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''Fok space''
 
''Fok space''
  
 
In the simplest and most often used case, a [[Hilbert space|Hilbert space]] consisting of infinite sequences of the form
 
In the simplest and most often used case, a [[Hilbert space|Hilbert space]] consisting of infinite sequences 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/f/f040/f040690/f0406901.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
= \{ f _ {0} , f _ {1} \dots f _ {n} ,\dots \} ,
 +
$$
  
 
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/f/f040/f040690/f0406902.png" /></td> </tr></table>
+
$$
 +
f _ {0}  \in  \mathbf C ,\ \
 +
f _ {1}  \in  L _ {2} ( \mathbf R  ^  \nu  , d  ^  \nu  x),\ \
 +
f _ {n}  \in  L _ {2}  ^ {s} (( \mathbf R  ^  \nu  )  ^ {n} ,\
 +
( d  ^  \nu  x)  ^ {n} ),
 +
$$
  
 
or
 
or
  
<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/f/f040/f040690/f0406903.png" /></td> </tr></table>
+
$$
 +
f _ {n}  \in  L _ {2}  ^ {a} (( \mathbf R  ^  \nu  )  ^ {n} ,\
 +
( d  ^  \nu  x)  ^ {n} ),\ \
 +
n = 2, 3 \dots \ \
 +
\nu = 1, 2 \dots
 +
$$
  
 
in which
 
in which
  
<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/f/f040/f040690/f0406904.png" /></td> </tr></table>
+
$$
 +
L _ {2}  ^ {s} (( \mathbf R  ^  \nu  )  ^ {n} , ( d  ^  \nu  x)  ^ {n} ) \ \
 +
( \textrm{ or } \ \
 +
L _ {2}  ^ {a} (( \mathbf R  ^  \nu  )  ^ {n} , ( d  ^  \nu  x)  ^ {n} ))
 +
$$
  
denotes the Hilbert space of symmetric (respectively, anti-symmetric) functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f0406905.png" /> variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f0406906.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f0406907.png" />. The scalar product of two sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f0406908.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f0406909.png" /> of the form (1) is equal to
+
denotes the Hilbert space of symmetric (respectively, anti-symmetric) functions in $  n $
 +
variables $  x _ {1} \dots x _ {n} \in \mathbf R  ^  \nu  $,
 +
$  n = 2, 3 ,\dots $.  
 +
The scalar product of two sequences $  F $
 +
and $  G $
 +
of the form (1) is equal to
  
<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/f/f040/f040690/f04069010.png" /></td> </tr></table>
+
$$
 +
( F, G)  = \
 +
f _ {0} \overline{g}\; _ {0} +
 +
\sum _ {n = 1 } ^  \infty 
 +
( f _ {n} , g _ {n} ) _ {L _ {2}  (( \mathbf R  ^  \nu  )  ^ {n} , ( d  ^  \nu  x)  ^ {n} ) } .
 +
$$
  
In the case when the sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069011.png" /> consist of symmetric functions, one speaks of a symmetric (or boson) Fock space, and in the case of sequences of anti-symmetric functions the Fock space is called anti-symmetric (or fermion). Fock spaces were first introduced by V.A. Fock [V.A. Fok] [[#References|[1]]] in this simplest case.
+
In the case when the sequences $  F $
 +
consist of symmetric functions, one speaks of a symmetric (or boson) Fock space, and in the case of sequences of anti-symmetric functions the Fock space is called anti-symmetric (or fermion). Fock spaces were first introduced by V.A. Fock [V.A. Fok] [[#References|[1]]] in this simplest case.
  
