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A linear weakly-continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040100/f0401001.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040100/f0401002.png" />, of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040100/f0401003.png" /> of basic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040100/f0401004.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040100/f0401005.png" />, that take values in a finite-dimensional vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040100/f0401006.png" />, to the set of operators (generally speaking, unbounded) defined on a dense linear manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040100/f0401007.png" /> of some Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040100/f0401008.png" />. Here it is assumed that both in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040100/f0401009.png" /> and in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040100/f04010010.png" /> certain representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040100/f04010011.png" /> (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040100/f04010012.png" />) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040100/f04010013.png" /> (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040100/f04010014.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040100/f04010015.png" />, of the inhomogeneous Lorentz group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040100/f04010016.png" /> act in such a way that the equation
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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/f/f040/f040100/f04010017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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A linear weakly-continuous mapping  $  f \rightarrow \phi _ {f} $,
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$  f \in D  ^ {L} ( \mathbf R  ^ {4} ) $,
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of the space  $  D  ^ {L} ( \mathbf R  ^ {4} ) $
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of basic functions  $  f ( x) $,
 +
$  x \in \mathbf R  ^ {4} $,
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that take values in a finite-dimensional vector space  $  L $,
 +
to the set of operators (generally speaking, unbounded) defined on a dense linear manifold  $  D _ {0} \in H $
 +
of some Hilbert space  $  H $.  
 +
Here it is assumed that both in  $  L $
 +
and in  $  H $
 +
certain representations  $  g \rightarrow T _ {g} $(
 +
in  $  L $)
 +
and  $  g \rightarrow U _ {g} $(
 +
in  $  H $),
 +
$  g \in G $,
 +
of the inhomogeneous Lorentz group  $  G $
 +
act in such a way that the equation
 +
 
 +
$$ \tag{* }
 +
U _ {g} \phi _ {f} U _ {g}  ^ {-} 1  = \
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\phi _ {\tau _ {g}  f } ,\ \
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g \in G,\ \
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f \in D  ^ {L} ( \mathbf R  ^ {4} ),
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$$
  
 
holds, where
 
holds, 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/f040100/f04010018.png" /></td> </tr></table>
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$$
 +
( \tau _ {g} f  ) ( x)  = \
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T _ {g} f ( g  ^ {-} 1 x),\ \
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x \in \mathbf R  ^ {4} .
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$$
  
Depending on the representation (scalar, vector, spinor, etc.) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040100/f04010019.png" />, the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040100/f04010020.png" /> is called, respectively, scalar, vector or spinor. A family of field operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040100/f04010021.png" /> together with representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040100/f04010022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040100/f04010023.png" /> for which condition (*) holds together with several general conditions (see [[#References|[1]]]) is called a quantum (or quantized) field.
+
Depending on the representation (scalar, vector, spinor, etc.) in $  L $,  
 +
the field $  \{ {\phi _ {f} } : {f \in D  ^ {L} ( \mathbf R  ^ {4} ) } \} $
 +
is called, respectively, scalar, vector or spinor. A family of field operators $  \{ {\phi _ {f} } : {f \in D  ^ {L} ( \mathbf R  ^ {4} ) } \} $
 +
together with representations $  \{ {T _ {g} } : {g \in G } \} $
 +
and $  \{ {U _ {g} } : {g \in G } \} $
 +
for which condition (*) holds together with several general conditions (see [[#References|[1]]]) is called a quantum (or quantized) field.
  
 
Aside from some models referring to the two-dimensional or three-dimensional world (see [[#References|[2]]], [[#References|[4]]]), one has constructed only (1983) simple examples of so-called free quantum fields [[#References|[3]]].
 
