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A family of closed linear operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a0125501.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a0125502.png" /> is some Hilbert space, acting on a [[Fock space|Fock space]] constructed from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a0125503.png" /> (i.e. on the symmetrization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a0125504.png" /> or anti-symmetrization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a0125505.png" /> of the space of tensors over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a0125506.png" />) such that on the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a0125507.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a0125508.png" />, consisting of the symmetrized <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a0125509.png" /> or anti-symmetrized <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255010.png" /> tensor product of a sequence of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255012.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255013.png" />, they are given by the formulas:
+
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<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/a/a012/a012550/a01255014.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
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 +
{{TEX|done}}
  
<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/a/a012/a012550/a01255015.png" /></td> </tr></table>
+
A family of closed linear operators  $  \{ {a (f) } : {f \in H } \} $,
 +
where  $  H $
 +
is some Hilbert space, acting on a [[Fock space|Fock space]] constructed from  $  H $(
 +
i.e. on the symmetrization  $  \Gamma  ^ {s} (H) $
 +
or anti-symmetrization  $  \Gamma  ^ {a} (H) $
 +
of the space of tensors over  $  H $)
 +
such that on the vector  $  ( f _ {1} \otimes {} \dots \otimes f _ {n} ) _  \alpha  \in \Gamma  ^  \alpha  (H) $,
 +
$  \alpha = s , a $,
 +
consisting of the symmetrized  $  ( \alpha = s ) $
 +
or anti-symmetrized  $  ( \alpha = a ) $
 +
tensor product of a sequence of elements  $  f _ {1} \dots f _ {n} \in H $,
 +
$  n = 1 , 2 \dots $
 +
in  $  H $,
 +
they are given by the formulas:
 +
 
 +
$$ \tag{1 }
 +
a (f) ( f _ {1} \otimes \dots
 +
\otimes f _ {n} ) _ {s\ } =
 +
$$
 +
 
 +
$$
 +
= \
 +
\sum _ { i=1 } ^ { n }  ( f , f _ {i} ) ( f _ {1} \otimes
 +
\dots \otimes f _ {i-1} \otimes f _ {i+1} \otimes \dots \otimes f _ {n} ) _ {s}  $$
  
 
in the symmetric case, and
 
in the symmetric case, 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/a/a012/a012550/a01255016.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
a (f) ( f _ {1} \otimes \dots
 +
\otimes f _ {n} ) _ {a\ } =
 +
$$
  
<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/a/a012/a012550/a01255017.png" /></td> </tr></table>
+
$$
 +
= \
 +
\sum _ { i=1 } ^ { n }  ( - 1 )  ^ {i-1} ( f , f _ {i} ) ( f _ {i} \otimes \dots \otimes f _ {i-1} \otimes f _ {i+1} \otimes \dots \otimes f _ {n} ) _ {a}  $$
  
in the anti-symmetric case; the empty vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255019.png" /> (i.e. the unit vector in the subspace of constants in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255020.png" />) is mapped to zero by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255021.png" />. In these formulas <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255022.png" /> is the inner product in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255023.png" />. The operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255024.png" /> dual to the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255025.png" /> are called creation operators; their action on the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255028.png" /> is given by the formulas
+
in the anti-symmetric case; the empty vector $  \Omega \in \Gamma  ^  \alpha  (H) $,
 +
$  \alpha = s , a $(
 +
i.e. the unit vector in the subspace of constants in $  \Gamma  ^  \alpha  (H) $)  
 +
is mapped to zero by a (f) $.  
 +
In these formulas $  ( \cdot , \cdot ) $
 +
is the inner product in $  H $.  
 +
The operators $  \{ {a  ^ {*} (f) } : {f \in H } \} $
 +
dual to the operators a (f) $
 +
are called creation operators; their action on the vectors $  ( f _ {1} \otimes \dots \otimes f _ {n} ) _  \alpha  $,
 +
$  \alpha = s , a $,  
 +
$  n = 1 , 2 \dots $
 +
is given by the formulas
  
