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geometric quantization

Context

Geometric quantization

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

Symplectic geometry

Contents

Idea

Geometric quantization is one formalization of the notion of quantization of a classical mechanical system/classical field theory to a quantum mechanical system/quantum field theory. In comparison to deformation quantization it focuses on spaces of states, hence on the Schrödinger picture of quantum mechanics.

Ingredients

With a symplectic manifold regarded as a classical mechanical system, geometric quantization produces quantization of this to a quantum mechanical system by

  1. realize the symplectic form as the curvature of a U(1)U(1)-principal bundle with connection (which requires the form to have integral periods): called the prequantum circle bundle;

  2. choose a polarization – a splitting of the abstract phase space into “coordinates” and “momenta”;

and then form

  1. a Hilbert space of states as the space of sections of the associated line bundle which depend only on the “coordinates” (not on the “momenta”);

  2. associate with every function on the symplectic manifold – every Hamiltonian – a linear operator on this Hilbert space.

History and variants

The approach is due to Alexandre Kirillov (“orbit method”), Bertram Kostant and Jean-Marie Souriau. See the References below. It is closely related to Berezin quantization? and the subject of coherent states.

In a long term project Alan Weinstein and many of his students have followed the idea that the true story behind geometric quantization crucially involves symplectic Lie groupoids: higher symplectic geometry. See geometric quantization of symplectic groupoids for more on this.

More generally, there is higher geometric quantization.

Overview

This overview is taken from (Baez).

Geometric quantization is a tool for understanding the relation between classical physics and quantum physics. Here’s a brief sketch of how it goes.

  1. We start with a classical phase space: mathematically, this is a manifold XX with a symplectic structure ω\omega.

  2. Then we do prequantization: this gives us a Hermitian line bundle LL over XX, equipped with a U(1)U(1) connection DD whose curvature equals iωi \omega. LL is called the prequantum line bundle.

    Warning: we can only do this step if ω\omega satisfies the Bohr–Sommerfeld condition, which says that ω/2π\omega/2\pi defines an integral cohomology class. If this condition holds, LL and DD are determined up to isomorphism, but not canonically.

  3. The Hilbert space H 0H_0 of square-integrable sections of LL is called the prequantum Hilbert space. This is not yet the Hilbert space of our quantized theory – it’s too big. But it’s a good step in the right direction. In particular, we can prequantize classical observables: there’s a map sending any smooth function on XX to an operator on H 0H_0. This map takes Poisson brackets to commutators, just as one would hope. The formula for this map involves the connection DD.

  4. To cut down the prequantum Hilbert space, we need to choose a polarization, say PP. What’s this? Well, for each point xXx \in X, a polarization picks out a certain subspace P xP_x of the complexified tangent space at xx. We define the quantum Hilbert space, HH, to be the space of all square-integrable sections of LL that give zero when we take their covariant derivative at any point xx in the direction of any vector in P xP_x. The quantum Hilbert space is a subspace of the prequantum Hilbert space.

    Warning: for PP to be a polarization, there are some crucial technical conditions we impose on the subspaces P xP_x. First, they must be isotropic: the complexified symplectic form ω\omega must vanish on them. Second, they must be Lagrangian: they must be maximal isotropic subspaces. Third, they must vary smoothly with xx. And fourth, they must be integrable.

  5. The easiest sort of polarization to understand is a real polarization. This is where the subspaces P xP_x come from subspaces of the tangent space by complexification. It boils down to this: a real polarization is an integrable distribution PP on the classical phase space where each space P xP_x is Lagrangian subspace of the tangent space T xXT_x X.

  6. To understand this rigamarole, one must study examples! First, it’s good to understand how good old Schrödinger quantization fits into this framework. Remember, in Schrödinger quantization? we take our classical phase space XX to be the cotangent bundle T *MT^* M of a manifold MM called the classical configuration space. We then let our quantum Hilbert space be the space of all square-integrable functions on MM.

    Modulo some technical trickery, we get this example when we run the above machinery and use a certain god-given real polarization on X=T*MX = T*M, namely the one given by the vertical vectors.

  7. It’s also good to study the Bargmann–Segal representation, which we get by taking X= nX = \mathbb{C}^n with its god-given symplectic structure (the imaginary part of the inner product) and using the god-given Kähler polarization. When we do this, our quantum Hilbert space consists of analytic functions on n\mathbb{C}^n which are square-integrable with respect to a Gaussian measure centered at the origin.

  8. The next step is to quantize classical observables, turning them into linear operators on the quantum Hilbert space HH. Unfortunately, we can’t quantize all such observables while still sending Poisson brackets to commutators, as we did at the prequantum level. So at this point things get trickier and my brief outline will stop. Ultimately, the reason for this problem is that quantization is not a functor from the category of symplectic manifolds to the category of Hilbert spaces – but for that one needs to learn a bit about category theory.

Basic Jargon

Here are some definitions of important terms. Unfortunately they are defined using other terms that you might not understand. If you are really mystified, you need to read some books on differential geometry and the math of classical mechanics before proceeding.

  • complexification: We can tensor a real vector space with the complex numbers and get a complex vector space; this process is called complexification. For example, we can complexify the tangent space at some point of a manifold, which amounts to forming the space of complex linear combinations of tangent vectors at that point.

  • distribution: The word “distribution” means many different things in mathematics, but here’s one: a “distribution” VV on a manifold XX is a choice of a subspace V xV_x of each tangent space T pXT_p X, where the choice depends smoothly on xx.

