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In physics
Physics

Physics is the natural science which examines basic concepts such as energy, force, and spacetime and all that derives from these, such as mass, charge, matter and its Motion ....
, the canonical commutation relation is the relation between canonical conjugate quantities (quantities which are related by definition such that one is the Fourier transform
Fourier transform

In mathematics, Fourier analysis is a subject area which grew out of the study of Fourier series. The subject began with trying to understand when it was possible to represent general functions by sums of simpler trigonometric functions....
 of another), for example:

between the position and momentum in the direction of a point particle in one dimension, where is the so-called commutator
Commutator

In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory....
 of and , is the imaginary unit
Imaginary unit

In mathematics, physics, and engineering, the imaginary unit is denoted by  or the Latin   or the Greek iota . It allows the real number system, to be extended to the complex number system,   Its precise definition is dependent upon the particular method of extension....
 and is the reduced Planck's constant . This relation is attributed to Max Born
Max Born

Max Born was a Germany physicist and mathematician who was instrumental in the development of quantum mechanics. He also made contributions to solid-state physics and optics and supervised the work of a number of notable physicists in the 1920s and 30s....
, and it implies the Heisenberg
Werner Heisenberg

Werner Heisenberg was a German Theoretical physics who made foundational contributions to quantum mechanics and is best known for asserting the uncertainty principle of quantum theory....
 uncertainty principle
Uncertainty principle

In quantum physics, the Werner Heisenberg uncertainty principle states that certain physical quantities, like the position and momentum, cannot both have precise values at the same time....
.

observation led Dirac
Paul Dirac

Paul Adrien Maurice Dirac, Order of Merit , Royal Society was a United Kingdom theoretical physicist. Dirac made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics....
 to postulate that, in general, the quantum counterparts of classical observables should satisfy

In 1927, Hermann Weyl
Hermann Weyl

Hermann Klaus Hugo Weyl was a Germany mathematician. Although much of his working life was spent in Z?rich, Switzerland and then Princeton, New Jersey, he is associated with the University of G?ttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski....
 showed that a literal correspondence between a quantum operator and a classical distribution in phase space
Phase space

In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space....
 could not hold.






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In physics
Physics

Physics is the natural science which examines basic concepts such as energy, force, and spacetime and all that derives from these, such as mass, charge, matter and its Motion ....
, the canonical commutation relation is the relation between canonical conjugate quantities (quantities which are related by definition such that one is the Fourier transform
Fourier transform

In mathematics, Fourier analysis is a subject area which grew out of the study of Fourier series. The subject began with trying to understand when it was possible to represent general functions by sums of simpler trigonometric functions....
 of another), for example:

between the position and momentum in the direction of a point particle in one dimension, where is the so-called commutator
Commutator

In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory....
 of and , is the imaginary unit
Imaginary unit

In mathematics, physics, and engineering, the imaginary unit is denoted by  or the Latin   or the Greek iota . It allows the real number system, to be extended to the complex number system,   Its precise definition is dependent upon the particular method of extension....
 and is the reduced Planck's constant . This relation is attributed to Max Born
Max Born

Max Born was a Germany physicist and mathematician who was instrumental in the development of quantum mechanics. He also made contributions to solid-state physics and optics and supervised the work of a number of notable physicists in the 1920s and 30s....
, and it implies the Heisenberg
Werner Heisenberg

Werner Heisenberg was a German Theoretical physics who made foundational contributions to quantum mechanics and is best known for asserting the uncertainty principle of quantum theory....
 uncertainty principle
Uncertainty principle

In quantum physics, the Werner Heisenberg uncertainty principle states that certain physical quantities, like the position and momentum, cannot both have precise values at the same time....
.

