Commutation and anti-commutation relationships, representation of

representation of commutation and anti-commutation relations

A linear weakly-continuous mapping $ f \rightarrow a _ {f} $, $ f \in L $, from a pre-Hilbert space $ L $ into a set of operators acting in some Hilbert space $ H $ such that either the commutation relations

$$ \tag{1 } a _ {f _ {1} } a _ {f _ {2} } ^ {*} - a _ {f _ {2} } ^ {*} a _ {f _ {1} } = \ ( f _ {1} , f _ {2} ) E ,\ \ a _ {f _ {1} } a _ {f _ {2} } - a _ {f _ {2} } a _ {f _ {1} } = 0 , $$

or the anti-commutation relations

$$ \tag{2 } a _ {f _ {1} } a _ {f _ {2} } ^ {*} + a _ {f _ {2} } ^ {*} a _ {f _ {1} } = \ ( f _ {1} , f _ {2} ) E ,\ \ a _ {f _ {1} } a _ {f _ {2} } + a _ {f _ {2} } a _ {f _ {1} } = 0 $$

hold, where $ a _ {f} ^ {*} $, $ f \in L $, is the adjoint of the operator $ a _ {f} $ in $ H $, $ E $ is the identity operator in $ H $ and $ ( \cdot , \cdot ) $ is the scalar product in $ L $.

In the case when $ L $ is finite-dimensional, all irreducible representations both of relation (1) and of (2) are unitarily equivalent. In the case of an infinite-dimensional space there are infinitely many distinct (not unitarily equivalent) irreducible representations of (1) and (2); for complete separable $ L $ they are described in [2]–[5].

Operators $ a _ {f} $, $ f \in L $, satisfying (1) and (2) form the basis of the so-called second quantization formalism (where $ a _ {f} $ is usually called the annihilation operator of a particle in state $ f \in L $ and $ a _ {f} ^ {*} $ is the creation operator of this particle), often used in the study of quantum physical systems with a large number of degrees of freedom. However in second quantization one uses mainly the so-called Fock [Fok] representation of the commutation and anti-commutation relations; these are irreducible representations with as index space $ L $ a separable Hilbert space, while in the space $ H $ there exists a so-called vacuum vector that is annihilated by all operators $ a _ {f} $, $ \sqrt f \in L $.

References

[1]	F.A. Berezin, "The method of second quantization" , Acad. Press (1966) (Translated from Russian) (Revised (augmented) second edition: Kluwer, 1989)
[2]	I.M. Gel'fand, N.Ya. Vilenkin, "Generalized functions. Applications of harmonic analysis" , 4 , Acad. Press (1968) (Translated from Russian)
[3]	V.Ya. Golodets, "A description of the representation of anticommutation relations" Uspekhi Mat. Nauk , 24 : 4 (148) (1969) pp. 3–64 (In Russian)
[4a]	L. Gårding, A. Wightman, "Representations of the anticommutation relations" Proc. Nat. Acad. Sci. USA , 40 : 7 (1954) pp. 617–621
[4b]	L. Gårding, A. Wightman, "Representations of the commutation relations" Proc. Nat. Acad. Sci. USA , 40 : 7 (1954) pp. 622–626
[5]	I.E. Segal, "Distributions in Hilbert space and canonical systems of operators" Trans. Amer. Math. Soc. , 88 : 1 (1958) pp. 12–41

Comments

In Western literature the relations in question are often called canonical commutation and anti-commutation relations, and one uses the abbreviation CCR and CAR to denote them.

Two standard ways to write the CCR are (in the case of one degree of freedom)

$$ [ p, q] = - i \hbar I \ \ ( \textrm{ and } \ [ p, I] = [ q, I] = 0) $$

(where $ \hbar $ is Planck's constant, often taken equal to 1 in theoretical and mathematical considerations; the Heisenberg commutation relations) and

$$ [ a, a ^ {*} ] = I \ \ ( \textrm{ and } \ [ a, I] = [ a ^ {*} , I] = 0) $$

where $ a $ is the annihilation operator and $ a ^ {*} $, its conjugate, is the creation operator. The relation between the two is exceedingly simple, viz.

$$ a = ( 2 \hbar ) ^ {-} 1/2 ( q + ip),\ \ a ^ {*} = ( 2 \hbar ) ^ {-} 1/2 ( q - ip) . $$

The result that for finite-dimensional $ L $ all (suitable) irreducible representations of the CCR are unitarily equivalent is the celebrated Stone–von Neumann theorem (also known as the von Neumann theorem, the von Neumann uniqueness theorem or the Stone–von Neumann uniqueness theorem). It only holds under suitable additional regularity assumptions, such as integrability of the representation involved to a representation of the associated group.

Thus, more precisely, consider the question of representing the relation $ pq - qp = - iI $ by means of operators on a Hilbert space (or, more generaly, $ p _ {k} q _ {l} - q _ {l} p _ {k} = - iI \delta _ {k,l} $, $ k, l = 1 \dots n $). Abstractly these relations define an $ ( 2n + 1) $- dimensional Lie algebra over $ \mathbf C $ with basis $ p _ {1} \dots p _ {n} $, $ q _ {1} \dots q _ {n} $, $ I $, called the Heisenberg Lie algebra or CCR algebra.

