nLab canonical commutation relation




Algebraic Quantum Field Theory

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)



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quantum mechanical system, quantum probability

free field quantization

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Operator algebra

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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, 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()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, hence often called “canonical anti-commutation relations”.

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



Original discussion:

More on the history of the notion:

Monograph on commutators in operator algebra:

See also:

Last revised on April 15, 2024 at 13:08:44. See the history of this page for a list of all contributions to it.