0.1 Introduction

Quantization is understood as the process of defining a formal correspondence between a quantum system operator (such as the quantum Hamiltonian operatorPlanetmathPlanetmath) or quantum algebra and a classical system operator (such as the HamiltonianPlanetmathPlanetmath) or a classical algebraPlanetmathPlanetmath, such as the Poisson algebraPlanetmathPlanetmath. Theoretical quantum physicists often proceed in two ‘stages’, so that both first and second quantization procedures were reported in QFT, for example. Generalized quantization procedures involve asymptotic morphismsMathworldPlanetmathPlanetmath and Wigner–Weyl–Moyal quantization procedures or noncommutativedeformationsMathworldPlanetmath’ of C*-algebras (http://planetmath.org/CAlgebra3) associated with quantum operators on Hilbert spacesMathworldPlanetmath (as in noncommutative geometryPlanetmathPlanetmath). The non-commutative algebra of quantum observable operators is a Clifford algebraPlanetmathPlanetmath (http://planetmath.org/CliffordAlgebra), and the associated C*-Clifford algebra (http://planetmath.org/CCliffordAlgebra) is a fundamental concept of modern mathematical treatments of quantum theoriesPlanetmathPlanetmath. Note, however, that classical systems, including Einstein’s general relativity are commutativePlanetmathPlanetmath (or Abelian) theories, whereas quantum theories are intrinsically non-commutative (or non-AbelianMathworldPlanetmath), most likely as a consequnece of the non-comutativity of quantum logicsPlanetmathPlanetmath and the Heisenberg uncertainty principle of quantum mechanics.

This definition is quite broad, and as a result there are many approaches to quantization, employing a varietyMathworldPlanetmathPlanetmath of techniques. It should be emphasized the result of quantization is not unique; in fact, methods of quantization usually possess inherent ambiguities, in the sense that, while performing quantization, one usually must make choices at certain points of the process.

Classical systems

Definition 1.

A classical system is a triplet (M,ω,H), where (M,ω) (the phase space) is a symplectic manifold and H (the Hamiltonian) is a smooth function on M.

In most physical examples the phase space M is the cotangent bundle T*X of a manifold X. In this case, X is called the configuration space.

Definition 2.
  1. 1.

    A classical state is a point x in M.

  2. 2.

    A classical observable is a function f on M.

In classical mechanics, one studies the time-evolution of a classical system. The time-evolution of an observable is described the equation

dfdt=-{H,f}, (1)

where {,} is the Poisson bracket. Equation (1) is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to the Hamilton equations.


A classical system is sometimes defined more generally as a triplet (M,π,H), where π is a Poisson structureMathworldPlanetmath on M.

Quantum systems

Definition 3.

A quantum system is a pair (,H^), where is a Hilbert space and H^ is a self-adjointPlanetmathPlanetmath linear operator on .

If (,H^) is a quantum system, is referred to as the (quantum) phase space and H^ is referred to as the Hamiltonian operator.

Definition 4.
  1. 1.

    A quantum state is a vector Ψ in .

  2. 2.

    A quantum observable is a self-adjoint linear operator A on .

The space of quantum observables is denoted 𝒪(). If A and B are in 𝒪(), then

(i)-1[A,B]:=(i)-1(AB-BA) (2)

is in 𝒪() (Planck’s constant appears as a scaling factor arising from physical considerations). The operationMathworldPlanetmath (i)-1[,] thus gives 𝒪() the structure of a Lie algebra.

The time evolution of a quantum observable is described by the equation

dAdt=i[H^,A]. (3)

Equation (3) is equivalent to the time-dependent Schrödinger’s equation

idΨdt=H^Ψ. (4)

The problem of quantization

The problem of quantization is to find a correspondence between a quantum system and a classical system; this is clearly not always possible. Thus, specific methods of quantization describe several ways of constructing a pair (,H^) from a triplet (M,ω,H). Furthermore, in order to give physical meaning to the observables in the quantum system, there should be a map

q:C(M)𝒪(), (5)

satisfying the following conditions:

  1. 1.

    q is a Lie algebra homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath,

  2. 2.



Note that q is not an algebra homomorphism. Much of the complexity of quantization lies in the fact that, while C(M) is a commutative algebra, its image in 𝒪() necessarily does not commute.

The following is a list of some well-known methods of quantization:

  • Geometric quantization

  • Deformation quantization

  • Path-integral quantization

A detailed example of geometric quantization on quantum Riemannian spaces can be found in ref. [1].


  • 1 Abhay Ashtekar and Jerzy Lewandowski. 2005. Quantum Geometry and Its Applications. http://cgpg.gravity.psu.edu/people/Ashtekar/articles/qgfinal.pdfAvailable PDF download.
Title quantization
Canonical name Quantization
Date of creation 2013-03-22 15:36:59
Last modified on 2013-03-22 15:36:59
Owner bci1 (20947)
Last modified by bci1 (20947)
Numerical id 27
Author bci1 (20947)
Entry type Topic
Classification msc 81S10
Classification msc 53D50
Classification msc 46L65
Classification msc 81R50
Classification msc 81R15
Synonym quantisation
Synonym canonical quantization
Synonym Weyl-quantization
Related topic CCliffordAlgebra
Related topic AsymptoticMorphismsAndWignerWeylMoyalQuantizationProcedures
Related topic HamiltonianEquations
Related topic PoissonBracket
Related topic SchrodingersWaveEquation
Related topic CanonicalQuantization
Related topic HamiltonianOperatorOfAQuantumSystem
Related topic QuantumSpaceTimes
Related topic QFTOrQuantumFieldTheories
Related topic QEDInTh
Defines classical system
Defines classical state
Defines classical observable
Defines quantum system
Defines quantum state
Defines quantum observable