quantization

quantization

0.1

Introduction

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 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 (

http://planetmath.org/CAlgebra3

) associated with quantum operators on

Hilbert spaces

(as in

noncommutative geometry

). The

non-commutative

algebra of quantum observable operators is a

Clifford algebra

(

http://planetmath.org/CliffordAlgebra

), and the associated

C

*

-Clifford algebra (

http://planetmath.org/CCliffordAlgebra

) 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

,

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.

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

d

⁢

f

d

⁢

t

=

-

{

H

,

f

}

,

(1)

where

{

⋅

,

⋅

}

is the

Poisson bracket

. Equation (

1

) is

equivalent

to the

Hamilton equations

.

Remark

.

A classical system is sometimes defined more generally as a triplet

(

M

,

π

,

H

)

, where

π

is a Poisson

structure

on

M

.

Quantum systems

Definition 3

.

A

quantum system

is a pair

(

ℋ

,

H

^

)

, where

ℋ

is a Hilbert space and

H

^

is a

self-adjoint

linear operator

on

ℋ

.

If

(

ℋ

,

H

^

)

is a quantum system,

ℋ

is referred to as the (quantum) phase space and

H

^

is referred to as the

Hamiltonian operator

.

Definition 4

.

1.

A

quantum state

is a vector

Ψ

in

ℋ

.

2.

A

quantum observable

is a self-adjoint linear operator

A

on

ℋ

.

The space of quantum observables is denoted

𝒪

⁢

(

ℋ

)

. If

A

and

B

are in

𝒪

⁢

(

ℋ

)

, then

(

i

⁢

ℏ

)

-

1

⁢

[

A

,

B

]

:=

(

i

⁢

ℏ

)

-

1

⁢

(

A

⁢

B

-

B

⁢

A

)

(2)

is in

𝒪

⁢

(

ℋ

)

(Planck’s

constant

ℏ

appears as a

scaling

factor arising from physical considerations). The

operation

(

i

⁢

ℏ

)

-

1

⁢

[

⋅

,

⋅

]

thus gives

𝒪

⁢

(

ℋ

)

the structure of a Lie algebra.

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

d

⁢

A

d

⁢

t

=

i

ℏ

⁢

[

H

^

,

A

]

.

(3)

Equation (

3

) is equivalent to the

time-dependent

Schrödinger’s equation

i

⁢

ℏ

⁢

d

⁢

Ψ

d

⁢

t

=

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

,

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

⁢

(

ℋ

)

,

(5)

satisfying the following conditions:

1.

q

is a Lie algebra

homomorphism

,

2.

q

⁢

(

H

)

=

H

^

.

Remark

.

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

⁢

(

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

]

.

References

1

Abhay Ashtekar and Jerzy Lewandowski. 2005.

Quantum Geometry

and Its Applications.

http://cgpg.gravity.psu.edu/people/Ashtekar/articles/qgfinal.pdf

Available 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

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