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[parent] quantization (Definition)

Quantization is understood as the process of defining a formal correspondence between a quantum system operator (such as the quantum Hamiltonian operator) or quantum algebra and a classical system operator (such as the Hamiltonian) or a classical algebra, such as the Poisson algebra. Theoretical quantum physicists often proceed in two `stages', so that both First Quantization and Second Quantization procedures were reported in QFT, for example. Generalized quantization procedures involve asymptotic morphisms and Wigner-Weyl-Moyal quantization procedures or noncommutative `deformations' of C*-algebras associated with quantum operators on Hilbert spaces (as in Noncommutative Geometry). The non-commutative algebra of quantum observable operators is a Clifford algebra, and the associated $ C^*$-Clifford algebra is a fundamental concept of modern mathematical treatments of quantum theories. Note, however, that classical systems, including Einstein's General Relativity are commutative (or Abelian) theories, whereas quantum theories are intrinsically non-commutative (or non-Abelian), most likely as a consequnece of the non-comutativity of quantum logics 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 variety 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, \omega, H)$, where $ (M, \omega)$ (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. A classical state is a point $ x$ in $ M$.
  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

$\displaystyle \frac{df}{dt} = -\{H,f\},$ (1)

where $ \{\cdot, \cdot\}$ is the Poisson bracket. Equation (1) is equivalent to the Hamilton equations.
Remark 1   A classical system is sometimes defined more generally as a triplet $ (M, \pi, H)$, where $ \pi$ is a Poisson structure on $ M$.

Quantum systems

Definition 3   A quantum system is a pair $ (\mathcal{H}, \hat{H})$, where $ \mathcal{H}$ is a Hilbert space and $ \hat{H}$ is a self-adjoint linear operator on $ \mathcal{H}$.

If $ (\mathcal{H}, \hat{H})$ is a quantum system, $ \mathcal{H}$ is referred to as the (quantum) phase space and $ \hat{H}$ is referred to as the Hamiltonian operator.

Definition 4  
  1. A quantum state is a vector $ \Psi$ in $ \mathcal{H}$.
  2. A quantum observable is a self-adjoint linear operator $ A$ on $ \mathcal{H}$.

The space of quantum observables is denoted $ \mathcal{O}(\mathcal{H})$. If $ A$ and $ B$ are in $ \mathcal{O}(\mathcal{H})$, then

$\displaystyle (i\hbar)^{-1}[A,B] := (i\hbar)^{-1}(AB - BA)$ (2)

is in $ \mathcal{O}(\mathcal{H})$ (Planck's constant $ \hbar$ appears as a scaling factor arising from physical considerations). The operation $ (i\hbar)^{-1}[\cdot,\cdot]$ thus gives $ \mathcal{O}(\mathcal{H})$ the structure of a Lie algebra.

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

$\displaystyle \frac{dA}{dt} = \frac{i}{\hbar} [\hat{H}, A].$ (3)

Equation (3) is equivalent to Schrödinger's equation
$\displaystyle i \hbar \frac{d\Psi}{dt} = \hat{H} \Psi.$ (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 $ (\mathcal{H}, \hat{H})$ from a triplet $ (M, \omega, H)$. Furthermore, in order to give physical meaning to the observables in the quantum system, there should be a map

$\displaystyle q\colon C^\infty(M) \to \mathcal{O}(\mathcal{H}),$ (5)

satisfying the following conditions:
  1. $ q$ is a Lie algebra homomorphism,
  2. $ q(H) = \hat{H}$.
Remark 2   Note that $ q$ is not an algebra homomorphism. Much of the complexity of quantization lies in the fact that, while $ C^\infty(M)$ is a commutative algebra, its image in $ \mathcal{O}(\mathcal{H})$ necessarily does not commute.

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

  • Canonical quantization
  • Geometric quantization
  • Deformation quantization
  • Path-integral quantization

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

Bibliography

1
Abhay Ashtekar and Jerzy Lewandowski. 2005. Quantum Geometry and Its Applications. Available PDF download.



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See Also: $C^*$-Clifford algebra, asymptotic morphisms and Wigner--Weyl--Moyal quantization procedures, Hamilton equations, Poisson bracket, Schrödinger's wave equation, canonical quantization, Hamiltonian operator, quantum space-times, quantum field theories, quantum electrodynamics

Other names:  quantisation, canonical quantization, Weyl-quantization
Also defines:  classical system, classical state, classical observable, quantum system, quantum state, quantum observable
Keywords:  quantization, symplectic, Poisson, Hamiltonian operator, Lie algebra, Clifford algebra

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Cross-references: image, homomorphism, map, order, Schrödinger's equation, Lie algebra, operation, factor, scaling, vector, linear operator, self-adjoint, structure, Hamilton equations, equivalent, Poisson bracket, equation, function, configuration space, manifold, cotangent bundle, smooth function, symplectic manifold, triplet, points, variety, quantum logics, non-Abelian, theories, abelian, commutative, non-commutative, noncommutative geometry, Hilbert spaces, deformations, noncommutative, asymptotic morphisms and Wigner--Weyl--Moyal quantization procedures, QFT, Poisson algebra, Hamiltonian, algebra, Hamiltonian operator, operator
There are 29 references to this entry.

This is version 17 of quantization, born on 2005-12-25, modified 2008-10-17.
Object id is 7540, canonical name is Quantization.
Accessed 6125 times total.

Classification:
AMS MSC: 81R15 (Quantum theory :: Groups and algebras in quantum theory :: Operator algebra methods)
 81R50 (Quantum theory :: Groups and algebras in quantum theory :: Quantum groups and related algebraic methods)
 46L65 (Functional analysis :: Selfadjoint operator algebras :: Quantizations, deformations)
 53D50 (Differential geometry :: Symplectic geometry, contact geometry :: Geometric quantization)
 81S10 (Quantum theory :: General quantum mechanics and problems of quantization :: Geometry and quantization, symplectic methods)
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