Canonical commutation relation
Encyclopedia
In physics
, the canonical commutation relation is the relation between canonical conjugate quantities (quantities which are related by definition such that one is the Fourier transform
of another), for example:
between the position x and momentum px in the x direction of a point particle in one dimension, where
is the commutator
of x and px, i is the imaginary unit
, and ħ is the reduced Planck's constant h /2π . This relation is attributed to Max Born
, and it was noted by E. Kennard (1927) to imply the Heisenberg
uncertainty principle
.
, all observables commute and the commutator
would be zero. However, an analogous relation exists, which is obtained by replacing the commutator with the Poisson bracket
multiplied by i ħ:
This observation led Dirac
to propose that the quantum counterparts
of classical observables f, g satisfy
In 1946, Hip Groenewold demonstrated that a general systematic correspondence between quantum commutators and Poisson brackets could not hold consistently. However, he did appreciate that such a
systematic correspondence does, in fact, exist between the quantum commutator and a deformation of the Poisson bracket, the Moyal bracket
, and, in general, quantum operators and classical observables and distributions in phase space
. He thus finally elucidated the correspondence mechanism, Weyl quantization
, that underlies an alternate equivalent mathematical approach to quantization known as deformation quantization.
, quantum observables such as x and p should be represented as self-adjoint operator
s on some Hilbert space
. It is relatively easy to see that two operators satisfying the canonical commutation relations cannot both be bounded
. The canonical commutation relations can be made tamer by writing them in terms of the (bounded) unitary operator
s
and 
, which admit finite-dimensional representations as well. The resulting braiding relations for these are 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
associated with these commutation relations is called the Heisenberg group.
,
valid for the quantization
of the simplest classical system, can be generalized to the case of an arbitrary Lagrangian
. We identify canonical coordinates (such as x in the example above, or a field φ(x) in the case of quantum field theory
) and canonical momenta πx (in the example above it is p, or more generally, some functions involving the derivative
s of the canonical coordinates with respect to time):
.
This definition of the canonical momentum ensures that one of the Euler-Lagrange equation
s has the form
.
The canonical commutation relations then amount to
,
where δij is the Kronecker delta.
Further, it can be easily shown that
.
. However, in the presence of an electromagnetic field
, the canonical momentum p is not gauge invariant. The correct gauge-invariant momentum (or "kinetic momentum") is
(SI units) 
(cgs units
),
where q is the particle's electric charge
, A is the vector potential
, and c is the speed of light
. Although the quantity pkin 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
for a quantized charged particle of mass m in a classical electromagnetic field is (in cgs units)
where A is the three-vector potential and
is the scalar potential
. 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
for so(3), where
is the Levi-Civita symbol
. 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
. The inequivalence of these two formulations shows up in the Zeeman effect
and the Aharonov-Bohm effect
.
where
is the Levi-Civita symbol
and simply reverses the sign of the answer under pairwise interchange of the indices. An analogous relation holds for the spin
operators.
All such nontrivial commutation relations for pairs of operators lead to corresponding uncertainty relations
, involving positive semi-definite expectation contributions by their respective commutators and anticommutators. In general, for two Hermitian operators
A and B, consider expectation values in a system in the state ψ, the variances around the corresponding expectation values being (ΔA)2 ≡ ⟨ (A −<A>)2 ⟩, etc.
Then
where [A,B] ≡ AB−BA is the commutator of A and B, and {A,B} ≡ AB+BA is the anticommutator.
This follows through use of the Cauchy–Schwarz inequality
, since
|⟨A2⟩| |⟨B2⟩| ≥ |⟨AB⟩|2, and
AB = ([A,B] + {A,B}) /2 ; and similarly for the shifted operators A−⟨A⟩ and B−⟨B⟩ .
Judicious choices for A and B yield Heisenberg's familiar uncertainty relation,
for x and p, as usual; or, here, Lx and Ly ,
in angular momentum multiplets, ψ = |l , m ⟩ , useful constraints such as l (l+1) ≥ m (m+1), and hence l ≥ m, among others.