<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/a/a012/a012550/a01255029.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
a ^ {*} (f) ( f _ {1} \otimes \dots
 +
\otimes f _ {n} ) _  \alpha  = \
 +
( f \otimes f _ {1} \otimes \dots
 +
\otimes f _ {n} ) _  \alpha  $$
  
 
and
 
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/a/a012/a012550/a01255030.png" /></td> </tr></table>
+
$$
 +
a  ^ {*} (f) \Omega  = f .
 +
$$
  
As a consequence of these definitions, for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255031.png" /> the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255033.png" />, the symmetrized or anti-symmetrized <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255034.png" />-th tensor power of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255035.png" />, is mapped by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255036.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255037.png" /> and by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255038.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255039.png" />.
+
As a consequence of these definitions, for each $  n > 0 $
 +
the subspace $  \Gamma _ {n}  ^  \alpha  (H) _  \alpha  ^ {\otimes n } $,
 +
$  \alpha = s , a $,  
 +
the symmetrized or anti-symmetrized $  n $-
 +
th tensor power of $  H $,  
 +
is mapped by a (f) $
 +
into $  \Gamma _ {n-1}  ^  \alpha  $
 +
and by a ^ {*} (f) $
 +
into $  \Gamma _ {n+1}  ^  \alpha  (H) $.
  
In quantum physics, the Fock space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255040.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255041.png" />, is interpreted as the state space of a system consisting of an arbitrary (finite) number of identical quantum particles, the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255042.png" /> is the state space of a single particle, the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255043.png" /> corresponds to the states of the system with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255044.png" /> particles, i.e. states in which there are just <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255045.png" /> particles. A state with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255046.png" /> particles is mapped by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255047.png" /> to a state with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255048.png" /> particles ( "annihilation"  of a particle), and by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255049.png" /> to a state with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255050.png" /> particles ( "creation"  of a particle).
+
In quantum physics, the Fock space $  \Gamma  ^  \alpha  (H) $,
 +
$  \alpha = s , a $,  
 +
is interpreted as the state space of a system consisting of an arbitrary (finite) number of identical quantum particles, the space $  H $
 +
is the state space of a single particle, the subspace $  \Gamma _ {n}  ^  \alpha  (H) $
 +
corresponds to the states of the system with $  n $
 +
particles, i.e. states in which there are just $  n $
 +
particles. A state with $  n $
 +
particles is mapped by a (f) $
 +
to a state with $  n-1 $
 +
particles ( "annihilation"  of a particle), and by a ^ {*} (f) $
 +
to a state with $  n + 1 $
 +
particles ( "creation"  of a particle).
  
The operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255051.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255052.png" /> form irreducible families of operators satisfying the following permutation relations: In the symmetric case (the commutation relations)
+
The operators a (f) $
 +
and a ^ {*} (f) $
 +
form irreducible families of operators satisfying the following permutation relations: In the symmetric case (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/a/a012/a012550/a01255053.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
$$ \tag{4 }
 +
a ( f _ {1} ) a ( f _ {2} ) - a
 +
( f _ {2} ) a ( f _ {1} ) =
 +
$$
  
<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/a/a012/a012550/a01255054.png" /></td> </tr></table>
+
$$
 +
= \
 +
a  ^ {*} ( f _ {1} ) a  ^ {*} ( f _ {2} ) - a
 +
^ {*} ( f _ {2} ) a  ^ {*} ( f _ {1} )  = 0 ,
 +
$$
  
<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/a/a012/a012550/a01255055.png" /></td> </tr></table>
+
$$
 +
a  ^ {*} ( f _ {2} ) a ( f _ {1} ) - a ( f _ {1} )
 +
a  ^ {*} ( f _ {2} )  = - ( f _ {1} , f _ {2} ) E ;
 +
$$
  
 
and in the anti-symmetric case (the anti-commutation relations)
 
and in the anti-symmetric case (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/a/a012/a012550/a01255056.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>
+
$$ \tag{5 }
 +
a ( f _ {1} ) a ( f _ {2} ) + a
 +
( f _ {2} ) a ( f _ {1} ) =
 +
$$
  