  • Hamiltonian vector field: Given a manifold XX with a symplectic structure ω\omega, any smooth function f:Xf: X \to \mathbb{R} can be thought of as a “Hamiltonian”, meaning physically that we think of it as the energy function and let it give rise to a flow on XX describing the time evolution of states. Mathematically speaking, this flow is generated by a vector field v(f)v(f) called the “Hamiltonian vector field” associated to ff. It is the unique vector field such that

    ω(,v(f))=df \omega(-, v(f)) = d f

    In other words, for any vector field uu on XX we have

    ω(u,v(f))=df(u)=uf \omega(u,v(f)) = d f(u) = u f

    The vector field v(f)v(f) is guaranteed to exist by the fact that ω\omega is nondegenerate.

  • integrable distribution: A distribution on a manifold XX is “integrable” if at least locally, there is a foliation of XX by submanifolds such that V xV_x is the tangent space of the submanifold containing the point xx.

  • integral cohomology class: Any closed pp-form on a manifold MM defines an element of the ppth de Rham cohomology of MM. This is a finite-dimensional vector space, and it contains a lattice called the ppth integral cohomology group of MM. We say a cohomology class is integral if it lies in this lattice. Most notably, if you take any U(1)U(1) connection on any Hermitian line bundle over MM, its curvature 22-form will define an integral cohomology class once you divide it by 2πi2 \pi i. This cohomology class is called the first Chern class, and it serves to determine the line bundle up to isomorphism.

  • Poisson brackets: Given a symplectic structure on a manifold MM and given two smooth functions on that manifold, say ff and gg, there’s a trick for getting a new smooth function {f,g}\{f,g\} on the manifold, called the Poisson bracket of ff and gg.

    This trick works as follows: given any smooth function ff we can take its differential dfd f, which is a 11-form. Then there is a unique vector field v(f)v(f), the Hamiltonian vector field associated to ff, such that

    ω(,v(f))=df \omega(-,v(f)) = d f

    Using this we define

    {f,g}=ω(v(f),v(g)) \{f,g\} = \omega(v(f),v(g))

    It’s easy to check that we also have {f,g}=dg(v(f))=v(f)g\{f,g\} = d g(v(f)) = v(f) g. So {f,g}\{f,g\} says how much gg changes as we differentiate it in the direction of the Hamiltonian vector field generated by ff.

    In the familiar case where MM is 2n\mathbb{R}^{2n} with momentum and position coordinates p ip_i, q iq_i, the Poisson brackets of ff and gg work out to be

    {f,g}= idfdp idgdq idfdq idgdp i \{f,g\} = \sum_i \frac{d f}{d p_i} \frac{d g}{d q_i} - \frac{d f}{d q_i}\frac{d g}{d p_i}
  • square-integrable sections: We can define an inner product on the sections of a Hermitian line bundle over a manifold XX with a symplectic structure. The symplectic structure defines a volume form which lets us do the necessary integral. A section whose inner product with itself is finite is said to be square-integrable. Such sections form a Hilbert space H 0H_0 called the “prequantum Hilbert space”. It is a kind of preliminary version of the Hilbert space we get when we quantize the classical system whose phase space is XX.

  • symplectic structure: A symplectic structure on a manifold MM is a closed 22-form ω\omega which is nondegenerate in the sense that for any nonzero tangent vector uu at any point of MM, there is a tangent vector uu at that point for which w(u,v)w(u,v) is nonzero.

  • U(1)U(1) connection: The group U(1)U(1) is the group of unit complex numbers. Given a complex line bundle LL with an inner product on each fiber L xL_x, a U(1)U(1) connection on LL is a connection such that parallel translation preserves the inner product.

  • vertical vectors: Given a bundle EE over a manifold MM, we say a tangent vector to some point of EE is vertical if it projects to zero down on MM.

Definition

Geometric quantization involves two steps

  1. Geometric prequantization

  2. Geometric quantization proper.

Geometric prequantization

Prequantum line bundle

Given the symplectic form ω\omega, a prequantum circle bundle for it is a circle bundle with connection whose curvature is ω\omega.

In other words, prequantization is a lift of ω\omega through the curvature-exact sequence of ordinary differential cohomology (see there).

The multiple of the Chern class of this line bundle is identified with the inverse Planck constant.

Prequantum states

(…)

A prequantum state is a section of the prequantum bundle.

Prequantum operators

Let :XBU(1) conn\nabla : X \to \mathbf{B} U(1)_{conn} be a prequantum line bundle EXE \to X with connection for ω\omega. Write Γ X(E)\Gamma_X(E) for its space of smooth sections, the prequantum space of states.

Definition

For fC (X,)f \in C^\infty(X, \mathbb{C}) a function on phase space, the corresponding quantum operator is the linear map

f^:Γ X(E)Γ X(E) \hat f : \Gamma_X(E) \to \Gamma_X(E)

given by

ψi v fψ+fψ, \psi \mapsto -i \nabla_{v_f} \psi + f \cdot \psi \,,

where

  • v fv_f is the Hamiltonian vector field corresponding to ff;

  • v f:Γ X(E)Γ X(E)\nabla_{v_f} : \Gamma_X(E) \to \Gamma_X(E) is the covariant derivative of sections along v fv_f for the given choice of prequantum connection;

  • f():Γ X(E)Γ X(E)f \cdot (-) : \Gamma_X(E) \to \Gamma_X(E) is the operation of degreewise multiplication pf sections.

Geometric quantization

Given a prequantum bundle as above, the actual step of genuine geometric quantization consists first of forming half its space of sections in a certain sense. Physically this means passing to the space of wavefunctions that depend only on canonical coordinates but not on canonical momenta. Second, subgroups of the group pf (exponentiated) prequantum operators are made to descend to this space of quantum states, these are the quantum operators or quantum observables.

Quantum states

Historically, the traditional way to formalize the formation of the space of quantum states is as a 3-step process

  1. choose a Polarization;

  2. choose a Metaplectic correction;

  3. form the induced space of quantum states as the space of polarized sections of the prequantum line bundle tensored a certain half-form bundle.

