# nLab canonical commutation relation

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Idea

In contexts related to quantum mechanics and quantum field theory, by the “canonical commutation relations” (CCR) one refers to the commutator relations in Weyl algebras, i.e. associative algebras generated from elements $\{a_k, a^\ast_k\}_{k \in K}$ subject to the “canonical” expressions for the commutators $[a,b] \coloneqq a \cdot b - b \cdot a$

$\underset{i,j \in K}{\forall} \left( [a_i, a_j] = 0 = [a^\ast_i, a^\ast_j] \right)$
$\underset{i,j \in K}{\forall}\left( [a_i, a^\ast_j] = diag((a_k))_{i, j} \right) \,,$

where $diag((a_k))$ is some diagonal matrix with entries $(a_k)_{k \in K}$.

The archetypical example is the deformation quantization of the simple phase space which is the symplectic vector space $\mathbb{R}^2$ equipped with the symplectic form $\omega = \left( \array{ 0 & -1 \\ 1 & 0 } \right)$.

The resulting algebra is equivalently the quotient of the universal enveloping algebra of the Heisenberg Lie algebra $h_2$ which identifies the central element with a multiple of the 1 (the multiplicative neutral element, see at polynomial Poisson algebra for more).

More concretely, in the quantization of a single particle propagating on the real line the Hilbert space of quantum states is identified with the the space of square integrable functions $L^2(\mathbb{R})$. On this the operators

$a \coloneqq \tfrac{1}{\sqrt{2}}\left(x + i \hbar \frac{\partial}{\partial x} \right) \phantom{AAAAA} a^\ast \coloneqq \tfrac{1}{\sqrt{2}}\left(x - i \hbar \frac{\partial}{\partial x}\right)$

act (where “$x$” denotes the operator that multiplies a function with the canonical coordinate function, and $\frac{\partial}{\partial x}$ is the operator that forms the derivative with respect to this coordinate).

These operators satisfy the canonical commutation relations with

$[a, a^\ast] = i \hbar$

If the particle being quantized here is equipped with Hamiltonian that represents the energy of a harmonic oscillator, then one may show that the operator $a$ has the interpretation of removing one quantum of energy from the oscillator, while $a^\ast$ has the interpretation of adding one quantum.

(Accordingly the CCR relations in this case have been argued to be related to the combinatorics of placing a ball into a box and removing a ball from a box.)

More generally, in the quantum field theory of the free scalar field on Minkowski spacetime of dimension $d+1 \in \mathbb{N}$, each Fourier mode amplitude $a_k$ of the field behaves independently like a harmonic oscillator and hence the Wick algebra of quantum observables of this free field is a Weyl algebra with a countable set $\{a_k, a^\ast_k\}_{k \in \mathbb{Z}^d}$ of generator, subject to the “canonical commutation relations”

$[a_k, a^\ast_{k'}] = i \hbar \delta_{k, k'}$

(where on the right we have the Kronecker delta). Now $a_k$ is interpreted as having the effect of “annihilating” a paticle/quantum in mode $k$, while $a_k^\ast$ has the effect of “creating” one.

Therefore operators satisfying the “canonical commutation relations” are often referred to as (particle) creation and annihilation operators.

One a curved spacetime these relations become more complicated, see at Wick algebra for more.

If the field in question is not a bosonic field but a fermionic field then all of the above has to be understood in superalgebra with the fermionic variabled in off super-degree. This yields anti-commutator relations as above, hence often called “canonical anti-commutation relations”.

Under passing to exponentials the canomical commutation relations are also called the Weyl relations.

## Properties

Last revised on July 10, 2018 at 07:35:22. See the history of this page for a list of all contributions to it.