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, 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 from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
of another), for example:
between the position x and momentum px in the x direction of a point particle in one dimension, where
is the commutatorCommutator
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.-Group theory:...
of x and px, i is the imaginary unit
Imaginary unit
In mathematics, the imaginary unit allows the real number system ℝ to be extended to the complex number system ℂ, which in turn provides at least one root for every polynomial . The imaginary unit is denoted by , , or the Greek...
, and ħ is the reduced Planck's constant h /2π . This relation is attributed to Max Born
Max Born
Max Born was a German-born 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 was noted by E. Kennard (1927) to imply the Heisenberg
Werner Heisenberg
Werner Karl Heisenberg was a German theoretical physicist 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 mechanics, the Heisenberg uncertainty principle states a fundamental limit on the accuracy with which certain pairs of physical properties of a particle, such as position and momentum, can be simultaneously known...
.
Relation to classical mechanics
By contrast, in classical physicsClassical physics
What "classical physics" refers to depends on the context. When discussing special relativity, it refers to the Newtonian physics which preceded relativity, i.e. the branches of physics based on principles developed before the rise of relativity and quantum mechanics...
, 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.-Group 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 binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time-evolution of a Hamiltonian dynamical system...
multiplied by i ħ:
This observation led Dirac
Paul Dirac
Paul Adrien Maurice Dirac, OM, FRS was an English theoretical physicist who made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics...
to propose that the quantum counterparts
of classical observables f, g satisfyIn 1946, Hip Groenewold demonstrated that a general systematic correspondence between quantum commutators and Poisson brackets could not hold consistently. However, he did appreciate that such a
systematic correspondence does, in fact, exist between the quantum commutator and a deformation of the Poisson bracket, the Moyal bracket
Moyal bracket
In physics, the Moyal bracket is the suitably normalized antisymmetrization of the phase-space star product.The Moyal Bracket was developed in about 1940 by José Enrique Moyal, but Moyal only succeeded in publishing his work in 1949 after a lengthy dispute with Dirac...
, and, in general, quantum operators and classical observables and distributions 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...
. He thus finally elucidated the correspondence 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" kernel function in phase space invertibly...
, that underlies an alternate equivalent mathematical approach to quantization known as deformation quantization.
Representations
According to the standard mathematical formulation of quantum mechanicsMathematical formulation of quantum mechanics
The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics. Such are distinguished from mathematical formalisms for theories developed prior to the early 1900s by the use of abstract mathematical structures, such as...
, quantum observables such as x and p should be represented as self-adjoint operator
Self-adjoint operator
In mathematics, on a finite-dimensional inner product space, a self-adjoint operator is an operator that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose...
s on some Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
. It is relatively easy to see that two operators satisfying the canonical commutation relations cannot both be bounded
Bounded operator
In functional analysis, a branch of mathematics, 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 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 satisfyingU^*U=UU^*=I...
s
and , which admit finite-dimensional representations as well. The resulting braiding relations for these are 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 groupGroup (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
associated with these commutation relations is called the Heisenberg group.
Generalizations
The simple formula,valid for the quantization
Canonical quantization
In physics, canonical quantization is a procedure for quantizing a classical theory while attempting to preserve the formal structure of the classical theory, to the extent possible. Historically, this was Werner Heisenberg's route to obtaining quantum mechanics...
of the simplest classical system, can be generalized to the case of an arbitrary Lagrangian
Lagrangian
The Lagrangian, L, 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 by Irish mathematician William Rowan Hamilton known as...
. We identify canonical coordinates (such as x in the example above, or a field φ(x) in the case of quantum field theoryQuantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
) and canonical momenta πx (in the example above it is p, 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 one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
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, Euler's equation, or Lagrange's equation, is a differential equation whose solutions are the functions for which a given functional is stationary...
s has the form
.The canonical commutation relations then amount to
,where δij is the Kronecker delta.