This we discuss at

This traditional route via polarizations and metaplectic corrections has the disadvantage that mathematically it is not a very natural operation. However, under mild conditions it turns out to be equivalent to the following mathematically very natural construction

  1. choose a KU-orientation, hence a spin^c structure of XX compatible with the given prequantum bundle;

  2. take the space of quantum states to be the push-forward in complex K-theory of the prequantum line bundle to the point, hence the index of the spin^c Dirac operator twisted by the prequantum line bundle.

This general geometric quantization by push-forward is discussed below at

In the special case that the prequantum line bundle admits a Kähler polarization this push-forward quantization has a direct expression in terms of the complex abelian sheaf cohomology and of the Dolbeault operator of the prequantum holomorphic line bundle. Now the choice of metaplectic correction is precisely a spin structure (a “Theta characteristic”) and the index is now that of the Dolbeault-Dirac operator which is equivalently just the Euler characteristic of the holomorphic abelian sheaf cohomology of the prequantum line bundle. This complex Dolbeault quantization case we discuss in

and

Quantum state space as space of polarized sections

For (X,ω)(X, \omega) a symplectic manifold, choose a Kähler polarization, hence an involutive Lagrangian subbundle 𝒫T 𝒞X\mathcal{P} \subset T_{\mathcal{C}} X such that 𝒫𝒫¯=0\mathcal{P} \cap \overline{\mathcal{P}} = 0. Choose moreover a metaplectic correction of 𝒫\mathcal{P}. This defines the half-density bundle Ω𝒫\sqrt{\Omega \mathcal{P}} along 𝒫\mathcal{P}.

Let now (L,)(L,\nabla) be a prequantum line bundle for (X,ω)(X,\omega).

Definition

The space of quantum states of the prequantum bundle defined by the choice of Kähler polarization 𝒫\mathcal{P} and metaplectic correction Ω𝒫\sqrt{\Omega \mathcal{P}} is the space of sections of the tensor product LΩ𝒫L \otimes \sqrt{\Omega \mathcal{P}} which are covariant derivative covariantly constant with respect to \nabla along 𝒫¯\overline{\mathcal{P}} and square integrable with respect to the induced integration over X/𝒫¯X/\overline{\mathcal{P}}:

pol:L 2({ψΓ(LΩ𝒫)| 𝒫¯ψ=0}). \mathcal{H}_{pol} \colon L^2 \left( \left\{ \psi \in \Gamma(L \otimes \sqrt{\Omega \mathcal{P}}) \;|\; \nabla_{\overline{\mathcal{P}}} \psi = 0 \right\} \right) \,.

For instance (Bates-Weinstein, def. 7.17).

Remark

In the case of Kähler polarization it is useful to write Ω n,0\Omega^{n,0} for the canonical line bundle of the polarization and Ω n,0\sqrt{\Omega^{n,0}} for the corresponding choice of metaplectic correction/Theta characteristic.

Quantum state space as Euler characteristic of prequantum sheaf cohomology

Let (X,ω)(X,\omega) be a compact topological space symplectic manifold, L ωL_\omega a prequantum line bundle, 𝒫\mathcal{P} a Kähler polarization and Ω n,0\sqrt{\Omega^{n,0}} a metaplectic correction.

Definition

The corresponding Euler quantum Hilbert space is the virtual vector space

Euler k=0 n(1) kH 0,k(X,L ωΩ n,0), \mathcal{H}_{Euler} \coloneqq \bigoplus_{k = 0}^{n} (-1)^k \, H^{0,k}(X,L_\omega \otimes \sqrt{\Omega^{n,0}}) \,,

which is the alternating direct sum of the Dolbeault cohomology space of XX with coefficients in the tensor product L ωΩ n,0L_\omega \otimes \sqrt{\Omega^{n,0}} of the prequantum line bundle with the metaplectic correction/Theta characteristic.

Remark

The Kodaira vanishing theorem asserts that if L ωΩ n,0 1L_\omega \sqrt{\Omega^{n,0}}^{-1} is a positive line bundle then all the higher cohomology groups in the above expression vanish. Therefore in this case the definition coincides with that via polarizations in def. 2 above.

(L ωΩ n,0 1positive)( pol Euler). \left( L_\omega \otimes \sqrt{\Omega^{n,0}}^{-1} \; positive \right) \Rightarrow \left( \mathcal{H}_{pol} \simeq \mathcal{H}_{Euler} \right) \,.
Quantum state space as index of Dolbeault-Dirac operator

Suppose again that (X,ω,L ω)(X,\omega, L_\omega) is equipped with a Kähler polarization.

We need the following general fact on spin structures over Kähler manifolds.

Proposition

A choice of spin structure on the Kähler manifold XX (of real dimension 2n2n) is equivalently a choice of square root (“Theta characteristic”) Ω n,0\sqrt{\Omega^{n,0}} of the canonical line bundle Ω n,0\Omega^{n,0}.

Given such a choice, there is a natural isomorphism between the spinor bundle S XS_X and the (anti-)holomorphic form bundle tensored by this square root

SΩ 0,Ω n,0. S \simeq \Omega^{0,\bullet} \otimes \sqrt{\Omega^{n,0}} \,.

Finally, the corresponding Dirac operator is the Dolbeault-Dirac operator ¯+¯ *\overline{\partial} + \overline{\partial}^\ast.

See at spin structure – Over a Kähler manifold.

Remark

It follows that the Dirac operator on XX which is twisted by the connection \nabla on the prequantum holomorphic line bundle L ωL_\omega is of the form

¯ +¯ *:Ω 0,even/odd(X,L ωΩ n,0)Ω 0,odd/even(X,L ωΩ n,0). \overline{\partial}_\nabla + \overline{\partial}^\ast_\nabla \;\colon\; \Omega^{0,even/odd}\left(X, \;L_\omega \otimes \sqrt{\Omega^{n,0}}\right) \to \Omega^{0,odd/even}\left(X, \; L_\omega \otimes \sqrt{\Omega^{n,0}}\right) \,.

