canonical commutation relation






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 k *} kK\{a_k, a^\ast_k\}_{k \in K} subject to the “canonical” expressions for the commutators [a,b]abba[a,b] \coloneqq a \cdot b - b \cdot a

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

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

The archetypical example is the deformation quantization of the simple phase space which is the symplectic vector space 2\mathbb{R}^2 equipped with the symplectic form ω=(0 1 1 0)\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 2h_2 which identifies the central element with a multiple of the 1 (the multiplicative neutral element).

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()L^2(\mathbb{R}). On this the operators

a12(x+ix)AAAAAa *12(xix) 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 “xx” denotes the operator that multiplies a function with the canonical coordinate function, and x\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 *]=i [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 aa has the interpretation of removing one quantum of energy from the oscillator, while a *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+1d+1 \in \mathbb{N}, each Fourier mode amplitude a ka_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 k *} k d\{a_k, a^\ast_k\}_{k \in \mathbb{Z}^d} of generator, subject to the “canonical commutation relations”

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

(where on the right we have the Kronecker delta). Now a ka_k is interpreted as having the effect of “annihilating” a paticle/quantum in mode kk, while a k *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, heence often called “canonical anti-commutation relations”.

Revised on August 4, 2017 07:11:31 by Urs Schreiber (