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''permutation relations''
 
''permutation relations''
  
Rules for permuting the product of two creation or annihilation operators. That is, for the [[Annihilation operators|annihilation operators]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072300/p0723001.png" /> and the adjoint [[Creation operators|creation operators]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072300/p0723002.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072300/p0723003.png" /> is some Hilbert space, acting in the symmetric [[Fock space|Fock space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072300/p0723004.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072300/p0723005.png" />, these relationships take the form
+
Rules for permuting the product of two creation or annihilation operators. That is, for the [[Annihilation operators|annihilation operators]] $  \{ {a( f  ) } : {f \in H } \} $
 +
and the adjoint [[Creation operators|creation operators]] $  \{ {a  ^  \star  ( f  ) } : {f \in H } \} $,  
 +
where $  H $
 +
is some Hilbert space, acting in the symmetric [[Fock space|Fock space]] $  F( H) $
 +
over $  H $,  
 +
these relationships take the form
  
<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/p/p072/p072300/p0723006.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
a( f _ {1} ) a ( f _ {2} ) - a( f _ {2} ) a( f _ {1} ) =
 +
$$
  
<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/p/p072/p072300/p0723007.png" /></td> </tr></table>
+
$$
 +
= \
 +
a  ^  \star  ( f _ {1} ) a  ^  \star  ( f _ {2} ) - a  ^  \star  ( f _ {2} ) a  ^  \star  ( f _ {1} )  = 0,
 +
$$
  
<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/p/p072/p072300/p0723008.png" /></td> </tr></table>
+
$$
 +
a( f _ {1} ) a  ^  \star  ( f _ {2} ) - a  ^  \star  ( f _ {2} ) a( f _ {1} )  = ( f _ {1} , f _ {2} ) E ,\  f _ {1} , f _ {2} \in H,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072300/p0723009.png" /> is the [[Inner product|inner product]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072300/p07230010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072300/p07230011.png" /> is the identity operator acting in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072300/p07230012.png" />. The relations (1) are also called the commutation relations. In the case of an anti-symmetric Fock space, the creation and annihilation operators permute in accordance with the rules
+
where $  ( \cdot , \cdot ) $
 +
is the [[Inner product|inner product]] in $  H $
 +
and $  E $
 +
is the identity operator acting in $  F( H) $.  
 +
The relations (1) are also called the commutation relations. In the case of an anti-symmetric Fock space, the creation and annihilation operators permute in accordance with the rules
  
<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/p/p072/p072300/p07230013.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
a( f _ {1} ) a( f _ {2} ) + a( f _ {2} ) a( f _ {1} ) =
 +
$$
  
<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/p/p072/p072300/p07230014.png" /></td> </tr></table>
+
$$
 +
= \
 +
a  ^  \star  ( f _ {1} ) a  ^  \star  ( f _ {2} ) + a  ^  \star  ( f _ {2} ) a  ^  \star  ( f _ {1} )  = 0,
 +
$$
  
<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/p/p072/p072300/p07230015.png" /></td> </tr></table>
+
$$
 +
a( f _ {1} ) a  ^  \star  ( f _ {2} ) + a  ^  \star  ( f _ {2} ) a( f _ {1} )  = ( f _ {1} , f _ {2} ) E ,\  f _ {1} , f _ {2} \in H,
 +
$$
  
 
which are called the anti-commutation relations.
 
which are called the anti-commutation relations.
  
In the case of an infinite-dimensional space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072300/p07230016.png" />, besides the creation and annihilation operators acting in Fock spaces over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072300/p07230017.png" /> there exist other irreducible representations not equivalent to them for the commutation and anti-commutation relations, i.e. other families of operators acting in some Hilbert space and satisfying the permutation rules (1) or (2) [[#References|[1]]], . In the case of a finite-dimensional Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072300/p07230018.png" />, all the irreducible representations of (1) or (2) are unitarily equivalent.
+
In the case of an infinite-dimensional space $  H $,  
 +
besides the creation and annihilation operators acting in Fock spaces over $  H $
 +
there exist other irreducible representations not equivalent to them for the commutation and anti-commutation relations, i.e. other families of operators acting in some Hilbert space and satisfying the permutation rules (1) or (2) [[#References|[1]]], . In the case of a finite-dimensional Hilbert space $  H $,  
 +
all the irreducible representations of (1) or (2) are unitarily equivalent.
  