Observe how from the point of view of just the Dolbeault operator, this is twisting not just by the prequantum line bundle itself but by the metaplectically corrected prequantum line bundle L ωΩ n,0L_\omega \otimes \sqrt{\Omega^{n,0}}, while from the point of view of the Dirac operator it is just twisting by L ωL_\omega, since tensoring with the square root line bundle Ω n,0\sqrt{\Omega^{n,0}} induces the isomorphism between the antiholomorphic differential form bundle and the actual spinor bundle over the Kähler manifold.

Definition

The Dolbeault-Dirac space of quantum states of (X,ω,L ω)(X,\omega, L_\omega) is the index of the L ωL_\omega-twisted Dolbeault-Dirac operator (the Todd genus)

DolbDirindex(¯ +¯ *). \mathcal{H}_{DolbDir} \coloneqq index( \overline{\partial}_\nabla + \overline{\partial}_\nabla^\ast ) \,.
Remark

This definition agrees with that by abelian sheaf cohomology in def. 3

Euler DolbDirac. \mathcal{H}_{Euler} \simeq \mathcal{H}_{DolbDirac} \,.
Remark

What before in the quantization prescription by polarization and by abelian sheaf cohomology was the “metaplectic correction” introduced “by hand” is now naturally part of the isomorphism of prop. 1 which identifies the Dirac operator with the Dolbeault-Dirac operator.

Quantum state spaces as index of the Spin cSpin^c-Dirac operator

Finally we come to the true definition of geometric quantization, the most general and at the same time most natural one which contains the above as special cases.

The actual definition is def. 9 below. Here we lead up to it by spelling out the ingredients.

We need the following general facts about spin^c Dirac operators.

Definition

For nn \in \mathbb{N}, the spin^c-Lie group Spin c(n)Spin^c(n) is equivalently

Here the top horizontal map is called the universal determinant line bundle map.

See at spin^c group for more details.

Remark

It follows that if we represent elements of Spin c(n)SO(n)× 2U(1)Spin^c(n) \simeq SO(n) \underset{\mathbb{Z}_2}{\times} U(1) as equivalence classes of pairs (g,c)(g,c), then the determinant line bundle map of def. 5 is given by

det:(g,c)2c, det \;\colon \; (g,c) \mapsto 2c \,,

where on the right we write the group operation in the abelian group U(1)U(1) additively.

This factor of 2 on the right is crucial in all of the following.

Definition

Let XX be an oriented smooth manifold. A smooth spin^c structure on XX is a lift τ^ X\hat \tau_X of the classifying map τ X:XBSO(n)\tau_X \colon X \to \mathbf{B} SO(n) of its tangent bundle through the left vertical map in def. 5

BSpin c(n) τ^ X X τ X BSO(n). \array{ && \mathbf{B} Spin^c(n) \\ & {}^{\mathllap{\hat \tau_X}}\nearrow & \downarrow \\ X &\stackrel{\tau_X}{\to}& \mathbf{B}SO(n) } \,.
Remark

In words this says that a spin^c structure on an oriented manifold XX is a choice of circle group-principal bundle or equivalenty of hermitian complex line bundle such that its first Chern class modulo 2 equals the second Stiefel-Whitney class w 2w_2 of XX. If this second Stiefel-Whitney class vanishes (as an element in H 2(X, 2)H^2(X,\mathbb{Z}_2)) this means that XX has a genuine spin structure. So in other words whenever the determinant line bundle of a spin^c structure (in the sens of def. 5) has a first Chern class that is divisible by 2, then there is an actual spin^c structure.

We can formalize this statement as follows: there is a commuting square of the form

B(Spin(n)×U(1)) ((2)p 2 BU(1) c 1mod2 BSO(n) w 2 B 2 2 \array{ \mathbf{B}( Spin(n) \times U(1) ) &\stackrel{((2 \cdot)\circ p_2}{\to}& \mathbf{B}U(1) \\ \downarrow && \downarrow^{\mathrlap{\mathbf{c}_1 \, mod\, 2}} \\ \mathbf{B}SO(n) &\stackrel{\mathbf{w}^2}{\to}& \mathbf{B}^2 \mathbb{Z}_2 }

similar to the one in def. 5 but crucially different in that here we have just the cartesian product of Spin(n)Spin(n) with U(1)U(1) in the top left. In the standard presentation of these objects this diagram commutes on the nose (filled by a canonical homotopy) simply because both ways to go from the top left to the bottom right hit 0 20 \in \mathbb{Z}_2: the left-bottom one because BSpin(n)\mathbf{B}Spin(n) is essentially by definition such that the second SW class trivializes on it, and the top-right one because first multiplying by 2 and then reducing mod 2 is 0.

Since the diagram commutes, the universal property of the above homotopy pullback says that there is a canonically induced map

(p 1,p 2):Spin(n)×U(1)Spin c(n). (p_1, p_2) \;\colon\; Spin(n) \times U(1) \to Spin^c(n) \,.

Of course this is just the evident quotient projection which on elements is simply the identity

(g,c)(g,c), (g,c) \mapsto (g,c) \,,

only that on the right we regard the pair (g,c)(g,c) as a placeholder for its equivalence class in the tensor product over 2\mathbb{Z}_2.

Moreover, the universal property of the homotopy pullback says that every spin^c structure τ^ c\hat \tau_c whose underlying determinant line bundle is such that its first Chern class is divisible by 2 actually factors through this map, hence that it is just the product of an ordinary SpinSpin-principal bundle with a circle bundle.

