

Poisson Manifold Quantization

A Poisson manifold is a smooth manifold MM equipped with a Poisson bivector field π∈Γ(∧2TM)\pi \in \Gamma(\wedge^2 TM) that satisfies the integrability condition [π,π]Sch=0[\pi, \pi]_{\mathrm{Sch}} = 0, where [⋅,⋅]Sch[\cdot, \cdot]_{\mathrm{Sch}} denotes the Schouten-Nijenhuis bracket. This structure induces a Poisson bracket on the algebra of smooth functions {f,g}=π(df,dg)\{f, g\} = \pi(df, dg) for f,g∈C∞(M)f, g \in C^\infty(M), which extends the Hamiltonian mechanics framework to more general phase spaces beyond symplectic manifolds. The sharp map π#:T∗M→TM\pi^\#: T^*M \to TM defined by π#(α)(β)=π(α,β)\pi^\#( \alpha ) ( \beta ) = \pi( \alpha, \beta ) plays a central role, associating covectors to vectors and highlighting the non-degeneracy loci of the Poisson structure. The Poisson bivector π\pi defines a Dirac structure on the generalized tangent bundle TM⊕T∗MTM \oplus T^*M, the graph of the bundle map π#:T∗M→TM\pi^\#: T^*M \to TM, i.e., the subbundle {(π#(α),α)∣α∈T∗M}\{(\pi^\#(α), α) \mid α \in T^*M\} of TM⊕T∗MTM \oplus T^*M, which is maximally isotropic and integrable under the Courant bracket. This leads to a foliation of MM into symplectic leaves: the maximal integral submanifolds LL where π∣L\pi|_L is invertible, inducing a symplectic form ωL\omega_L via ωL(π#(df),π#(dg))={f,g}\omega_L(\pi^\#(df), \pi^\#(dg)) = \{f, g\}. On these leaves, the Poisson structure restricts to a genuine symplectic geometry, while transversally, the structure may degenerate, capturing the full complexity of the Poisson manifold. Prequantization proceeds by requiring the periods of ωL\omega_L over closed surfaces in each leaf to be integer multiples of 2πℏ2\pi \hbar, allowing the construction of line bundles over the leaves or, more globally, via the integration of the associated Poisson Lie algebroid to a symplectic groupoid Σ⇉M\Sigma \rightrightarrows M. Leafwise prequantum line bundles L→ML \to M with connections whose curvatures match ωL/iℏ\omega_L / i\hbar on leaves, or multiplicative line bundles over Σ\Sigma, provide the prequantum data. Quantization of the Poisson manifold restricts to the quantization of each symplectic leaf, yielding a pre-Hilbert space of polarized sections of the prequantum bundle over LL, typically using Kähler or real polarizations adapted to the leaf geometry. To account for the transverse directions and ensure a consistent global quantization, transverse corrections are incorporated, often via half-form bundles ΩP1/2\Omega^{1/2}_P over the polarized groupoid and convolution algebras of sections. This construction, polarized and twisted by the prequantum line bundle, completes to a C∗C^*-algebra representing the quantum observables, with transverse measures derived from symplectic potentials or KK-theory pushforwards along the foliation. The resulting quantization functor maps the Poisson algebra C∞(M)C^\infty(M) to operators on the leafwise Hilbert spaces, preserving the Poisson bracket in the semiclassical limit. A prototypical example is the dual g∗\mathfrak{g}^* of a Lie algebra g\mathfrak{g}, endowed with the Kirillov-Kostant-Souriau (KKS) Poisson structure, where the Poisson bivector is defined such that {f,g}(μ)=⟨μ,[df(μ),dg(μ)]g⟩\{f, g\}(\mu) = \langle \mu, [\mathrm{d}f(\mu), \mathrm{d}g(\mu)]_{\mathfrak{g}} \rangle for f,g∈C∞(g∗)f, g \in C^\infty(\mathfrak{g}^*), with the sharp map π#(df)μ\pi^\#(\mathrm{d}f)_\mu corresponding to the infinitesimal coadjoint action generated by df(μ)∈g\mathrm{d}f(\mu) \in \mathfrak{g}, making g∗\mathfrak{g}^* a linear . The symplectic leaves are the coadjoint orbits, each quantized via geometric quantization to irreducible representations of the corresponding , realizing the orbit method. For finite-dimensional g\mathfrak{g}, this yields the on the algebra of polynomial functions when considering the constant bivector case, unifying with .

Relation to Deformation Quantization

Deformation quantization provides an algebraic approach to quantization by deforming the commutative algebra of smooth functions C∞(M)C^\infty(M) on a Poisson manifold MM into a non-commutative associative algebra. Specifically, it constructs a star product ⋆\star on C∞(M)[[ℏ]]C^\infty(M)[[\hbar]], where ℏ\hbar is a formal parameter, such that the commutator satisfies [f,g]⋆iℏ→{f,g}\frac{[f,g]_\star}{i\hbar} \to \{f,g\} as ℏ→0\hbar \to 0, with {f,g}\{f,g\} denoting the Poisson bracket. On symplectic manifolds, Fedosov's yields an explicit deformation quantization by selecting a symplectic connection and defining the star product through the Weyl of that connection, ensuring the result is independent of the choice up to equivalence. For polarizable symplectic manifolds, geometric quantization yields a that carries a representation of the of observables defined by the star product from deformation quantization, as exemplified by the Berezin-Toeplitz operators on Kähler manifolds, where the quantization map identifies the deformed up to equivalence. While both methods achieve compatible quantizations on such manifolds, they differ fundamentally in their perspectives: geometric quantization is inherently geometric and fiberwise, relying on a choice of polarization to construct sections of associated line bundles, whereas deformation quantization is algebraic and global, deforming the function algebra without reference to a specific Hilbert space or polarization. Kontsevich's formality theorem extends deformation quantization to general Poisson manifolds by establishing an L∞L_\infty-quasi-isomorphism from the Gerstenhaber algebra of multivector fields to Hochschild cochains, yielding a canonical star product that parallels the leafwise geometric quantization approach on the symplectic leaves of the Poisson structure.

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