View source for Commutation and anti-commutation relationships, representation of

''representation of commutation and anti-commutation relations''

A linear weakly-continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c0233601.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c0233602.png" />, from a pre-Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c0233603.png" /> into a set of operators acting in some Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c0233604.png" /> such that either the commutation relations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c0233605.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

or the anti-commutation relations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c0233606.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

hold, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c0233607.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c0233608.png" />, is the adjoint of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c0233609.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336011.png" /> is the identity operator in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336013.png" /> is the scalar product in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336014.png" />.

In the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336015.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336016.png" /> they are described in [[#References|[2]]]–[[#References|[5]]].

Operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336018.png" />, satisfying (1) and (2) form the basis of the so-called second quantization formalism (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336019.png" /> is usually called the annihilation operator of a particle in state <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336021.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336022.png" /> a separable Hilbert space, while in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336023.png" /> there exists a so-called vacuum vector that is annihilated by all operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336025.png" />.

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



====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)

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336026.png" /></td> </tr></table>

(where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336027.png" /> is Planck's constant, often taken equal to 1 in theoretical and mathematical considerations; the Heisenberg commutation relations) and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336028.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336029.png" /> is the annihilation operator and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336030.png" />, its conjugate, is the creation operator. The relation between the two is exceedingly simple, viz.

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336031.png" /></td> </tr></table>

The result that for finite-dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336032.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336033.png" /> by means of operators on a Hilbert space (or, more generaly, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336035.png" />). Abstractly these relations define an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336036.png" />-dimensional [[Lie algebra|Lie algebra]] over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336037.png" /> with basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336039.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336040.png" />, called the Heisenberg Lie algebra or CCR algebra.

One particular representation of these relations is the Schrödinger representation, given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336042.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336043.png" /> is short for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336044.png" />) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336045.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336046.png" />. This particular representation is integrable to a unitary representation with (in the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336047.png" />) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336048.png" /> given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336050.png" /> given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336051.png" />. The integrated unitary operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336053.png" /> satisfy the Weyl commutation relations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336054.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

Define a Schrödinger couple to be a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336055.png" /> of self-adjoint operators on a countably infinite-dimensional Hilbert space such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336056.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336057.png" /> for some unitary operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336058.png" />. Then one form of the von Neumann uniqueness theorem says that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336059.png" /> is a pair of self-adjoint operators on a Hilbert space such that the unitary groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336060.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336061.png" /> satisfy the Weyl commutation relations (*), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336062.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336063.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336064.png" /> be closed symmetric operators on a Hilbert space with domains of definition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336065.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336066.png" />, respectively, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336067.png" /> is dense. Suppose, moreover, that there exists a linear set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336068.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336069.png" /> that is dense and such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336070.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336071.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336072.png" /> is essentially self-adjoint. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336073.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336074.png" /> are self-adjoint and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336075.png" /> is a Schrödinger couple or a direct sum of such couples.

Thus, though it is true that if two unitary one-parameter groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336076.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336077.png" /> satisfy the Weyl commutation relation (*) then these infinitesimal generators satisfy the Heisenberg commutation relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336078.png" />, the converse is not true. An example is given by the Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336079.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336080.png" /> is the Riemann surface of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336081.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336082.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336083.png" /> (cf. [[#References|[a2]]], p. 275).

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

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

In the case of infinite degree of freedom (quantum field theory; infinite-dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336084.png" />) 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. [[#References|[a5]]], Sect 3.c, and [[#References|[a6]]] for a statement and discussions). Loosely speaking, Haag's theorem says that if a quantized field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336085.png" /> and its derivative at a given time may be mapped unitarily on a free field and its canonical conjugate, i.e. are  "Fock" , then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336086.png" /> is itself a free field. Cf. [[Haag theorem|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. [[#References|[a7]]] for a specific example).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  O. Bratteli,   D.W. Robinson,   "Operator algebras and quantum statistical mechanics" , '''II''' , Springer  (1979)  pp. Chapt. 5.2</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Reed,   B. Simon,   "Methods of modern mathematical physics" , '''1. Functional analysis''' , Acad. Press  (1972)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  P.E.T. Jorgensen,   R.T. Moore,   "Operator commutator relations" , Reidel  (1984)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  C.R. Putnam,   "Commutation properties of Hilbert space operators and related topics" , Springer  (1967)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  G.E. Emch,   "Algebraic methods in statistical mechanics and quantum field theory" , Wiley  (1972)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  L. Streit,   "A generalization of Haag's theorem"  ''Nuovo Cimento'' , '''62A'''  (1969)  pp. 673–680</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  J.-P. Eckmann,   "Representation of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336087.png" /> in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023360/c02336088.png" /> model: independence of the space cut-off"  ''Comm. Math. Phys.'' , '''25'''  (1972)  pp. 1–61</TD></TR></table>