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 $U(1)$-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 – an 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 $X$ with a symplectic structure $\omega$.

2. Then we do prequantization: this gives us a Hermitian line bundle $L$ over $X$, equipped with a $U(1)$ connection $D$ whose curvature equals $i\omega$. $L$ is called the prequantum line bundle.

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

3. The Hilbert space $H_0$ of square-integrable sections of $L$ 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 $X$ to an operator on $H_0$. This map takes Poisson brackets to commutators, just as one would hope. The formula for this map involves the connection $D$.

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

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

5. The easiest sort of polarization to understand is a real polarization. This is where the subspaces $P_x$ come from subspaces of the tangent space by complexification. It boils down to this: a real polarization is an integrable distribution $P$ on the classical phase space where each space $P_x$ is Lagrangian subspace of the tangent space $T_{x}X$.

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 $X$ to be the cotangent bundle $T^{*}M$ of a manifold $M$ called the classical configuration space. We then let our quantum Hilbert space be the space of all square-integrable functions on $M$.

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

7. It’s also good to study the Bargmann–Segal representation, which we get by taking $X={ℂ}^n$ 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 ${ℂ}^n$ 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 $H$. 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” $V$ on a manifold $X$ is a choice of a subspace $V_x$ of each tangent space $T_{p}X$, where the choice depends smoothly on $x$.

• Hamiltonian vector field: Given a manifold $X$ with a symplectic structure $\omega$, any smooth function $f:X\to ℝ$ 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 $X$ describing the time evolution of states. Mathematically speaking, this flow is generated by a vector field $v(f)$ called the “Hamiltonian vector field” associated to $f$. It is the unique vector field such that

  $$\omega (-,v(f))=df$$\omega(-, v(f)) = d f

  In other words, for any vector field $u$ on $X$ we have

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

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

• integrable distribution: A distribution on a manifold $X$ is “integrable” if at least locally, there is a foliation of $X$ by submanifolds such that $V_x$ is the tangent space of the submanifold containing the point $x$.

• integral cohomology class: Any closed $p$-form on a manifold $M$ defines an element of the $p$th de Rham cohomology of $M$. This is a finite-dimensional vector space, and it contains a lattice called the $p$th integral cohomology group of $M$. We say a cohomology class is integral if it lies in this lattice. Most notably, if you take any $U(1)$ connection on any Hermitian line bundle over $M$, its curvature $2$-form will define an integral cohomology class once you divide it by $2\pi i$. 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 $M$ and given two smooth functions on that manifold, say $f$ and $g$, there’s a trick for getting a new smooth function $\{f,g\}$ on the manifold, called the Poisson bracket of $f$ and $g$.

  This trick works as follows: given any smooth function $f$ we can take its differential $df$, which is a $1$-form. Then there is a unique vector field $v(f)$, the Hamiltonian vector field associated to $f$, such that

  $$\omega (-,v(f))=df$$\omega(-,v(f)) = d f

  Using this we define

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

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

  In the familiar case where $M$ is ${ℝ}^{2n}$ with momentum and position coordinates $p_i$, $q_i$, the Poisson brackets of $f$ and $g$ work out to be

  $$\{f,g\}=\sum _i\frac{df}{d{p}_i}\frac{dg}{d{q}_i}-\frac{df}{d{q}_i}\frac{dg}{d{p}_i}$$\{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 $X$ 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 $H_0$ 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 $X$.

• symplectic structure: A symplectic structure on a manifold $M$ is a closed $2$-form $\omega$ which is nondegenerate in the sense that for any nonzero tangent vector $u$ at any point of $M$, there is a tangent vector $u$ at that point for which $w(u,v)$ is nonzero.

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

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

Definition

Geometric quantization involves two steps

1. Geometric prequantization

2. Geometric quantization proper.

Geometric prequantization

Prequantum line bundle

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

In other words, prequantization is a lift of $\omega$ 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 $\nabla :X\to BU(1{)}_{\mathrm{conn}}$ be a prequantum line bundle $E\to X$ with connection for $\omega$. Write ${\Gamma }_X(E)$ for its space of smooth sections, the prequantum space of states.

Geometric quantization

Given a prequantum bundle as above, the remaining step of genuine geometric quantization consists of forming half its space of sections in a certain sense. See at polarization for the physical intuition behind this

A traditional way to formalize this is as a 3-step process

1. choose a Polarization

2. choose a Metaplectic correction

3. form the induced Space of states as the space of polarized sections sensored a certain half-form bundle.

Another way which works more generally and coincides with this prescription under mild conditions is to

1. choose a spin^c structure compatible with the given prequantum bundle

2. form the fiber integration in differential K-theory of the prequantum bundle.

This spin^c quantization is discussed further at geometric quantization by push-forward.

Polarizations

For $(X,\omega )$ a symplectic manifold, a polarization is a foliation of $X$ by Lagrangian submanifolds with respect to $\omega$.

After a choice of prequantum line bundle $\nabla$ lifting $\omega$, a Bohr-Sommerfeld leaf of a polarization is a leaf on which the prequantum line bundle is not just flat, but also trivializable as a circle bundle.

Metaplectic correction

For the moment see at metaplectic correction (in geometric quantization).

${\mathrm{Spin}}^c$-structure

Quantum state space

(space of sections of the prequantum line bundle whose covariant derivative along the polarizaiton leaves vanishes…)

For the moment see at space of states (in geometric quantization)_.