Aside from some models referring to the two-dimensional or three-dimensional world (see [[#References|[2]]], [[#References|[4]]]), one has constructed only (1983) simple examples of so-called free quantum fields [[#References|[3]]].
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Jost,  "The general theory of quantized fields" , Amer. Math. Soc.  (1965)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B. Simon,  "The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040100/f04010024.png" />-Euclidean (quantum) field theory" , Princeton Univ. Press  (1974)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.N. Bogolyubov,  D.V. Shirkov,  "Introduction to the theory of quantized fields" , Interscience  (1959)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> , ''Euclidean quantum field theory. The Markov approach'' , Moscow  (1978)  (In Russian; translated from English)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Jost,  "The general theory of quantized fields" , Amer. Math. Soc.  (1965)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B. Simon,  "The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040100/f04010024.png" />-Euclidean (quantum) field theory" , Princeton Univ. Press  (1974)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.N. Bogolyubov,  D.V. Shirkov,  "Introduction to the theory of quantized fields" , Interscience  (1959)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> , ''Euclidean quantum field theory. The Markov approach'' , Moscow  (1978)  (In Russian; translated from English)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.J.M. Bongaarts,  "The mathematical structure of free quantum fields. Gaussian fields"  E.A. de Kerf (ed.)  H.G.J. Pijls (ed.) , ''Proc. Seminar. Mathematical structures in field theory'' , CWI, Amsterdam  (1987)  pp. 1–50</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.J.M. Bongaarts,  "The mathematical structure of free quantum fields. Gaussian fields"  E.A. de Kerf (ed.)  H.G.J. Pijls (ed.) , ''Proc. Seminar. Mathematical structures in field theory'' , CWI, Amsterdam  (1987)  pp. 1–50</TD></TR></table>

Revision as of 19:39, 5 June 2020


A linear weakly-continuous mapping $ f \rightarrow \phi _ {f} $, $ f \in D ^ {L} ( \mathbf R ^ {4} ) $, of the space $ D ^ {L} ( \mathbf R ^ {4} ) $ of basic functions $ f ( x) $, $ x \in \mathbf R ^ {4} $, that take values in a finite-dimensional vector space $ L $, to the set of operators (generally speaking, unbounded) defined on a dense linear manifold $ D _ {0} \in H $ of some Hilbert space $ H $. Here it is assumed that both in $ L $ and in $ H $ certain representations $ g \rightarrow T _ {g} $( in $ L $) and $ g \rightarrow U _ {g} $( in $ H $), $ g \in G $, of the inhomogeneous Lorentz group $ G $ act in such a way that the equation

$$ \tag{* } U _ {g} \phi _ {f} U _ {g} ^ {-} 1 = \ \phi _ {\tau _ {g} f } ,\ \ g \in G,\ \ f \in D ^ {L} ( \mathbf R ^ {4} ), $$

holds, where

$$ ( \tau _ {g} f ) ( x) = \ T _ {g} f ( g ^ {-} 1 x),\ \ x \in \mathbf R ^ {4} . $$

Depending on the representation (scalar, vector, spinor, etc.) in $ L $, the field $ \{ {\phi _ {f} } : {f \in D ^ {L} ( \mathbf R ^ {4} ) } \} $ is called, respectively, scalar, vector or spinor. A family of field operators $ \{ {\phi _ {f} } : {f \in D ^ {L} ( \mathbf R ^ {4} ) } \} $ together with representations $ \{ {T _ {g} } : {g \in G } \} $ and $ \{ {U _ {g} } : {g \in G } \} $ for which condition (*) holds together with several general conditions (see [1]) is called a quantum (or quantized) field.

Aside from some models referring to the two-dimensional or three-dimensional world (see [2], [4]), one has constructed only (1983) simple examples of so-called free quantum fields [3].

References

[1] R. Jost, "The general theory of quantized fields" , Amer. Math. Soc. (1965)
[2] B. Simon, "The -Euclidean (quantum) field theory" , Princeton Univ. Press (1974)
[3] N.N. Bogolyubov, D.V. Shirkov, "Introduction to the theory of quantized fields" , Interscience (1959) (Translated from Russian)
[4] , Euclidean quantum field theory. The Markov approach , Moscow (1978) (In Russian; translated from English)

Comments

References

[a1] P.J.M. Bongaarts, "The mathematical structure of free quantum fields. Gaussian fields" E.A. de Kerf (ed.) H.G.J. Pijls (ed.) , Proc. Seminar. Mathematical structures in field theory , CWI, Amsterdam (1987) pp. 1–50
How to Cite This Entry:
Field operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Field_operator&oldid=15468
This article was adapted from an original article by R.A. Minlos (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article