In the general case of an arbitrary Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069012.png" />, the Fock space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069013.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069014.png" />) constructed over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069015.png" /> is the symmetrized (or anti-symmetrized) tensor exponential of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069016.png" />, that is, the space
+
In the general case of an arbitrary Hilbert space $  H $,  
 +
the Fock space $  \Gamma  ^ {s} ( H) $(
 +
or $  \Gamma  ^ {a} ( H) $)  
 +
constructed over $  H $
 +
is the symmetrized (or anti-symmetrized) tensor exponential of $  H $,  
 +
that is, the space
  
<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/f/f040/f040690/f04069017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
\Gamma  ^  \alpha  ( H)  \equiv \
 +
\mathop{\rm Exp} _  \alpha  H  = \oplus
 +
\sum _ {n = 0 } ^  \infty 
 +
( H ^ {\otimes n } ) _  \alpha  ,\ \
 +
\alpha = s, a,
 +
$$
  
where the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069018.png" /> denotes the direct orthogonal sum of Hilbert spaces, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069020.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069022.png" />, is for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069023.png" /> the symmetrized or for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069024.png" /> the anti-symmetrized <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069025.png" />-th tensor power of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069026.png" />. In the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069027.png" /> definition (2) is equivalent to the definition of a Fock space given at the beginning of the article, if one identifies the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069029.png" /> so that the tensor product
+
where the symbol $  \oplus $
 +
denotes the direct orthogonal sum of Hilbert spaces, $  ( H ^ {\otimes 0 } ) _  \alpha  = \mathbf C  ^ {1} $,  
 +
$  ( H ^ {\otimes 1 } ) _  \alpha  = H $,  
 +
and $  ( H ^ {\otimes n } ) _  \alpha  $,  
 +
$  n > 1 $,  
 +
is for $  \alpha = s $
 +
the symmetrized or for $  \alpha = a $
 +
the anti-symmetrized $  n $-
 +
th tensor power of $  H $.  
 +
In the case $  H = L _ {2} ( \mathbf R  ^  \nu  , d  ^  \nu  x) $
 +
definition (2) is equivalent to the definition of a Fock space given at the beginning of the article, if one identifies the spaces $  L _ {2}  ^  \alpha  (( \mathbf R  ^  \nu  )  ^ {n} , ( d  ^  \nu  x)  ^ {n} ) $
 +
and $  ( L _ {2} ( \mathbf R  ^  \nu  , d  ^  \nu  x)) _  \alpha  ^ {\otimes n } $
 +
so that the tensor product
  
<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/f/f040/f040690/f04069030.png" /></td> </tr></table>
+
$$
 +
( f _ {1} \otimes \dots \otimes f _ {n} ) _  \alpha  \
 +
\in  ( L _ {2} ( \mathbf R  ^  \nu  , d  ^  \nu  x)) _  \alpha  ^ {\otimes n }
 +
$$
  
 
of the sequence of functions
 
of the sequence of functions
  
<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/f/f040/f040690/f04069031.png" /></td> </tr></table>
+
$$
 +
f _ {1} \dots f _ {n}  \in \
 +
L _ {2} ( \mathbf R  ^  \nu  , d  ^  \nu  x)
 +
$$
  
 
corresponds to the function
 
corresponds to the function
  
<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/f/f040/f040690/f04069032.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
{
 +
\frac{1}{\sqrt n! }
 +
}
 +
\sum _  \sigma
 +
( \pm  1) ^ { \mathop{\rm sign}  \sigma }
 +
\prod _ {i = 1 } ^ { n }
 +
f ( x _ {\sigma ( i) }  )  \in \
 +
L _ {2}  ^  \alpha  (( \mathbf R  ^  \nu  )  ^  \nu  , ( d  ^  \nu  x)  ^ {n} ),
 +
$$
  
where the summation is taken over all permutations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069033.png" /> of the indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069034.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069035.png" /> is the sign of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069036.png" />, and the sign <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069037.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069038.png" /> in (3) corresponds to the symmetric or anti-symmetric case.
+
where the summation is taken over all permutations $  \sigma $
 +
of the indices $  1 \dots n $;  
 +
$  \mathop{\rm sign}  \sigma $
 +
is the sign of $  \sigma $,  
 +
and the sign $  + 1 $
 +
or $  - 1 $
 +
in (3) corresponds to the symmetric or anti-symmetric case.
  