B(Spin(n)×U(1)) p 2 BU(1) (p 1,p 2) B(2) BSpin c(n) det BU(1) (pb) c 1mod2 BSO(n) w 2 B 2 2. \array{ \mathbf{B}(Spin(n) \times U(1)) &\stackrel{p_2}{\to}& \mathbf{B}U(1) \\ {}^{\mathllap{(p_1,p_2)}}\downarrow && \downarrow^{\mathbf{B}(2 \cdot)} \\ \mathbf{B}Spin^c(n) &\stackrel{det}{\to}& \mathbf{B}U(1) \\ \downarrow &(pb)& \downarrow^{\mathrlap{c_1\,mod\,2}} \\ \mathbf{B}SO(n) &\stackrel{\mathbf{w}_2}{\to}& \mathbf{B}^2 \mathbb{Z}_2 } \,.

This is the crucial relation by which the K-theoretic quantization will harmonize with the above Euler- and Dolbeault- quantization, discussed below in \ref{spring}.

Definition

For nn \in \mathbb{N} an even number, write

Δ n 2 n/2 \Delta_n \simeq \mathbb{C}^{2^{n/2}}

for the canonical complex spin representation, which decomposes into two chiral irreducible representations

Δ nΔ n +Δ n . \Delta_n \simeq \Delta_n^+ \oplus \Delta_n^- \,.

Then the canonical spin^c-representation

ρ:Spin c(n)×Δ nΔ n \rho \colon Spin^c(n) \times \Delta_n \to \Delta_n

is the one given in the components of remark 6 by

((g,c),ψ)(g(ψ))c ((g,c), \psi) \mapsto (g(\psi)) \cdot c

(first act with the spin-component in the usual way and then multiply by cU(1)c \in U(1) \hookrightarrow \mathbb{C}).

As discussed at representation we may think of this as a morphism of smooth groupoids (stacks) of the form

ρ:BSpin c(n)ModVect. \rho \colon \mathbf{B} Spin^c(n) \to \mathbb{C}\mathbf{Mod} \simeq \mathbb{C}\mathbf{Vect} \,.
Definition

Given a spin^c structure τ^ X:XBSpin c(n)\hat \tau_X \colon X \to \mathbf{B}Spin^c(n), def. 6, the associated spinor bundle is the one modulated by

S c:Xτ^ XBSpin c(n)ρVect. S^c \colon X \stackrel{\hat \tau_X}{\to} \mathbf{B}Spin^c(n) \stackrel{\rho}{\to} \mathbb{C}\mathbf{Vect} \,.

Crucial for the comparison of the K-theoretic quantization to be defined in a moment and the above Euler/Dolbeault quantization is the following:

Remark

By remark 7 it follows that in the case that the determinant line bundle of the spin^c structure is divisible by 2, hence in the case that c 1(det(τ^ X))=0mod2H 2(X, 2)c_1(det(\hat \tau_X)) = 0 \,mod\, 2 \in H^2(X,\mathbb{Z}_2), then the Spin cSpin^c-spinor bundle of def. 8 is just the tensor product of the ordinary underlying spinor bundle with half the determinant line bundle:

(c 1(det(τ^ X))=0mod2)(S cSdet(τ^ X)). \left( c_1(det(\hat \tau_X)) = 0 \,mod\, 2 \right) \;\Rightarrow\; \left(S^c \simeq S \otimes \sqrt{det(\hat \tau_X)}\right) \,.

Let now (X,ω)(X,\omega) be a (pre-)symplectic manifold and let (L ω,)(L_{\omega}, \nabla) be a prequantum line bundle.

Definition

Given a spin^c structure τ^ X\hat \tau_X on XX whose underlying determinant line bundle coincides, up to equivalence, with (L ω) 2(L_\omega)^{\otimes^2}, then the spin^c quantization of this prequantum data is the index of the corresponding spin^c Dirac operator D cD_{\mathbf{c}}:

Kindex(D c). \mathcal{H}_{K} \coloneqq index(D_{\mathbf{c}}) \,.
Remark

This is equivalently the push-forward in complex K-theory of the prequantum line bundle to the point.

Moreover, on cocycles this push-forward is expressed by the traditional construction of Hilbert spaces of quantum states:

let XX be a manifold equipped with a Riemannian metric, a spinor bundle and a Dirac operator D:Γ(S)Γ(S)D \colon \Gamma(S) \to \Gamma(S). Then the corresponding K-homology cycle is:

  1. the Hilbert space L 2(S)L^2(S) of square integrable sections of the spinor bundle;

  2. equipped with the action by fiberwise multiplication of the C*-algebra C 0(X)C_0(X) of functions vanishing at infinity

    C 0(X)(L 2(S)) C_0(X) \to \mathcal{B}(L^2(S))

    (which is by bounded operators)

  3. and equipped with the Dirac operator DD as the 2\mathbb{Z}_2-graded Fredholm operator defining the K-homology class.

Together with is equivalent an element

[D,L 2(S)]KK(C 0(X),). [D, L^2(S)] \in KK(C_0(X), \mathbb{C}) \,.

Postcomposition of K-theory classes [ξ]KK(,C 0(X))[\xi] \in KK(\mathbb{C}, C_0(X)) with this map is the push-forward/index map in K-theory.

Remark

This definition does not assume any choice of polarization, nor any choice of complex structure etc. On the other hand, every choice of almost complex structure (hence in particular of Kähler polarization) does induce a spin^c structure, as discussed there.

So then we can compare:

Proposition

If a choice of Kähler polarization exists on the compact prequantum geometry (X,L ω)(X,L_\omega), then the K-theoretic quantization of def. 9 coincides with the Dolbeault-Dirac quantization of def. 4 and hence, by remark 4, with the Euler-characteristic definition, def. 3, so that all the respectives spaces of quantum states agree:

((X,ω)Kaehler)( K DolbDirac Euler). \left(\left(X,\omega\right) \; Kaehler\right) \;\Rightarrow\; \left( \mathcal{H}_K \simeq \mathcal{H}_{DolbDirac} \simeq \mathcal{H}_{Euler} \right) \,.