<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/a/a012/a012550/a01255057.png" /></td> </tr></table>
+
$$
 +
= \
 +
a  ^ {*} ( f _ {1} ) a  ^ {*} ( f _ {2} ) + a
 +
^ {*} ( f _ {2} ) a  ^ {*} ( f _ {1} )  = 0 ,
 +
$$
  
<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/a/a012/a012550/a01255058.png" /></td> </tr></table>
+
$$
 +
a ( f _ {1} ) a  ^ {*} ( f _ {2} ) + a  ^ {*} ( f _ {2} ) a ( f _ {1} )  = ( f _ {1} , f _ {2} ) E ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255059.png" /> is the identity operator in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255060.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255061.png" />. Besides the families of operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255062.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255063.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255064.png" />, described here, there exist in the case of an infinite-dimensional space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255065.png" /> also other irreducible representations of the commutation and anti-commutation relations (4) and (5), not equivalent to those given above. Sometimes they are also called creation and annihilation operators. In the case of a finite-dimensional space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255066.png" />, all irreducible representations of the commutation or anti-commutation relations are unitarily equivalent.
+
where $  E $
 +
is the identity operator in $  \Gamma  ^ {s} (H) $
 +
or $  \Gamma  ^ {a} (H) $.  
 +
Besides the families of operators a (f) $
 +
and a ^ {*} (f) $,  
 +
$  f \in H $,  
 +
described here, there exist in the case of an infinite-dimensional space $  H $
 +
also other irreducible representations of the commutation and anti-commutation relations (4) and (5), not equivalent to those given above. Sometimes they are also called creation and annihilation operators. In the case of a finite-dimensional space $  H $,  
 +
all irreducible representations of the commutation or anti-commutation relations are unitarily equivalent.
  
The operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255067.png" /> are in many connections convenient  "generators"  in the set of all linear operators acting in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255068.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255069.png" />, and the representation of such operators as the sum of arbitrary creation and annihilation operators (the normal form of an operator) is very useful in applications. The connection with this formalism bears the name method of second quantization, cf. [[#References|[1]]].
+
The operators $  \{ {a (f) , a  ^ {*} (f) } : {f \in H } \} $
 +
are in many connections convenient  "generators"  in the set of all linear operators acting in the space $  \Gamma  ^  \alpha  (H) $,
 +
$  \alpha = s , a $,  
 +
and the representation of such operators as the sum of arbitrary creation and annihilation operators (the normal form of an operator) is very useful in applications. The connection with this formalism bears the name method of second quantization, cf. [[#References|[1]]].
  
In the particular, but for applications important, case in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255070.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255071.png" /> (or in a more general case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255072.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255073.png" /> is a measure space), the family of operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255074.png" /> defines two operator-valued generalized functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255075.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255076.png" /> such that
+
In the particular, but for applications important, case in which $  H = L _ {2} ( \mathbf R  ^  \nu  , d  ^  \nu  x ) $,
 +
$  \nu = 1 , 2 , . . . $(
 +
or in a more general case $  H = L _ {2} ( M , Q ) $,  
 +
where $  ( M , Q ) $
 +
is a measure space), the family of operators $  \{ {a (f) , a  ^ {*} (f) } : {f \in L _ {2} ( \mathbf R  ^  \nu  , d  ^  \nu  x ) } \} $
 +
defines two operator-valued generalized functions a (x) $
 +
and a ^ {*} (x) $
 +
such that
  
<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/a/a012/a012550/a01255077.png" /></td> </tr></table>
+
$$
 +
a (f)  = \int\limits _ {\mathbf R  ^  \nu  }
 +
a (x) f (x)  d  ^  \nu  x ,\ \
 +
a  ^ {*} (f)  = \int\limits _ {\mathbf R  ^  \nu  }
 +
a  ^ {*} (x) \overline{f}\; (x)  d  ^  \nu  x .
 +
$$
  