In quantum mechanics, the Fock spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069039.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069040.png" /> serve as the state spaces of quantum-mechanical systems consisting of an arbitrary (but finite) number of identical particles such that the state space of each separate particle is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069041.png" />. Here, depending on which of the Fock spaces — the symmetric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069042.png" /> or the anti-symmetric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069043.png" /> — describes this system, the particles themselves are called bosons or fermions, respectively. For every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069044.png" /> the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069045.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069046.png" />, is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069048.png" />-particle subspace: The vectors in it describe those states in which there are exactly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069049.png" /> particles; the unit vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069051.png" /> (in the notation of (1): <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069052.png" />), called the vacuum vector, describes the state of the system in which there are no particles.
+
In quantum mechanics, the Fock spaces $  \Gamma  ^ {s} ( H) $
 +
or $  \Gamma  ^ {a} ( H) $
 +
serve as the state spaces of quantum-mechanical systems consisting of an arbitrary (but finite) number of identical particles such that the state space of each separate particle is $  H $.  
 +
Here, depending on which of the Fock spaces — the symmetric $  \Gamma  ^ {s} ( H) $
 +
or the anti-symmetric $  \Gamma  ^ {a} ( H) $—  
 +
describes this system, the particles themselves are called bosons or fermions, respectively. For every $  n = 1, 2 \dots $
 +
the subspace $  \Gamma _ {n}  ^  \alpha  ( H) \equiv ( H ^ {\otimes n } ) _  \alpha  \subset  \Gamma  ^  \alpha  ( H) $,
 +
$  \alpha = s, a $,  
 +
is called the $  n $-
 +
particle subspace: The vectors in it describe those states in which there are exactly $  n $
 +
particles; the unit vector $  \Omega \in ( H ^ {\otimes 0 } ) _  \alpha  \subset  \Gamma  ^  \alpha  ( H) $,
 +
$  \alpha = s, a $(
 +
in the notation of (1): $  \Omega = \{ 1, 0 \dots 0 ,\dots \} $),  
 +
called the vacuum vector, describes the state of the system in which there are no particles.
  
In studying linear operators acting on the Fock spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069054.png" />, one often applies a special formalism called the method of second quantization. It is based on introducing two families of linear operators on each of the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069055.png" />: the so-called [[Annihilation operators|annihilation operators]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069056.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069057.png" />, and the family of operators adjoint to them <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069058.png" />, called [[Creation operators|creation operators]]. The annihilation operators are given as the closures of the operators acting on the vectors
+
In studying linear operators acting on the Fock spaces $  \Gamma  ^ {s} ( H) $
 +
and $  \Gamma  ^ {a} ( H) $,  
 +
one often applies a special formalism called the method of second quantization. It is based on introducing two families of linear operators on each of the spaces $  \Gamma  ^  \alpha  ( H) $:  
 +
the so-called [[Annihilation operators|annihilation operators]] $  \{ {a _  \alpha  ( f  ) } : {f \in H } \} $,  
 +
$  \alpha = s, a $,  
 +
and the family of operators adjoint to them $  \{ {a _  \alpha  ^ {*} ( f  ) } : {f \in H } \} $,  
 +
called [[Creation operators|creation operators]]. The annihilation operators are given as the closures of the operators acting on the vectors
  
<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/f/f040/f040690/f04069059.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
$$ \tag{4 }
 +
( f _ {1} \otimes \dots \otimes f _ {n} ) _  \alpha  \
 +
\in  \Gamma  ^  \alpha  ( H),\ \
 +
\alpha = s, a,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069060.png" /> are the symmetrized (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069061.png" />) or anti-symmetrized <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069062.png" /> tensor products of the sequences of vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069063.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069064.png" /> according to the formulas
+
where $  ( f _ {1} \otimes \dots \otimes f _ {n} ) _  \alpha  $
 +
are the symmetrized (for $  \alpha = s $)  
 +
or anti-symmetrized $  ( \alpha = a) $
 +
tensor products of the sequences of vectors $  f _ {1} \dots f _ {n} \in H $,  
 +
$  n = 1, 2 \dots $
 +
according to 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/f/f040/f040690/f04069065.png" /></td> </tr></table>
+
$$
 +
a _  \alpha  ( f  )
 +
( f _ {1} \otimes \dots \otimes f _ {n} ) _  \alpha  = \
 +
\sum _ {i = 1 } ^ { n }
 +
(- 1) ^ {g _  \alpha  ( i) }
 +
( f _ {i} , f ) \times
 +
$$
  