This is to some extent discussed for instance in (Hochs 08, lemma 3.32, Paradan 09, prop. 2.2).

Proof

Let Ω n,0\sqrt{\Omega^n,0} be a choice of square root (Theta characteristic) of the canonical line bundle, which according to 1 is a choice of spin structure in the Kähler manifold. Then by that same proposition the corresponding spinor bundle is isomorphic to

SΩ 0,Ω n,0 S \simeq \Omega^{0,\bullet} \otimes \sqrt{\Omega^{n,0}}

and so the corresponding L ωL_\omega-twisted spinor bundle is

SL ω (Ω 0,Ω n,0)L ω Ω 0,(Ω n,0L ω), \begin{aligned} S \otimes L_\omega & \simeq \left(\Omega^{0,\bullet} \otimes \sqrt{\Omega^{n,0}}\right) \otimes L_\omega \\ & \simeq \Omega^{0,\bullet} \otimes \left(\sqrt{\Omega^{n,0}} \otimes L_\omega\right) \end{aligned} \,,

where we have re-bracheted just for emphasis of how the metaplectic correction appears as part of the spinor bundle.

At the same time, by remark 8 we have that under this assumption that an actual spin structure exists, the Spin cSpin^c-spinor bundle is isomorphic to

S cSL ω. S^c \simeq S \otimes L_\omega \,.

So the two spinor bundles agree. It is now sufficient to observe that under this identification both the Dolbeault-Dirac operator as well as the spin^c Dirac operator have the same symbol to conclude that they have the same index.

Remark

The assumption in the above that we do have a spin structure is only for comparison with the Euler-/ Dolbeault-quantization. It is not necessary for the K-theoretic geometric quantization by spin^c structure.

In the general case the determinant line bundle of the spin^c structure may not admit a square root. (Its failure of having a square root will be compensated precisely by the failure of there being a genuine spin structure.) Still, by the above discussion the index of the corresponding spin^c structure is like a quantization of a would-be square root of the determinant line bundle in this case.

Quantum operators / observables

Given a Hamiltonian action on the symplectic manifold by a Lie group GG, we can apply the above K-theoretic quantization by push-foward in GG-equivariant K-theory, to

K G(*)K(*//G)R(G) K_G(\ast) \simeq K(\ast//G) \simeq R(G)

This is the representation ring of GG and hence yields not just a Hilbert space, but a Hilbert space equipped with a GG-action. This is the action by quantum operators, quantizing the GG-actions. Generalized orientation theory give the necessary condition for this quantization to exist: XX needs to be oriented not just in K-theory (spin^c structure) but in GG-equivariant K-theory (equivariant spin^c structure).

So the geometric quantization of observables is essentially what mathematically is known as Dirac induction.

Properties

Functorial dependence on choices

Chapter 3 of

  • Lauridsen, Aspects of quantum mathematics – Hitchin connections and the AJ conjecture, PhD thesis Aarhus 2010 (pdf)

Compatibility of quantization with symplectic reduction

On the relation between geometric quantization and symplectic reduction:

Characteristic central extensions

To a large extent geometric quantization is realized by central extension of various Lie groups arising in classical mechanics/symplectic geometry.

higher and integrated Kostant-Souriau extensions:

(∞-group extension of ∞-group of bisections of higher Atiyah groupoid for 𝔾\mathbb{G}-principal ∞-connection)

(Ω𝔾)FlatConn(X)QuantMorph(X,)HamSympl(X,) (\Omega \mathbb{G})\mathbf{FlatConn}(X) \to \mathbf{QuantMorph}(X,\nabla) \to \mathbf{HamSympl}(X,\nabla)
nngeometrystructureunextended structureextension byquantum extension
\inftyhigher prequantum geometrycohesive ∞-groupHamiltonian symplectomorphism ∞-groupmoduli ∞-stack of (Ω𝔾)(\Omega \mathbb{G})-flat ∞-connections on XXquantomorphism ∞-group
1symplectic geometryLie algebraHamiltonian vector fieldsreal numbersHamiltonians under Poisson bracket
1Lie groupHamiltonian symplectomorphism groupcircle groupquantomorphism group
22-plectic geometryLie 2-algebraHamiltonian vector fieldsline Lie 2-algebraPoisson Lie 2-algebra
2Lie 2-groupHamiltonian 2-plectomorphismscircle 2-groupquantomorphism 2-group
nnn-plectic geometryLie n-algebraHamiltonian vector fieldsline Lie n-algebraPoisson Lie n-algebra
nnsmooth n-groupHamiltonian n-plectomorphismscircle n-groupquantomorphism n-group

(extension are listed for sufficiently connected XX)

Examples

Cotangent bundles – Schrödinger representation

For the particle propagating on the line with respect to some standard action functional, the phase space is the cotangent bundle T * 2T^* \mathbb{R} \simeq \mathbb{R}^2, where the isomorphism is given by choosing standard coordinates, {q,p}\{q,p\}.

The symplectic form is the canonical volume form, which in these coordinates reads

ω=dpdq. \omega = d p \wedge d q \,.

A prequantum line bundle for this is given by the trivial line bundle equipped with the connection that is given by the globally defined 1-form

θ:=qdp. \theta := q \wedge d p \,.

A section of this complex line bundle is canonically identified simply with a \mathbb{C}-valued smooth function on 2\mathbb{R}^2.

A choice of foliation of phase space is given by constant-qq-slices

Λ q= 2. \Lambda_q = \subset \mathbb{R}^2 \,.

The polarization condition is that the covariant derivative along the leaves vanishes

/pψ=0, \nabla_{\partial/\partial p} \psi = 0 \,,

which with the above choice of connection is equivalently

pψ+iqψ=0. \frac{\partial}{\partial p} \psi + i q \psi = 0 \,.