The introduction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255078.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255079.png" /> turns out to be convenient for the formalism of second quantization (e.g. it allows one directly to consider operators of the form
+
The introduction of a (x) $
 +
and a ^ {*} (x) $
 +
turns out to be convenient for the formalism of second quantization (e.g. it allows one directly to consider operators of 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/a/a012/a012550/a01255080.png" /></td> </tr></table>
+
$$
 +
\int\limits _ {( \mathbf R  ^  \nu  )  ^ {m+n} }
 +
K ( x _ {1} \dots x _ {n} ; \
 +
y _ {1} \dots y _ {m} ) a  ^ {*}
 +
( x _ {1} ) \dots a ^ {*} ( x _ {n} ) \times
 +
$$
  
<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/a/a012/a012550/a01255081.png" /></td> </tr></table>
+
$$
 +
\times
 +
a ( y _ {1} ) \dots a ( y _ {n} ) \
 +
d  ^  \nu  x _ {1} \dots d  ^  \nu  x _ {n}  d  ^  \nu  y _ {1} \dots d  ^  \nu  y _ {m} ,
 +
$$
  
<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/a/a012/a012550/a01255082.png" /></td> </tr></table>
+
$$
 +
n , m = 1 , 2 \dots
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255083.png" /> is a certain  "sufficiently-good"  function), without having to recourse to their decomposition as a series in the monomials
+
where $  K ( x _ {1} \dots x _ {n} ;  y _ {1} \dots y _ {m} ) $
 +
is a certain  "sufficiently-good"  function), without having to recourse to their decomposition as a series in the monomials
  
<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/a/a012/a012550/a01255084.png" /></td> </tr></table>
+
$$
 +
a ^ {*} ( f _ {1} ) \dots a  ^ {*}
 +
( f _ {n} ) a ( g _ {1} ) \dots a
 +
( g _ {m} ) ,
 +
$$
  
 
where
 
where
  
<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/a/a012/a012550/a01255085.png" /></td> </tr></table>
+
$$
 +
f _ {1} \dots f _ {n} ,\
 +
g _ {1} \dots g _ {m} \
 +
\in  L _ {2} ( \mathbf R  ^  \nu  ,\
 +
d  ^  \nu  x ) .
 +
$$
  
 
====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">[2]</TD> <TD valign="top">  R.L. Dobrushin,  R.A. Minlos,  ''Uspekhi Mat. Nauk'' , '''32''' :  2  (1977)  pp. 67–122</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L. Gårding,  A. Wightman,  ''Proc. Nat. Acad. Sci. U.S.A.'' , '''40''' :  7  (1954)  pp. 617–626</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">[2]</TD> <TD valign="top">  R.L. Dobrushin,  R.A. Minlos,  ''Uspekhi Mat. Nauk'' , '''32''' :  2  (1977)  pp. 67–122</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L. Gårding,  A. Wightman,  ''Proc. Nat. Acad. Sci. U.S.A.'' , '''40''' :  7  (1954)  pp. 617–626</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Glimm,  A. Jaffe,  "Quantum physics, a functional integral point of view" , Springer  (1981)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N.N. Bogolyubov,  A.A. Logunov,  I.T. Todorov,  "Introduction to axiomatic quantum field theory" , Benjamin  (1975)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. de Boer,  "Construction operator formalism in many particle systems"  J. de Boer (ed.)  G.E. Uhlenbeck (ed.) , ''Studies in statistical mechanics'' , North-Holland  (1965)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Glimm,  A. Jaffe,  "Quantum physics, a functional integral point of view" , Springer  (1981)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N.N. Bogolyubov,  A.A. Logunov,  I.T. Todorov,  "Introduction to axiomatic quantum field theory" , Benjamin  (1975)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. de Boer,  "Construction operator formalism in many particle systems"  J. de Boer (ed.)  G.E. Uhlenbeck (ed.) , ''Studies in statistical mechanics'' , North-Holland  (1965)</TD></TR></table>