<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/f/f040/f040690/f04069066.png" /></td> </tr></table>
+
$$
 +
\times
 +
( f _ {1} \otimes \dots \otimes f _ {i - 1 }  \otimes f _ {i + 1 }  \otimes \dots \otimes f _ {n} ) _  \alpha  ,
 +
$$
  
<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/f/f040/f040690/f04069067.png" /></td> </tr></table>
+
$$
 +
\alpha  = s , a,\  a _  \alpha  ( f  ) \Omega  = 0,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069068.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069069.png" />. The creation operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069070.png" /> act on the vectors (3) according to the formulas
+
where $  g _ {s} ( i) = 0 $
 +
and $  g _ {a} ( i) = i - 1 $.  
 +
The creation operators $  a _  \alpha  ^ {*} ( f ) $
 +
act on the vectors (3) according to 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/f/f040/f040690/f04069071.png" /></td> </tr></table>
+
$$
 +
a _  \alpha  ^ {*} ( f  )
 +
( f _ {1} \otimes \dots \otimes f _ {n} ) _  \alpha  = \
 +
( f \otimes f _ {1} \otimes \dots \otimes f _ {n} ) _  \alpha  ,
 +
$$
  
<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/f/f040/f040690/f04069072.png" /></td> </tr></table>
+
$$
 +
a _  \alpha  ^ {*} ( f  ) \Omega  = f.
 +
$$
  
Here for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069073.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069074.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069075.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069076.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069077.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069078.png" /> that is, states of the physical system with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069079.png" /> particles are mapped by the annihilation operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069080.png" /> to states with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069081.png" /> particles, and by the creation operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069082.png" /> to states with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069083.png" /> particles. Creation and annihilation operators occur in many cases of a similar system as  "generators"  in the collection of all operators (bounded and unbounded) acting on Fock spaces. The representation of such operators in the form of a sum (finite or infinite) of operators of the form
+
Here for every f \in H $,  
 +
$  a _  \alpha  ( f ) $:  
 +
$  \Gamma _ {n}  ^  \alpha  ( H) \rightarrow \Gamma _ {n - 1 }  ^  \alpha  ( H) $,
 +
$  n = 1, 2 \dots $
 +
and  $  a _  \alpha  ^ {*} ( f ): \Gamma _ {n}  ^  \alpha  ( H) \rightarrow \Gamma _ {n + 1 }  ^ {a} ( H) $,  
 +
$  n = 0, 1 \dots $
 +
that is, states of the physical system with $  n $
 +
particles are mapped by the annihilation operators $  a _  \alpha  ( f ) $
 +
to states with $  ( n - 1) $
 +
particles, and by the creation operators $  a _  \alpha  ^ {*} ( f ) $
 +
to states with $  ( n + 1) $
 +
particles. Creation and annihilation operators occur in many cases of a similar system as  "generators"  in the collection of all operators (bounded and unbounded) acting on Fock spaces. The representation of such operators in the form of a sum (finite or infinite) of 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/f/f040/f040690/f04069084.png" /></td> </tr></table>
+
$$
 +
a _  \alpha  ^ {*} ( f _ {1} ) \dots
 +
a _  \alpha  ^ {*} ( f _ {n} )
 +
a _  \alpha  ( g _ {1} ) \dots
 +
a _  \alpha  ( g _ {m} )
 +
$$
  
<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/f/f040/f040690/f04069085.png" /></td> </tr></table>
+
$$
 +
( f _ {1} \dots f _ {n} , g _ {1} \dots g _ {m} \in H,\  n, m = 0, 1 ,\dots)
 +
$$
  
 
— the so-called normal form of an operator — and methods of dealing with operators based on such a representation (computing functions of them, reducing operators to the  "simplest"  form, various examples of approximation, etc.) also constitute the content of the formalism of second quantization mentioned above (see [[#References|[2]]]).
 
— the so-called normal form of an operator — and methods of dealing with operators based on such a representation (computing functions of them, reducing operators to the  "simplest"  form, various examples of approximation, etc.) also constitute the content of the formalism of second quantization mentioned above (see [[#References|[2]]]).
  