Relation to classical mechanics


By contrast, in classical physics
Classical physics

Classical physics is a general term used to describe the branches of physics based on principles developed before the rise of general theory of relativity and Quantum mechanics, usually including special theory of relativity....
 all observables commute and the commutator
Commutator

In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory....
 would be zero; however, an analogous relation exists, which is obtained by replacing the commutator with the Poisson bracket
Poisson bracket

In mathematics and classical mechanics, the Poisson bracket is an important operator in Hamiltonian mechanics, playing a central role in the definition of the time-evolution of a dynamical system in the Hamiltonian formulation....
 and the constant with :

This observation led Dirac
Paul Dirac

Paul Adrien Maurice Dirac, Order of Merit , Royal Society was a United Kingdom theoretical physicist. Dirac made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics....
 to postulate that, in general, the quantum counterparts of classical observables should satisfy

In 1927, Hermann Weyl
Hermann Weyl

Hermann Klaus Hugo Weyl was a Germany mathematician. Although much of his working life was spent in Z?rich, Switzerland and then Princeton, New Jersey, he is associated with the University of G?ttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski....
 showed that a literal correspondence between a quantum operator and a classical distribution in phase space
Phase space

In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space....
 could not hold. However, he did propose a mechanism, Weyl quantization
Weyl quantization

In mathematics and physics, in the area of quantum mechanics, Weyl quantization is a method for systematically associating a "quantum mechanical" Hermitian operator with a "classical" distribution in phase space invertibly....
, that underlies a mathematical approach to quantization known as deformation quantization.

Representations


According to the standard mathematical formulation of quantum mechanics
Mathematical formulation of quantum mechanics

The mathematical formulation of quantum mechanics is the body of mathematical formalisms which permits a rigorous description of quantum mechanics....
, quantum observables such as and should be represented as self-adjoint operator
Self-adjoint operator

In mathematics, on a finite-dimensional inner product space, a self-adjoint operator is one that is its own Adjoint of an operator, or, equivalently, one whose matrix is Hermitian matrix, where a Hermitian matrix is one which is equal to its own conjugate transpose....
s on some Hilbert space
Hilbert space

The mathematics concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra from the two-dimensional plane and three-dimensional space to infinite-dimensional spaces....
. It is relatively easy to see that two operator
Operator

In mathematics, an operator is a function which operates on another function. Often, an "operator" is a function which acts on functions to produce other functions ; or it may be a generalization of such a function, as in linear algebra, where some of the terminology reflects the origin of the subject in operations on the functions which ar...
s satisfying the canonical commutation relations cannot both be bounded
Bounded operator

In functional analysis , a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L to that of v is bounded set by the same number, over all non-zero vectors v in X....
. The canonical commutation relations can be made tamer by writing them in terms of the (bounded) unitary operator
Unitary operator

In functional analysis, a branch of mathematics, a unitary operator is a bounded linear operator U : H ? H on a Hilbert space H satisfying...
s and . The result is the so-called Weyl relations. The uniqueness of the canonical commutation relations between position and momentum is guaranteed by the Stone-von Neumann theorem. The group
Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an Binary operation that combines any two of its element to form a third element....
 associated with the commutation relations is called the Heisenberg group
Heisenberg group

In mathematics, the term Heisenberg group, named after Werner Heisenberg, refers to the group of 3×3 triangular matrix of the formor its generalizations....
.

Generalizations


The simple formula

,

valid for the quantization
Quantization

Quantization is the procedure of constraining something from a continuous set of values to a discrete set . Quantization in specific domains is discussed in:...
 of the simplest classical system, can be generalized to the case of an arbitrary Lagrangian
Lagrangian

The Lagrangian, , of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics known as Lagrangian mechanics....
 . We identify canonical coordinates (such as in the example above, or a field in the case of quantum field theory
Quantum field theory

Quantum field theory or QFT provides a theoretical framework for constructing quantum mechanics models of systems classically described by field or of Many-body problem....
) and canonical momenta (in the example above it is , or more generally, some functions involving the derivative
Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much a quantity is changing at a given point....
s of the canonical coordinates with respect to time).

This definition of the canonical momentum ensures that one of the Euler-Lagrange equation
Euler-Lagrange equation

In calculus of variations, the Euler?Lagrange equation, or Lagrange's equation, is a differential equation whose solutions are the function s for which a given functional is stationary point....
s has the form

The canonical commutation relations then say

where is the Kronecker delta
Kronecker delta

In mathematics, the Kronecker delta or Kronecker's delta, named after Leopold Kronecker , is a Function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise....
.