Properties

Compatibility of quantization with symplectic reduction

On the relation between geometric quantization and symplectic reduction:

• Guillemin-Sternberg geometric quantization conjecture

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 $𝔾$-principal ∞-connection)

$n$	geometry	structure	unextended structure	extension by	quantum extension
$\mathrm{\infty }$	higher prequantum geometry	cohesive ∞-group	Hamiltonian symplectomorphism ∞-group	moduli ∞-stack of $(\Omega 𝔾)$-flat ∞-connections on $X$	quantomorphism ∞-group
1	symplectic geometry	Lie algebra	Hamiltonian vector fields	real numbers	Hamiltonians under Poisson bracket
1		Lie group	Hamiltonian symplectomorphism group	circle group	quantomorphism group
2	2-plectic geometry	Lie 2-algebra	Hamiltonian vector fields	line Lie 2-algebra	Poisson Lie 2-algebra
2		Lie 2-group	Hamiltonian 2-plectomorphisms	circle 2-group	quantomorphism 2-group
$n$	n-plectic geometry	Lie n-algebra	Hamiltonian vector fields	line Lie n-algebra	Poisson Lie n-algebra
$n$		smooth n-group	Hamiltonian n-plectomorphisms	circle n-group	quantomorphism n-group

(extension are listed for sufficiently connected $X$)

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 $T^{*}ℝ\simeq {ℝ}^2$, where the isomorphism is given by choosing standard coordinates, $\{q,p\}$.

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

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

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

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

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

which with the above choice of connection is equivalently

The solutions to this differential equation are of the form

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

Since we have the hamiltonian vector fields

the action of the quantum operators $\hat{q}$ and $\hat{p}$ on these states is

and

Kähler manifolds

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

• the prequantum line bundle is a holomorphic line bundle since its curvature is $\omega \mathrm{in}{T}_{(1,1)}^{*}X$;

• a polarization is given by the anti-holomorphic tangent bundle $T^{0,1}X$;

• a polarized section is a holomorphic section;

• the dimension of the space of states (in geometric quantization) is given by the Riemann-Roch theorem, see (Hitchin).

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

Theta functions

See at theta function.

Quantization of Chern-Simons theory

The geometric quantization of Chern-Simons theory leads to invariants for 3-manifolds. A detailed discussion is in (Witten 89).

Quantization of the bosonic string $\sigma$-model

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

Related concepts

• prequantum geometry

• reductions deformations resolutions in physics

• Weinstein symplectic category

• prequantum field theory

  • prequantum circle n-bundle

• quantization

  • deformation quantization

  • geometric quantization, higher geometric quantization

    • geometric quantization of symplectic groupoids

    • spin-c quantization

    • geometric quantization of non-integral forms

  • path integral

  • semiclassical approximation

duality between algebra and geometry in physics:

algebra	geometry
Poisson algebra	Poisson manifold
deformation quantization	geometric quantization
algebra of observables	space of states
Heisenberg picture	Schrödinger picture
AQFT	FQFT
higher algebra	higher geometry
Poisson n-algebra	n-plectic manifold
En-algebras	higher symplectic geometry
BD-BV quantization	higher geometric quantization
factorization algebra of observables	extended quantum field theory
factorization homology	cobordism representation

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

• Nicholas Woodhouse, Geometric Quantization, Oxford University Press (1997)

Introductions and lecture notes include

• William Gordon Ritter, Geometric Quantization (arXiv:math-ph/0208008)

• Matthias Blau, Symplectic Geometry and Geometric Quantization (pdf)

• Eugene Lerman, Geometric quantization; a crash course (arXiv:1206.2334)

Lecture notes with an emphasis on semiclassical states are in

• Sean Bates, Alan Weinstein, Lectures on the geometry of quantization, pdf

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

• John Baez, Geometric Quantization (web)

Some useful talk notes include

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

${\mathrm{Spin}}^{ℂ}$-Quantization

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)

• Andrei Tyurin, Quantization, classical and quantum field theory and theta functions (arXiv:math/0210466v1)

• 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)

Relation to other formalisms

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

• Eli Hawkins, The Correspondence between Geometric Quantization and Formal Deformation Quantization (arXiv:math/9811049)

• Eli Hawkins, Geometric Quantization of Vector Bundles (arXiv:math/9808116)

• Christoph Nölle, Geometric and deformation quantization (arXiv:0903.5336)

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.

• Takashi Kimura, BRST Quantization and Poisson Reduction, PhD Thesis (1990) (web)

• José Figueroa-O’Farrill, Takashi Kimura, Geometric BRST quantization (pdf)

  Geometric BRST quantization. I. Prequantization, Comm. Math. Phys. Volume 136, Number 2 (1991), 209-229. (Euclid)

• Gijs Tuynman, Geometric quantization of the BRST charge Comm. Math. Phys. Volume 150, Number 2 (1992), 237-265. (Euclid, pdf)

• Ronald Fulp, BRST Extension of Geometric Quantization, Foundations of Physics Volume 37, Number 1 (2007), (arXiv:math/0604270)

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 $U$ 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)

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

• Eli Hawkins, A Groupoid Approach to Quantization (arXiv:math.SG/0612363)

Discussion of geometric prequantization in fully fledged higher geometry is in

• Domenico Fiorenza, Chris Rogers, Urs Schreiber,

  Higher geometric prequantum theory

  L-∞ algebras of local observables from higher prequantum bundles