In the case of a symmetric Fock space over a real space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069086.png" /> there is a canonical isomorphism between that space and the Hilbert space of square-integrable functionals in a Gaussian linear random process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069087.png" /> defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069088.png" /> such that
+
In the case of a symmetric Fock space over a real space $  H $
 +
there is a canonical isomorphism between that space and the Hilbert space of square-integrable functionals in a Gaussian linear random process $  \{ {\xi _ {f} } : {f \in H } \} $
 +
defined on $  H $
 +
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/f/f040/f040690/f04069089.png" /></td> </tr></table>
+
$$
 +
M _ {\xi _ {f}  }  = 0,\ \
 +
M ( \xi _ {f _ {1}  } \xi _ {f _ {2}  } )  = \
 +
( f _ {1} , f _ {2} ) _ {H} ,\ \
 +
f, f _ {1} , f _ {2} \in H.
 +
$$
  
This isomorphism, called the Itô–Segal–Wick mapping, is uniquely determined by the condition that for any orthonormal system of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069090.png" /> and any collection of non-negative integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069091.png" /> the vector
+
This isomorphism, called the Itô–Segal–Wick mapping, is uniquely determined by the condition that for any orthonormal system of elements f _ {1} \dots f _ {k} \in H $
 +
and any collection of non-negative integers $  n _ {1} \dots n _ {k} $
 +
the vector
  
<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/f/f040/f040690/f04069092.png" /></td> </tr></table>
+
$$
 +
f _ {1} \otimes \dots \otimes
 +
f _ {1} \otimes \dots \otimes
 +
f _ {k} \otimes \dots \otimes
 +
f _ {k}  \in \
 +
\Gamma  ^ {s} ( H)
 +
$$
  
(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069093.png" /> times <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069094.png" /> times <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069095.png" />) is mapped to the functional
+
( $  n _ {1} $
 +
times f _ {1} \dots n _ {k} $
 +
times f _ {k} $)  
 +
is mapped to the functional
  
<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/f/f040/f040690/f04069096.png" /></td> </tr></table>
+
$$
 +
\prod _ {i = 1 } ^ { k }
 +
H _ {n _ {i}  }
 +
( \xi _ {f _ {i}  } ),
 +
$$
  
where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069097.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069098.png" /> are the Hermite polynomials with leading coefficient one (see [[#References|[3]]], [[#References|[4]]]).
+
where the $  H _ {n} ( \cdot ) $,
 +
$  n = 0, 1 \dots $
 +
are the Hermite polynomials with leading coefficient one (see [[#References|[3]]], [[#References|[4]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V. Fock,  ''Z. Phys.'' , '''75'''  (1932)  pp. 622–647</TD></TR><TR><TD valign="top">[2]</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">[3]</TD> <TD valign="top">  R.L. Dobrushin,  R.A. Minlos,  "Polynomials in linear random functions"  ''Russian Math. Surveys'' , '''32''' :  2  (1971)  pp. 71–127  ''Uspekhi Mat. Nauk'' , '''32''' :  2  (1977)  pp. 67–122</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  B. Simon,  "The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069099.png" /> Euclidean (quantum) field theory" , Princeton Univ. Press  (1974)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V. Fock,  ''Z. Phys.'' , '''75'''  (1932)  pp. 622–647</TD></TR><TR><TD valign="top">[2]</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">[3]</TD> <TD valign="top">  R.L. Dobrushin,  R.A. Minlos,  "Polynomials in linear random functions"  ''Russian Math. Surveys'' , '''32''' :  2  (1971)  pp. 71–127  ''Uspekhi Mat. Nauk'' , '''32''' :  2  (1977)  pp. 67–122</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  B. Simon,  "The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f04069099.png" /> Euclidean (quantum) field theory" , Princeton Univ. Press  (1974)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
The number operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f040690100.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f040690101.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f040690102.png" /> runs through an orthonormal basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f040690103.png" />, has as eigen spaces with eigen value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f040690104.png" /> the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040690/f040690105.png" />. It is interpreted as giving the number of particles.
+
The number operator $  \sum _ {f} a _  \alpha  ^ {*} ( f  ) a _  \alpha  ( f ) $,  
 +
$  \alpha = a , s $,  
 +
where f $
 +
runs through an orthonormal basis of $  H $,  
 +
has as eigen spaces with eigen value $  n $
 +
the spaces $  \Gamma _ {n}  ^  \alpha  ( H) $.  
 +
It is interpreted as giving the number of particles.
  