Latest revision as of 18:47, 5 April 2020


A family of closed linear operators $ \{ {a (f) } : {f \in H } \} $, where $ H $ is some Hilbert space, acting on a Fock space constructed from $ H $( i.e. on the symmetrization $ \Gamma ^ {s} (H) $ or anti-symmetrization $ \Gamma ^ {a} (H) $ of the space of tensors over $ H $) such that on the vector $ ( f _ {1} \otimes {} \dots \otimes f _ {n} ) _ \alpha \in \Gamma ^ \alpha (H) $, $ \alpha = s , a $, consisting of the symmetrized $ ( \alpha = s ) $ or anti-symmetrized $ ( \alpha = a ) $ tensor product of a sequence of elements $ f _ {1} \dots f _ {n} \in H $, $ n = 1 , 2 \dots $ in $ H $, they are given by the formulas:

$$ \tag{1 } a (f) ( f _ {1} \otimes \dots \otimes f _ {n} ) _ {s\ } = $$

$$ = \ \sum _ { i=1 } ^ { n } ( f , f _ {i} ) ( f _ {1} \otimes \dots \otimes f _ {i-1} \otimes f _ {i+1} \otimes \dots \otimes f _ {n} ) _ {s} $$

in the symmetric case, and

$$ \tag{2 } a (f) ( f _ {1} \otimes \dots \otimes f _ {n} ) _ {a\ } = $$

$$ = \ \sum _ { i=1 } ^ { n } ( - 1 ) ^ {i-1} ( f , f _ {i} ) ( f _ {i} \otimes \dots \otimes f _ {i-1} \otimes f _ {i+1} \otimes \dots \otimes f _ {n} ) _ {a} $$

in the anti-symmetric case; the empty vector $ \Omega \in \Gamma ^ \alpha (H) $, $ \alpha = s , a $( i.e. the unit vector in the subspace of constants in $ \Gamma ^ \alpha (H) $) is mapped to zero by $ a (f) $. In these formulas $ ( \cdot , \cdot ) $ is the inner product in $ H $. The operators $ \{ {a ^ {*} (f) } : {f \in H } \} $ dual to the operators $ a (f) $ are called creation operators; their action on the vectors $ ( f _ {1} \otimes \dots \otimes f _ {n} ) _ \alpha $, $ \alpha = s , a $, $ n = 1 , 2 \dots $ is given by the formulas

$$ \tag{3 } a ^ {*} (f) ( f _ {1} \otimes \dots \otimes f _ {n} ) _ \alpha = \ ( f \otimes f _ {1} \otimes \dots \otimes f _ {n} ) _ \alpha $$

and

$$ a ^ {*} (f) \Omega = f . $$

As a consequence of these definitions, for each $ n > 0 $ the subspace $ \Gamma _ {n} ^ \alpha (H) _ \alpha ^ {\otimes n } $, $ \alpha = s , a $, the symmetrized or anti-symmetrized $ n $- th tensor power of $ H $, is mapped by $ a (f) $ into $ \Gamma _ {n-1} ^ \alpha $ and by $ a ^ {*} (f) $ into $ \Gamma _ {n+1} ^ \alpha (H) $.

In quantum physics, the Fock space $ \Gamma ^ \alpha (H) $, $ \alpha = s , a $, is interpreted as the state space of a system consisting of an arbitrary (finite) number of identical quantum particles, the space $ H $ is the state space of a single particle, the subspace $ \Gamma _ {n} ^ \alpha (H) $ corresponds to the states of the system with $ n $ particles, i.e. states in which there are just $ n $ particles. A state with $ n $ particles is mapped by $ a (f) $ to a state with $ n-1 $ particles ( "annihilation" of a particle), and by $ a ^ {*} (f) $ to a state with $ n + 1 $ particles ( "creation" of a particle).

The operators $ a (f) $ and $ a ^ {*} (f) $ form irreducible families of operators satisfying the following permutation relations: In the symmetric case (the commutation relations)

$$ \tag{4 } a ( f _ {1} ) a ( f _ {2} ) - a ( f _ {2} ) a ( f _ {1} ) = $$

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

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

and in the anti-symmetric case (the anti-commutation relations)

$$ \tag{5 } a ( f _ {1} ) a ( f _ {2} ) + a ( f _ {2} ) a ( f _ {1} ) = $$

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

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

where $ E $ is the identity operator in $ \Gamma ^ {s} (H) $ or $ \Gamma ^ {a} (H) $. Besides the families of operators $ a (f) $ and $ a ^ {*} (f) $, $ f \in H $, described here, there exist in the case of an infinite-dimensional space $ H $ also other irreducible representations of the commutation and anti-commutation relations (4) and (5), not equivalent to those given above. Sometimes they are also called creation and annihilation operators. In the case of a finite-dimensional space $ H $, all irreducible representations of the commutation or anti-commutation relations are unitarily equivalent.