The solutions to this differential equation are of the form

ψ(q,p)=ψ(q,0)exp(ipq). \psi(q,p) = \psi(q,0) \exp(i p q) \,.

The space of these quantum states is (noncanonically) identified with the space of complex functions on the line by evaluating at p=0p = 0.

Since we have the hamiltonian vector fields

v q=p v_q = \frac{\partial}{\partial p}
v p=q v_p = -\frac{\partial}{\partial q}

the action of the quantum operators q^\hat q and p^\hat p on these states is

q^ψ=i /pψ+qψ=qψ \hat q \psi = - i \nabla_{\partial/\partial p}\psi + q \psi = q \psi

and

p^ψ=i /qψ+pψ=iqψ. \hat p \psi = i \nabla_{\partial/ \partial q} \psi + p \psi = i \frac{\partial}{\partial q} \psi \,.

Kähler manifolds

If the symplectic manifold (X,ω)(X, \omega) happens to be also a Kähler manifold there are natural choices of prequantization:

Applied to a symplectic vector space this yields the Bargmann-Fock representation? of the Heisenberg group.

The 2-sphere

Consider the 2-sphere S 2S^2 with its canonical round volume form as a symplectic manifold. This admits a complex structure JJ such that the subgroup of the diffeomorphism group which preserves JJ is the special orthogonal group SO(3)SO(3). The corresponding Kähler metric is a multiple of the standard round Riemannian metric on the 2-sphere.

By the identification BU(1)K(,2)B U(1) \simeq K(\mathbb{Z},2) the prequantum bundles \mathcal{L} on S 2S^2 are classified by \mathbb{Z}.

The Hilbert space is

=H 0(S 2,), \mathcal{H} = H^0(S^2, \mathcal{L}) \,,

the space of holomorphic sections of such a holomorphic line bundle. The universal cover SU(2)SU(2) of SO(3)SO(3) naturally acts on this Hilbert space. The canonical coordinate functions x,y,z:S 2x,y,z \colon S^2 \to \mathbb{R} naturally act on \mathcal{H} and as such form a Lie algebra representation of the special orthogonal Lie algebra.

A way to reduce the number of choices in this example of geometric quantization is to proceed by quantization via the A-model, see (Gukov-Witten 08, p. 6 onwards).

For more see geometric quantization of the 2-sphere.

Tori

Theta functions

See at theta function.

Quantization of Chern-Simons theory

The geometric quantization of Chern-Simons theory leads to invariants for 3-manifolds.(Witten 89).

See at quantization of Chern-Simons theory for more.

Quantization of loop groups / of the WZW model

See at quantization of loop groups.

Quantization in Gromov-Witten theory

Geometric quantization appears in Gromov-Witten theory. See (Clader-Priddis-Shoemaker 13).

Quantization of the bosonic string σ\sigma-model

For discussion of the geometric quantization of the bosonic string 2d sigma-model see at string – Symplectic geometry and geometric quantization .

duality between algebra and geometry in physics:

algebrageometry
Poisson algebraPoisson manifold
deformation quantizationgeometric quantization
algebra of observablesspace of states
Heisenberg pictureSchrödinger picture
AQFTFQFT
higher algebrahigher geometry
Poisson n-algebran-plectic manifold
En-algebrashigher symplectic geometry
BD-BV quantizationhigher geometric quantization
factorization algebra of observablesextended quantum field theory
factorization homologycobordism representation

partition functions in quantum field theory as indices/genera/orientations in generalized cohomology theory:

ddpartition function in dd-dimensional QFTsuperchargeindex in cohomology theorygenuslogarithmic coefficients of Hirzebruch series
0push-forward in ordinary cohomology: integration of differential formsorientation
1spinning particleDirac operatorKO-theory indexA-hat genusBernoulli numbersAtiyah-Bott-Shapiro orientation MSpinKOM Spin \to KO
endpoint of 2d Poisson-Chern-Simons theory stringSpin^c Dirac operator twisted by prequantum line bundlespace of quantum states of boundary phase space/Poisson manifoldTodd genusBernoulli numbersAtiyah-Bott-Shapiro orientation MSpin cKUM Spin^c \to KU
endpoint of type II superstringSpin^c Dirac operator twisted by Chan-Paton gauge fieldD-brane chargeTodd genusBernoulli numbersAtiyah-Bott-Shapiro orientation MSpin cKUM Spin^c \to KU
2type II superstringDirac-Ramond operatorsuperstring partition function in NS-R sectorOchanine elliptic genusSO orientation of elliptic cohomology
heterotic superstringDirac-Ramond operatorsuperstring partition functionWitten genusEisenstein seriesstring orientation of tmf
self-dual stringM5-brane charge

References

General

Original references include

  • Jean-Marie Souriau, Structure des systemes dynamiques Dunod, Paris (1970)

    Translated and reprinted as (see section V.18 for geometric quantization):

    Jean-Marie Souriau, Structure of dynamical systems - A symplectic view of physics, Brikhäuser (1997)

  • Bertram Kostant, Quantization and unitary representations, in Lectures in modern analysis and applications III, Lecture Notes in Math. 170 (1970), Springer Verlag, 87—208

  • Bertram Kostant, On the definition of quantization, in Géométrie Symplectique et Physique Mathématique, Colloques Intern. CNRS, vol. 237, Paris (1975) 187—210

  • Victor Guillemin, Shlomo Sternberg, Geometric Asymptotics, Math. Surveys no. 14, Amer. Math. Soc. (1977) (web)

  • Alexandre Kirillov, Geometric quantization Dynamical systems – 4, Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Fund. Napr., 4, VINITI, Moscow, 1985, 141–176 (web)

A fairly comprehensive textbook with modern developments is

Introductions and lecture notes include

Lecture notes with an emphasis on semiclassical states are in

A careful discussion of the polarization-step from prequantization to quantization is in

  • J. Śniatycki, Wave functions relative to a real polarization, Internat. J. Theoret. Phys., 14(4):277–288 (1975)

  • J. Śniatycki, Geometric Quantization and Quantum Mechanics, volume 30 of Applied Mathematical Sciences. Springer-Verlag, New York (1980)

The special case of Kähler quantization is discussed for instance in

  • Nigel Hitchin, Flat connections and geometric quantization, Comm. Math. Phys. 131 (1990), no. 2, 347–380.