 
Fock spaces also play an important role in stochastic integration (including fermionic integration), cf. [[Stochastic integral|Stochastic integral]], and in [[White noise|white noise]] analysis.
 
Fock spaces also play an important role in stochastic integration (including fermionic integration), cf. [[Stochastic integral|Stochastic integral]], and in [[White noise|white noise]] analysis.

Latest revision as of 19:39, 5 June 2020


Fok space

In the simplest and most often used case, a Hilbert space consisting of infinite sequences of the form

$$ \tag{1 } F = \{ f _ {0} , f _ {1} \dots f _ {n} ,\dots \} , $$

where

$$ f _ {0} \in \mathbf C ,\ \ f _ {1} \in L _ {2} ( \mathbf R ^ \nu , d ^ \nu x),\ \ f _ {n} \in L _ {2} ^ {s} (( \mathbf R ^ \nu ) ^ {n} ,\ ( d ^ \nu x) ^ {n} ), $$

or

$$ f _ {n} \in L _ {2} ^ {a} (( \mathbf R ^ \nu ) ^ {n} ,\ ( d ^ \nu x) ^ {n} ),\ \ n = 2, 3 \dots \ \ \nu = 1, 2 \dots $$

in which

$$ L _ {2} ^ {s} (( \mathbf R ^ \nu ) ^ {n} , ( d ^ \nu x) ^ {n} ) \ \ ( \textrm{ or } \ \ L _ {2} ^ {a} (( \mathbf R ^ \nu ) ^ {n} , ( d ^ \nu x) ^ {n} )) $$

denotes the Hilbert space of symmetric (respectively, anti-symmetric) functions in $ n $ variables $ x _ {1} \dots x _ {n} \in \mathbf R ^ \nu $, $ n = 2, 3 ,\dots $. The scalar product of two sequences $ F $ and $ G $ of the form (1) is equal to

$$ ( F, G) = \ f _ {0} \overline{g}\; _ {0} + \sum _ {n = 1 } ^ \infty ( f _ {n} , g _ {n} ) _ {L _ {2} (( \mathbf R ^ \nu ) ^ {n} , ( d ^ \nu x) ^ {n} ) } . $$

In the case when the sequences $ F $ consist of symmetric functions, one speaks of a symmetric (or boson) Fock space, and in the case of sequences of anti-symmetric functions the Fock space is called anti-symmetric (or fermion). Fock spaces were first introduced by V.A. Fock [V.A. Fok] [1] in this simplest case.

In the general case of an arbitrary Hilbert space $ H $, the Fock space $ \Gamma ^ {s} ( H) $( or $ \Gamma ^ {a} ( H) $) constructed over $ H $ is the symmetrized (or anti-symmetrized) tensor exponential of $ H $, that is, the space

$$ \tag{2 } \Gamma ^ \alpha ( H) \equiv \ \mathop{\rm Exp} _ \alpha H = \oplus \sum _ {n = 0 } ^ \infty ( H ^ {\otimes n } ) _ \alpha ,\ \ \alpha = s, a, $$

where the symbol $ \oplus $ denotes the direct orthogonal sum of Hilbert spaces, $ ( H ^ {\otimes 0 } ) _ \alpha = \mathbf C ^ {1} $, $ ( H ^ {\otimes 1 } ) _ \alpha = H $, and $ ( H ^ {\otimes n } ) _ \alpha $, $ n > 1 $, is for $ \alpha = s $ the symmetrized or for $ \alpha = a $ the anti-symmetrized $ n $- th tensor power of $ H $. In the case $ H = L _ {2} ( \mathbf R ^ \nu , d ^ \nu x) $ definition (2) is equivalent to the definition of a Fock space given at the beginning of the article, if one identifies the spaces $ L _ {2} ^ \alpha (( \mathbf R ^ \nu ) ^ {n} , ( d ^ \nu x) ^ {n} ) $ and $ ( L _ {2} ( \mathbf R ^ \nu , d ^ \nu x)) _ \alpha ^ {\otimes n } $ so that the tensor product