Further, it can be easily shown that
.Gauge invariance
Canonical quantization is applied, by definition, on canonical coordinatesCanonical 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. Canonical coordinates arise naturally in physics in the study of Hamiltonian mechanics...
. However, in the presence of an electromagnetic field
Electromagnetic field
An electromagnetic field is a physical field produced by moving electrically charged objects. It affects the behavior of charged objects in the vicinity of the field. The electromagnetic field extends indefinitely throughout space and describes the electromagnetic interaction...
, the canonical momentum p is not gauge invariant. The correct gauge-invariant momentum (or "kinetic momentum") is
(SI units) (cgs unitsGaussian units
Gaussian units comprise a metric system of physical units. This system is the most common of the several electromagnetic unit systems based on cgs units. It is also called the Gaussian unit system, Gaussian-cgs units, or often just cgs units...
),
where q is the particle's electric charge
Electric charge
Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...
, A is the vector potential
Magnetic potential
The term magnetic potential can be used for either of two quantities in classical electromagnetism: the magnetic vector potential, A, and the magnetic scalar potential, ψ...
, and c is the speed of light
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
. Although the quantity pkin 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 (quantum mechanics)
In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...
for a quantized charged particle of mass m in a classical electromagnetic field is (in cgs units)
where A is the three-vector potential and
is the scalar potentialScalar potential
A scalar potential is a fundamental concept in vector analysis and physics . The scalar potential is an example of a scalar field...
. 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. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
for so(3), where
is the Levi-Civita symbolLevi-Civita symbol
The Levi-Civita symbol, also called the permutation symbol, antisymmetric symbol, or alternating symbol, is a mathematical 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 magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
. 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 mechanical phenomenon in which an electrically charged particle is affected by an electromagnetic field , despite being confined to a region in which both the magnetic field B and electric field E are...
.
Angular momentum operators
where
is the Levi-Civita symbolLevi-Civita symbol
The Levi-Civita symbol, also called the permutation symbol, antisymmetric symbol, or alternating symbol, is a mathematical symbol used in particular in tensor calculus...
and simply reverses the sign of the answer under pairwise interchange of the indices. An analogous relation holds for the spin
Spin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
operators.
All such nontrivial commutation relations for pairs of operators lead to corresponding uncertainty relations
Uncertainty principle
In quantum mechanics, the Heisenberg uncertainty principle states a fundamental limit on the accuracy with which certain pairs of physical properties of a particle, such as position and momentum, can be simultaneously known...
, involving positive semi-definite expectation contributions by their respective commutators and anticommutators. In general, for two Hermitian operators
Self-adjoint operator
In mathematics, on a finite-dimensional inner product space, a self-adjoint operator is an operator that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose...
A and B, consider expectation values in a system in the state ψ, the variances around the corresponding expectation values being (ΔA)2 ≡ ⟨ (A −<A>)2 ⟩, etc.
Then
where [A,B] ≡ AB−BA is the commutator of A and B, and {A,B} ≡ AB+BA is the anticommutator.
This follows through use of the Cauchy–Schwarz inequality
Cauchy–Schwarz inequality
In mathematics, the Cauchy–Schwarz inequality , is a useful inequality encountered in many different settings, such as linear algebra, analysis, probability theory, and other areas...
, since
|⟨A2⟩| |⟨B2⟩| ≥ |⟨AB⟩|2, and
AB = ([A,B] + {A,B}) /2 ; and similarly for the shifted operators A−⟨A⟩ and B−⟨B⟩ .
Judicious choices for A and B yield Heisenberg's familiar uncertainty relation,
for x and p, as usual; or, here, Lx and Ly ,
in angular momentum multiplets, ψ = |l , m ⟩ , useful constraints such as l (l+1) ≥ m (m+1), and hence l ≥ m, among others.