Also, it can be shown,

,

Gauge invariance

Canonical quantization is performed, by definition, on canonical coordinates
Canonical coordinates

In mathematics and classical mechanics, canonical coordinates are particular sets of coordinates on the phase space, or equivalently, on the cotangent manifold of a manifold....
. However, in the presence of an electromagnetic field
Electromagnetic field

The electromagnetic field is a physical field produced by electric charge. It affects the behavior of charged objects in the vicinity of the field....
, the canonical momentum p is not gauge invariant. The correct gauge-invariant momentum (or "kinetic momentum") is (in SI-units)

where e is the electric charge
Electric charge

Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. Electrically charged matter is influenced by, and produces, electromagnetic fields....
, and A is the vector potential
Vector potential

In vector calculus, a vector potential is a vector field whose Curl is a given vector field. This is analogous to a scalar potential, which is a scalar field whose negative gradient is a given vector field....
 and c is the speed of light
Speed of light

The speed of light in an free space is an important physical constant usually written as c, with a value of 299,792,458 metres per second....
. Although the quantity is the "physical momentum" in that it is the quantity to be identified with momentum in laboratory experiments, it does not satisfy the canonical commutation relations; only the canonical momentum does that. This can be seen as follows.

The non-relativistic Hamiltonian
Hamiltonian

Hamiltonian may refer toIn mathematics:* Hamiltonian system* Hamiltonian path, in graph theory* Hamiltonian group, in group theory* Hamiltonian ...
 for a quantized charged particle of mass m in a classical electromagnetic field is

where A is the three-vector potential and is the scalar potential
Scalar potential

A scalar potential is a fundamental concept in vector analysis and physics . Given a vector field F, its scalar potential V is a scalar field whose negative gradient is F,...
. This form of the Hamiltonian, as well as the Schroedinger equation , the Maxwell equations and the Lorentz force law are invariant under the gauge transformation

where

and is the gauge function.

The canonical angular momentum is

and obeys the canonical quantization relations

defining the Lie algebra
Lie algebra

In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds....
 for so(3), where is the Levi-Civita symbol
Levi-Civita symbol

The Levi-Civita symbol, also called the permutation symbol, antisymmetric symbol, or alternating symbol, is a mathematics symbol used in particular in tensor calculus....
. Under gauge transformations, the angular momentum transforms as

The gauge-invariant angular momentum (or "kinetic angular momentum") is given by

which has the commutation relations

where

is the magnetic field
Magnetic field

A magnetism field is a vector field which can exert a magnetic force on moving electric charges and on magnetic dipoles . When placed in a magnetic field, magnetic dipoles tend to align their axes parallel to the magnetic field....
. The inequivalence of these two formulations shows up in the Zeeman effect
Zeeman effect

The Zeeman effect is the splitting of a spectral line into several components in the presence of a static magnetic field. It is analogous to the Stark effect, the splitting of a spectral line into several components in the presence of an electric field....
 and the Aharonov-Bohm effect
Aharonov-Bohm effect

The Aharonov?Bohm effect, sometimes called the Ehrenberg?Siday?Aharonov?Bohm effect, is a quantum mechanics phenomenon by which a charged particle is affected by electromagnetic fields in regions from which the particle is excluded....
.

Examples


Angular momentum operator
Angular momentum operator

In quantum mechanics, the angular momentum operator is an operator analogous to classical angular momentum. The angular momentum operator plays a central role in the theory of atomic physics and other quantum problems involving rotational symmetry....
s

Here is the Levi-Civita symbol
Levi-Civita symbol

The Levi-Civita symbol, also called the permutation symbol, antisymmetric symbol, or alternating symbol, is a mathematics symbol used in particular in tensor calculus....
 and simply reverses the sign of the answer under acyclic permutation of the indices. An analogous relation holds for the spin
Spin (physics)

In quantum mechanics, spin is a fundamental property of atomic nucleus, hadrons, and elementary particles. For particles with non-zero spin, spin direction is an important intrinsic degrees of freedom ....
 operators.

See also

  • Canonical quantization
    Canonical quantization

    In physics, canonical quantization is one of many procedures for quantization a classical theory. Historically, this was the earliest method to be used to build quantum mechanics....
  • CCR algebra
  • Lie derivative
    Lie derivative

    In mathematics, the Lie derivative, named after Sophus Lie by Wladyslaw Slebodzinski, evaluates the change of one vector field along the flow of another vector field....


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