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

of the sequence of functions

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

corresponds to the function

$$ \tag{3 } { \frac{1}{\sqrt n! } } \sum _ \sigma ( \pm 1) ^ { \mathop{\rm sign} \sigma } \prod _ {i = 1 } ^ { n } f ( x _ {\sigma ( i) } ) \in \ L _ {2} ^ \alpha (( \mathbf R ^ \nu ) ^ \nu , ( d ^ \nu x) ^ {n} ), $$

where the summation is taken over all permutations $ \sigma $ of the indices $ 1 \dots n $; $ \mathop{\rm sign} \sigma $ is the sign of $ \sigma $, and the sign $ + 1 $ or $ - 1 $ in (3) corresponds to the symmetric or anti-symmetric case.

In quantum mechanics, the Fock spaces $ \Gamma ^ {s} ( H) $ or $ \Gamma ^ {a} ( H) $ serve as the state spaces of quantum-mechanical systems consisting of an arbitrary (but finite) number of identical particles such that the state space of each separate particle is $ H $. Here, depending on which of the Fock spaces — the symmetric $ \Gamma ^ {s} ( H) $ or the anti-symmetric $ \Gamma ^ {a} ( H) $— describes this system, the particles themselves are called bosons or fermions, respectively. For every $ n = 1, 2 \dots $ the subspace $ \Gamma _ {n} ^ \alpha ( H) \equiv ( H ^ {\otimes n } ) _ \alpha \subset \Gamma ^ \alpha ( H) $, $ \alpha = s, a $, is called the $ n $- particle subspace: The vectors in it describe those states in which there are exactly $ n $ particles; the unit vector $ \Omega \in ( H ^ {\otimes 0 } ) _ \alpha \subset \Gamma ^ \alpha ( H) $, $ \alpha = s, a $( in the notation of (1): $ \Omega = \{ 1, 0 \dots 0 ,\dots \} $), called the vacuum vector, describes the state of the system in which there are no particles.

In studying linear operators acting on the Fock spaces $ \Gamma ^ {s} ( H) $ and $ \Gamma ^ {a} ( H) $, one often applies a special formalism called the method of second quantization. It is based on introducing two families of linear operators on each of the spaces $ \Gamma ^ \alpha ( H) $: the so-called annihilation operators $ \{ {a _ \alpha ( f ) } : {f \in H } \} $, $ \alpha = s, a $, and the family of operators adjoint to them $ \{ {a _ \alpha ^ {*} ( f ) } : {f \in H } \} $, called creation operators. The annihilation operators are given as the closures of the operators acting on the vectors

$$ \tag{4 } ( f _ {1} \otimes \dots \otimes f _ {n} ) _ \alpha \ \in \Gamma ^ \alpha ( H),\ \ \alpha = s, a, $$

where $ ( f _ {1} \otimes \dots \otimes f _ {n} ) _ \alpha $ are the symmetrized (for $ \alpha = s $) or anti-symmetrized $ ( \alpha = a) $ tensor products of the sequences of vectors $ f _ {1} \dots f _ {n} \in H $, $ n = 1, 2 \dots $ according to the formulas

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

$$ \times ( f _ {1} \otimes \dots \otimes f _ {i - 1 } \otimes f _ {i + 1 } \otimes \dots \otimes f _ {n} ) _ \alpha , $$

$$ \alpha = s , a,\ a _ \alpha ( f ) \Omega = 0, $$

where $ g _ {s} ( i) = 0 $ and $ g _ {a} ( i) = i - 1 $. The creation operators $ a _ \alpha ^ {*} ( f ) $ act on the vectors (3) according to the formulas

