In physics, the canonical commutation relation is the relation A black hole concept drawing by NASA. Physics (from the Greek, φυσικός (physikos), natural, and φύσις (physis), nature) is the science of the natural world dealing with the fundamental constituents of the universe, the forces they exert on one another, and the results produced by these forces. ...

among the position x and momentum p of a point particle in one dimension, where [x,p] = xp − px is the so-called commutator of x and p, i is the imaginary unit and is the reduced Planck's constant h / 2π. This relation is attributed to Heisenberg, and it implies his uncertainty principle. For an electrical switch that periodically reverses the current see commutator (electric) In mathematics, the commutator gives an indication of how poorly a certain binary operation fails to be commutative. ... In mathematics, the imaginary unit i (sometimes also represented by the Latin j or the Greek iota, but in this article i will be used exclusively) allows the real number system to be extended to the complex number system . ... A commemoration plaque for Max Planck on his discovery of Plancks constant, in front of Humboldt University, Berlin. ... Werner Heisenberg Werner Karl Heisenberg (December 5, 1901 – February 1, 1976) was a celebrated German physicist and Nobel laureate, one of the founders of quantum mechanics. ... In quantum physics, the Heisenberg uncertainty principle states that one cannot assign with full precision values for certain pairs of observable variables, including the position and momentum, of a single particle at the same time even in theory. ...

Relation to classical mechanics

By contrast, in classical physics 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 and the constant with 1: Classical physics is physics based on principles developed before the rise of quantum theory, including the special theory of relativity. ... For an electrical switch that periodically reverses the current see commutator (electric) In mathematics, the commutator gives an indication of how poorly a certain binary operation fails to be commutative. ... 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. ...

{x,p} = 1

This observation led Dirac to postulate that, in general, the quantum counterparts of classical observables f,g should satisfy Paul Adrien Maurice Dirac Paul Adrien Maurice Dirac, OM (IPA: [dɪræk]) (August 8, 1902 – October 20, 1984) was a British theoretical physicist and a founder of the field of quantum physics. ...

Representations

According to the standard mathematical formulation of quantum mechanics, quantum observables such as x and p should be represented as self-adjoint operators 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 operators e ^{− ikx} and e ^{− iap}. 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 associated with the commutation relations is called the Heisenberg group. One of the remarkable characteristics of the mathematical formulation of quantum mechanics, which distinguishes it from mathematical formulations of theories developed prior to the early 1900s, is its use of abstract mathematical structures, such as Hilbert spaces and operators on these spaces. ... On a finite-dimensional inner product space, a self-adjoint operator is one 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. ... In mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. ... In mathematics, an operator is some kind of function; if it comes with a specified type of operand as function domain, it is no more than another way of talking of functions of a given type. ... In mathematics, the operator norm is a norm defined on the space of bounded operators between two Banach spaces. ... In functional analysis, a unitary operator is a bounded linear operator U on a Hilbert space satisfying U*U=UU*=I where I is the identity operator. ... In mathematics and in theoretical physics, the Stone-von Neumann theorem is any one of a number of different formulations of the uniqueness of the canonical commutation relations between position and momentum operators. ... In mathematics and in theoretical physics, the Stone-von Neumann theorem is any one of a number of different formulations of the uniqueness of the canonical commutation relations between position and momentum operators. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, the Heisenberg group is a group of 3×3 upper triangular matrices of the form Elements a,b,c can be taken from some (arbitrary) commutative ring. ...

Generalizations

The simple formula

,

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 π_x (in the example above it is p, or more generally, some functions involving the derivatives of the canonical coordinates with respect to time). Generally, quantization is the state of being constrained to a set of discrete values, rather than varying continuously. ... A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a function of the dynamical variables and concisely describes the equations of motion of the system. ... Quantum field theory (QFT) is the application of quantum mechanics to fields. ... In mathematics, the derivative is one of the two central concepts of calculus. ...

This definition of the canonical momentum ensures that one of the Euler-Lagrange equations has the form The Euler-Lagrange Equation is the major formula of the Calculus of variations. ...

The canonical commutation relations then say

where δ_ij is the Kronecker delta. In mathematics, the Kronecker delta or Kroneckers delta, named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. ...

See also

canonical quantization CCR algebra In physics, canonical quantization is one of many procedures for quantizing a classical theory. ... In quantum field theory, if V is a real vector space equipped with a nonsingular real antisymmetric bilinear form (,) (i. ...