One particular representation of these relations is the Schrödinger representation, given by $ q _ {l} \mapsto Q _ {l} $, $ ( Q _ {l} f ) ( x) = x _ {l} f ( x) $( where $ x $ is short for $ ( x _ {1} \dots x _ {n} ) $) and $ p _ {k} \mapsto P _ {k} $, $ ( P _ {k} f ) ( x) = (- i ( \partial / \partial x _ {k} ) f ) ( x) $. This particular representation is integrable to a unitary representation with (in the case $ n = 1 $) $ U _ {t} = e ^ {itP} $ given by $ ( U _ {t} f ) ( x) = f ( x + t) $ and $ V _ {s} = e ^ {isQ} $ given by $ ( V _ {s} f ) ( x) = e ^ {isx} f ( x) $. The integrated unitary operators $ U _ {t} $ and $ V _ {s} $ satisfy the Weyl commutation relations

$$ \tag{* } U _ {t} V _ {s} = \ e ^ {its} V _ {s} U _ {t} . $$

Define a Schrödinger couple to be a pair $ ( p, q) $ of self-adjoint operators on a countably infinite-dimensional Hilbert space such that $ p = U PU ^ {*} $, $ q = UQU ^ {*} $ for some unitary operator $ U $. Then one form of the von Neumann uniqueness theorem says that if $ ( p, q) $ is a pair of self-adjoint operators on a Hilbert space such that the unitary groups $ U _ {t} = e ^ {itP} $ and $ V _ {s} = e ^ {isQ} $ satisfy the Weyl commutation relations (*), then $ ( p, q) $ is a Schrödinger couple or a direct sum of such couples.

There are other, weaker, assumptions which guarantee uniqueness, such as the following one due to B. Rellich and J. Dixmier. Let $ p $ and $ q $ be closed symmetric operators on a Hilbert space with domains of definition $ D _ {p} $ and $ D _ {q} $, respectively, such that $ D _ {p} \cap D _ {q} $ is dense. Suppose, moreover, that there exists a linear set $ \Omega $ in $ D _ {p} \cap D _ {q} $ that is dense and such that $ pq - qp = - iI $ on $ \Omega $ and $ ( p ^ {2} + q ^ {2} ) \mid _ \Omega $ is essentially self-adjoint. Then $ p $ and $ q $ are self-adjoint and $ ( p, q) $ is a Schrödinger couple or a direct sum of such couples.

Thus, though it is true that if two unitary one-parameter groups $ U _ {t} $, $ V _ {s} $ satisfy the Weyl commutation relation (*) then these infinitesimal generators satisfy the Heisenberg commutation relation $ pq - qp = - iI $, the converse is not true. An example is given by the Hilbert space $ L _ {2} ( M) $ where $ M $ is the Riemann surface of $ \sqrt z $ and $ p = - i ( \partial / \partial x) $, $ q = x + i ( \partial / \partial y) $( cf. [a2], p. 275).

For a great deal more information concerning representations of the CCR cf., e.g., [a1], [a2], Sect. VIII.5, [a3], the classic [a4], and [a5], Chapt. 3.

For more details about the Fock representation of the CCR and CAR cf. Fock space

In the case of infinite degree of freedom (quantum field theory; infinite-dimensional $ L $) the Fock representation may very well be the wrong one to work with. In the case of interacting fields it is even typically the wrong one. This is an essential consequence of Haag's theorem (cf. [a5], Sect 3.c, and [a6] for a statement and discussions). Loosely speaking, Haag's theorem says that if a quantized field $ B ( x) $ and its derivative at a given time may be mapped unitarily on a free field and its canonical conjugate, i.e. are "Fock" , then $ B ( x) $ is itself a free field. Cf. Haag theorem for more details. Often Fock representations are used as a starting point and suitable non-Fock representations are constructed as weak limits (cf. [a7] for a specific example).

References

[a1]	O. Bratteli, D.W. Robinson, "Operator algebras and quantum statistical mechanics" , II , Springer (1979) pp. Chapt. 5.2
[a2]	M. Reed, B. Simon, "Methods of modern mathematical physics" , 1. Functional analysis , Acad. Press (1972)
[a3]	P.E.T. Jorgensen, R.T. Moore, "Operator commutator relations" , Reidel (1984)
[a4]	C.R. Putnam, "Commutation properties of Hilbert space operators and related topics" , Springer (1967)
[a5]	G.E. Emch, "Algebraic methods in statistical mechanics and quantum field theory" , Wiley (1972)
[a6]	L. Streit, "A generalization of Haag's theorem" Nuovo Cimento , 62A (1969) pp. 673–680
[a7]	J.-P. Eckmann, "Representation of the in the model: independence of the space cut-off" Comm. Math. Phys. , 25 (1972) pp. 1–61