 
====References====
 
====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">[2a]</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">[2b]</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></table>
 
<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">[2a]</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">[2b]</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></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 08:05, 6 June 2020


permutation relations

Rules for permuting the product of two creation or annihilation operators. That is, for the annihilation operators $ \{ {a( f ) } : {f \in H } \} $ and the adjoint creation operators $ \{ {a ^ \star ( f ) } : {f \in H } \} $, where $ H $ is some Hilbert space, acting in the symmetric Fock space $ F( H) $ over $ H $, these relationships take the form

$$ \tag{1 } a( f _ {1} ) a ( f _ {2} ) - a( f _ {2} ) a( f _ {1} ) = $$

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

$$ a( f _ {1} ) a ^ \star ( f _ {2} ) - a ^ \star ( f _ {2} ) a( f _ {1} ) = ( f _ {1} , f _ {2} ) E ,\ f _ {1} , f _ {2} \in H, $$

where $ ( \cdot , \cdot ) $ is the inner product in $ H $ and $ E $ is the identity operator acting in $ F( H) $. The relations (1) are also called the commutation relations. In the case of an anti-symmetric Fock space, the creation and annihilation operators permute in accordance with the rules

$$ \tag{2 } a( f _ {1} ) a( f _ {2} ) + a( f _ {2} ) a( f _ {1} ) = $$

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

$$ a( f _ {1} ) a ^ \star ( f _ {2} ) + a ^ \star ( f _ {2} ) a( f _ {1} ) = ( f _ {1} , f _ {2} ) E ,\ f _ {1} , f _ {2} \in H, $$

which are called the anti-commutation relations.

In the case of an infinite-dimensional space $ H $, besides the creation and annihilation operators acting in Fock spaces over $ H $ there exist other irreducible representations not equivalent to them for the commutation and anti-commutation relations, i.e. other families of operators acting in some Hilbert space and satisfying the permutation rules (1) or (2) [1], . In the case of a finite-dimensional Hilbert space $ H $, all the irreducible representations of (1) or (2) are unitarily equivalent.

References

[1] F.A. Berezin, "The method of second quantization" , Acad. Press (1966) (Translated from Russian) (Revised (augmented) second edition: Kluwer, 1989)
[2a] L. Gårding, A. Wightman, "Representations of the anticommutation relations" Proc. Nat. Acad. Sci. USA , 40 : 7 (1954) pp. 617–621
[2b] L. Gårding, A. Wightman, "Representations of the commutation relations" Proc. Nat. Acad. Sci. USA , 40 : 7 (1954) pp. 622–626

Comments

The abbreviations CCR and CAR, which stand for canonical commutation relations and canonical anti-commutation relations are often used for relations (1) and (2). One also speaks of CCR algebras and CAR algebras.

References

[a1] N.N. Bogolyubov, A.A. Logunov, A.I. Oksak, I.T. Todorov, "General principles of quantum field theory" , Kluwer (1990) pp. 265ff; 295 (Translated from Russian)
[a2] G.G. Emch, "Algebraic methods in statistical mechanics and quantum field theory" , Wiley (Interscience) (1972)
[a3] S.S. [S.S. Khorozhii] Horuzhy, "Introduction to algebraic quantum field theory" , Kluwer (1990) pp. 256ff (Translated from Russian)
[a4] O.I. [O.I. Zav'yalov] Zavialov, "Renormalized quantum field theory" , Kluwer (1990) pp. 3ff (Translated from Russian)
How to Cite This Entry:
Permutation relationships. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Permutation_relationships&oldid=48161
This article was adapted from an original article by R.A. Minlos (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article