Discussion with an emphasis of quantizing classical field theory on curved spacetime is in

  • Nicholas Woodhouse, Geometric quantization and quantum field theory in curved space-times, Reports on Mathematical Physics, Volume 12, Issue 1, (1977), Pages 45–56

(For more on geometric quantization of quantum field theories see also at Quantization of multisymplectic geometry.)

Aspects at least of geometric prequantization are usefully discussed also in section II of

  • Jean-Luc Brylinski, Loop spaces, characteristic classes and geometric quantization, Birkhäuser (1993)

Further reviews include

  • A. Echeverria-Enriquez, M.C. Munoz-Lecanda, N. Roman-Roy, C. Victoria-Monge, Mathematical Foundations of Geometric Quantization Extracta Math. 13 (1998) 135-238 (arXiv:math-ph/9904008)

The above “Overview” and “Basic Jargon” sections are taken from

Some useful talk notes include

  • Eva Miranda, From action-angle coordinates to geometric quantization and back (2011) (pdf)

Discussion with an eye towards the philosophy of physics is in

Holographic quantization

The geometric quantization via the A-model is discussed in

See also the geometric quantization of symplectic groupoids below, around (Hawkins).

Spin Spin^{\mathbb{C}}-Quantization

  • Peter Hochs, Quantisation commutes with reduction for cocompact Hamiltonian group actions 2008 (pdf)

See the references at geometric quantization by push-forward.

Examples

The basic example of geometric quantization of a symplectic vector space is discussed in pretty much every text on the matter for instance Nohara, starting with example 2.6.

Discussion of geometric quantization of Chern-Simons theory is for instance in

  • Edward Witten (lecture notes by Lisa Jeffrey), New results in Chern-Simons theory, pages 81 onwards in: S. Donaldson, C. Thomas (eds.) Geometry of low dimensional manifolds 2: Symplectic manifolds and Jones-Witten theory (1989) (pdf)
  • Scott Axelrod, Steve Della Pietra, and Edward Witten, Geometric quantization of Chern-Simons gauge theory, J. Differential Geom. Volume 33, Number 3 (1991), 787-902. (EUCLID)

Discussion of geometric quantization of abelian varieties, toric varieties, flag varieties and its relation to theta functions is in

  • Yuichi Nohara, Independence of polarization in geometric quantization (pdf)
  • N.J. Hitchin, Flat connections and geometric quantization, Comm. Math. Phys. 131, n 2 (1990), 347-380, euclid

An appearance of geometric quantization in mirror symmetry is pointed out in

  • Andrei Tyurin, Geometric quantization and mirror symmetry, (math.AG/9902027)

For discussion of the geometric quantization of the bosonic string 2d sigma-model see at string – Symplectic geometry and geometric quantization.

Discussion of geometric quantization of self-dual higher gauge theory is in

  • Samuel Monnier, Geometric quantization and the metric dependence of the self-dual field theory (arXiv:1011.5890)

Discussion in the context of Gromov-Witten theory:

  • Emily Clader, Nathan Priddis, Mark Shoemaker, Geometric Quantization with Applications to Gromov-Witten Theory (arXiv:1309.1150)

Relation to deformation quantization

Discussion of the relation of geometric quantization to deformation quantization is in

For a basic comparative review of both see also section 1 of

(which then develops geometric quantization via the A-model).

Relation to path integral quantization

Relation to path integral quantization is discussed in

  • Laurent Charles, Feynman path integral and Toeplitz Quantization (pdf)

Geometric BRST quantization

The Lie algebroid version of an action groupoid is given (dually) by a BRST complex. Quantization over a BRST complex is hence quantization over an infinitesimal action groupoid. (See at higher geometric quantization).

Geometric quantization over BRST complexes is discussed in the following articles.

Supergeometric version

One can consider geometric quantization in supergeometry.

  • S.-M. Fei, H. -Y. Guo und Y. Yu, Symplectic geometry and geometric quantization on supermanifold with UU numbers, Z. Phys, C - Particles and Fields 45, 339-344 (1989)

  • Gijs M. Tuynman, Super Symplectic Geometry and Prequantization (2003) (arXiv:math-ph/0306049)

Of presymplectic manifolds

Discussion of quantization of presymplectic manifolds is in

  • C. Günther, Presymplectic manifolds and the quantization of relativistic particles, Salamanca 1979, Proceedings, Differential Geometrical Methods In Mathematical Physics, 383-400 (1979)

  • Izu Vaisman, Geometric quantization on presymplectic manifolds, Monatshefte für Mathematik, vol. 96, no. 4, pp. 293-310, 1983

  • Ana Canas da Silva, Yael Karshon, Susan Tolman, Quantization of Presymplectic Manifolds and Circle Actions (arXiv:dg-ga/9705008)

Of generalized complex manifolds

Discussion in generalized complex geometry is in

In higher differential geometry

Geometric quantization in or with tools of higher differential geometry, notably with/over Lie groupoids is discussed in the following references.

  • Rogier Bos, Groupoids in geometric quantization PhD Thesis (2007) pdf

The geometric quantization of symplectic groupoids is accomplished in

Discussion of geometric prequantization in fully fledged higher geometry is in

Revised on August 17, 2014 13:43:46 by Urs Schreiber (89.204.155.233)