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

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

Here for every $ f \in H $, $ a _ \alpha ( f ) $: $ \Gamma _ {n} ^ \alpha ( H) \rightarrow \Gamma _ {n - 1 } ^ \alpha ( H) $, $ n = 1, 2 \dots $ and $ a _ \alpha ^ {*} ( f ): \Gamma _ {n} ^ \alpha ( H) \rightarrow \Gamma _ {n + 1 } ^ {a} ( H) $, $ n = 0, 1 \dots $ that is, states of the physical system with $ n $ particles are mapped by the annihilation operators $ a _ \alpha ( f ) $ to states with $ ( n - 1) $ particles, and by the creation operators $ a _ \alpha ^ {*} ( f ) $ to states with $ ( n + 1) $ particles. Creation and annihilation operators occur in many cases of a similar system as "generators" in the collection of all operators (bounded and unbounded) acting on Fock spaces. The representation of such operators in the form of a sum (finite or infinite) of operators of the form

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

$$ ( f _ {1} \dots f _ {n} , g _ {1} \dots g _ {m} \in H,\ n, m = 0, 1 ,\dots) $$

— the so-called normal form of an operator — and methods of dealing with operators based on such a representation (computing functions of them, reducing operators to the "simplest" form, various examples of approximation, etc.) also constitute the content of the formalism of second quantization mentioned above (see [2]).

In the case of a symmetric Fock space over a real space $ H $ there is a canonical isomorphism between that space and the Hilbert space of square-integrable functionals in a Gaussian linear random process $ \{ {\xi _ {f} } : {f \in H } \} $ defined on $ H $ such that

$$ M _ {\xi _ {f} } = 0,\ \ M ( \xi _ {f _ {1} } \xi _ {f _ {2} } ) = \ ( f _ {1} , f _ {2} ) _ {H} ,\ \ f, f _ {1} , f _ {2} \in H. $$

This isomorphism, called the Itô–Segal–Wick mapping, is uniquely determined by the condition that for any orthonormal system of elements $ f _ {1} \dots f _ {k} \in H $ and any collection of non-negative integers $ n _ {1} \dots n _ {k} $ the vector

$$ f _ {1} \otimes \dots \otimes f _ {1} \otimes \dots \otimes f _ {k} \otimes \dots \otimes f _ {k} \in \ \Gamma ^ {s} ( H) $$

( $ n _ {1} $ times $ f _ {1} \dots n _ {k} $ times $ f _ {k} $) is mapped to the functional

$$ \prod _ {i = 1 } ^ { k } H _ {n _ {i} } ( \xi _ {f _ {i} } ), $$

where the $ H _ {n} ( \cdot ) $, $ n = 0, 1 \dots $ are the Hermite polynomials with leading coefficient one (see [3], [4]).

References

[1] V. Fock, Z. Phys. , 75 (1932) pp. 622–647
[2] F.A. Berezin, "The method of second quantization" , Acad. Press (1966) (Translated from Russian) (Revised (augmented) second edition: Kluwer, 1989)
[3] R.L. Dobrushin, R.A. Minlos, "Polynomials in linear random functions" Russian Math. Surveys , 32 : 2 (1971) pp. 71–127 Uspekhi Mat. Nauk , 32 : 2 (1977) pp. 67–122
[4] B. Simon, "The Euclidean (quantum) field theory" , Princeton Univ. Press (1974)

Comments

The number operator $ \sum _ {f} a _ \alpha ^ {*} ( f ) a _ \alpha ( f ) $, $ \alpha = a , s $, where $ f $ runs through an orthonormal basis of $ H $, has as eigen spaces with eigen value $ n $ the spaces $ \Gamma _ {n} ^ \alpha ( H) $. It is interpreted as giving the number of particles.

Fock spaces also play an important role in stochastic integration (including fermionic integration), cf. Stochastic integral, and in white noise analysis.

References

[a1] N.N. [N.N. Bogolyubov] Bogolubov, A.A. Logunov, I.T. Todorov, "Introduction to axiomatic quantum field theory" , Benjamin (1975) (Translated from Russian)
[a2] P.J.M. Bongaarts, "The mathematical structure of free quantum fields. Gaussian systems" E.A. de Kerf (ed.) H.G.J. Pijls (ed.) , Proc. Seminar. Mathematical structures in field theory , CWI, Amsterdam (1984–1986) pp. 1–50
How to Cite This Entry:
Fock space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fock_space&oldid=13259
This article was adapted from an original article by R.A. Minlos (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article