The operators $ \{ {a (f) , a ^ {*} (f) } : {f \in H } \} $ are in many connections convenient "generators" in the set of all linear operators acting in the space $ \Gamma ^ \alpha (H) $, $ \alpha = s , a $, and the representation of such operators as the sum of arbitrary creation and annihilation operators (the normal form of an operator) is very useful in applications. The connection with this formalism bears the name method of second quantization, cf. [1].

In the particular, but for applications important, case in which $ H = L _ {2} ( \mathbf R ^ \nu , d ^ \nu x ) $, $ \nu = 1 , 2 , . . . $( or in a more general case $ H = L _ {2} ( M , Q ) $, where $ ( M , Q ) $ is a measure space), the family of operators $ \{ {a (f) , a ^ {*} (f) } : {f \in L _ {2} ( \mathbf R ^ \nu , d ^ \nu x ) } \} $ defines two operator-valued generalized functions $ a (x) $ and $ a ^ {*} (x) $ such that

$$ a (f) = \int\limits _ {\mathbf R ^ \nu } a (x) f (x) d ^ \nu x ,\ \ a ^ {*} (f) = \int\limits _ {\mathbf R ^ \nu } a ^ {*} (x) \overline{f}\; (x) d ^ \nu x . $$

The introduction of $ a (x) $ and $ a ^ {*} (x) $ turns out to be convenient for the formalism of second quantization (e.g. it allows one directly to consider operators of the form

$$ \int\limits _ {( \mathbf R ^ \nu ) ^ {m+n} } K ( x _ {1} \dots x _ {n} ; \ y _ {1} \dots y _ {m} ) a ^ {*} ( x _ {1} ) \dots a ^ {*} ( x _ {n} ) \times $$

$$ \times a ( y _ {1} ) \dots a ( y _ {n} ) \ d ^ \nu x _ {1} \dots d ^ \nu x _ {n} d ^ \nu y _ {1} \dots d ^ \nu y _ {m} , $$

$$ n , m = 1 , 2 \dots $$

where $ K ( x _ {1} \dots x _ {n} ; y _ {1} \dots y _ {m} ) $ is a certain "sufficiently-good" function), without having to recourse to their decomposition as a series in the monomials

$$ a ^ {*} ( f _ {1} ) \dots a ^ {*} ( f _ {n} ) a ( g _ {1} ) \dots a ( g _ {m} ) , $$

where

$$ f _ {1} \dots f _ {n} ,\ g _ {1} \dots g _ {m} \ \in L _ {2} ( \mathbf R ^ \nu ,\ d ^ \nu x ) . $$

References

[1] F.A. Berezin, "The method of second quantization" , Acad. Press (1966) (Translated from Russian) (Revised (augmented) second edition: Kluwer, 1989)
[2] R.L. Dobrushin, R.A. Minlos, Uspekhi Mat. Nauk , 32 : 2 (1977) pp. 67–122
[3] L. Gårding, A. Wightman, Proc. Nat. Acad. Sci. U.S.A. , 40 : 7 (1954) pp. 617–626

Comments

References

[a1] J. Glimm, A. Jaffe, "Quantum physics, a functional integral point of view" , Springer (1981)
[a2] N.N. Bogolyubov, A.A. Logunov, I.T. Todorov, "Introduction to axiomatic quantum field theory" , Benjamin (1975) (Translated from Russian)
[a3] J. de Boer, "Construction operator formalism in many particle systems" J. de Boer (ed.) G.E. Uhlenbeck (ed.) , Studies in statistical mechanics , North-Holland (1965)
How to Cite This Entry:
Annihilation operators. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Annihilation_operators&oldid=45188
This article was adapted from an original article by R.A. Minlos (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article