In physics, the canonical commutation relation is the relation
![[x,p] = i\hbar](a/../upload/math/3315cfa4d6b0728b9fdb59593c67fa63.png)
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. This relation is attributed to Heisenberg, and it implies his uncertainty principle.
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:
- {x,p} = 1
This observation led Dirac to postulate that, in general, the quantum counterparts 
of classical observables f,g should satisfy
![[\hat f,\hat g]= i\hbar\widehat{\{f,g\}}.\,](a/../upload/math/a91946f8e4965a214da7b97b8ed9540e.png)
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 gaurenteed by the Stone-von Neumann theorem. The group associated with the commutation relations is called the Heisenberg group.
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 πx (in the example above it is p, or more generally, some functions involving the derivatives of the canonical coordinates with respect to time).
This definition of the canonical momentum ensures that one of the Euler-Lagrange equations has the form
The canonical commutation relations then say
![[x_i,\pi_j] = i\hbar\delta_{ij}](a/../upload/math/bdec8e6bc7ab92f6a507394fd59333a1.png)
where δij is the Kronecker symbol.
See also
canonical quantization
CCR algebra