nLab canonical commutation relation

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

• The Stone-von Neumann theorem says that for finitely many generators the canonical commutation relations (in the form of the Weyl relations) have, up to isomorphism, a unique irreducible unitary representation: the Schrödinger representation.

  Haag's theorem says that this uniqueness fails for infinitely many generators.

Related concepts

• Weyl relation, Weyl algebra

• Heisenberg algebra

• Stone-von Neumann theorem

• Wick algebra

• singleton representation

References

Original discussion:

• Hermann Weyl, (36) in: Quantenmechanik und Gruppentheorie, Zeitschrift für Physik 46 (1927) 1–46 [doi:10.1007/BF02055756]

More on the history of the notion:

• Severino C. Coutinho, Introduction to: A primer of algebraic D-modules, London Math. Soc. Stud. Texts 33, Cambridge University Press (1995) [doi:10.1017/CBO9780511623653]

• Severino C. Coutinho, The Many Avatars of a Simple Algebra, The American Mathematical Monthly 104 7 (1997) 593-604 [doi:10.2307/2975052, jstor:2975052]

Monograph on commutators in operator algebra:

• Calvin R. Putnam, Commutation Properties of Hilbert Space Operators and Related Topics, Ergebnisse der Mathematik und ihrer Grenzgebiete 36, Springer (1967) [doi:10.1007/978-3-642-85938-0]

See also:

• Elisa Ercolessi, Giuseppe Marmo, Giuseppe Morandi, From the Equations of Motion to the Canonical Commutation Relations, Riv. Nuovo Cim. 33 (2010) 401-590 [arXiv:1005.1164, doi:10.1393/ncr/i2010-10057-x]