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''operator of mass''
 
''operator of mass''
  
 
The operator taking account of the interaction of a particle with its own field and other fields. Let the state of a system be described by the quantity
 
The operator taking account of the interaction of a particle with its own field and other fields. Let the state of a system be described by the quantity
  
<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/m/m062/m062600/m0626001.png" /></td> </tr></table>
+
$$
 +
\Psi ( x)  = \Psi _ {0} ( x) \psi ( x) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062600/m0626002.png" /> is the field operator acting on the wave function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062600/m0626003.png" /> (the state vector) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062600/m0626004.png" /> is a four-dimensional coordinate vector. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062600/m0626005.png" /> satisfies the equation
+
where $  \psi ( x) $
 +
is the field operator acting on the wave function $  \Psi _ {0} $(
 +
the state vector) and $  x $
 +
is a four-dimensional coordinate vector. If $  \Psi ( x) $
 +
satisfies the equation
  
<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/m/m062/m062600/m0626006.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
[ L ( x) + M ( x) ] \Psi ( x)  = 0 ,
 +
$$
  
where the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062600/m0626007.png" /> corresponds to a free particle and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062600/m0626008.png" /> accounts for its interaction with the particle's own field and other fields, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062600/m0626009.png" /> is called the mass operator. The mass operator is an integral operator with kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062600/m06260010.png" />:
+
where the operator $  L ( x) $
 +
corresponds to a free particle and $  M ( x) $
 +
accounts for its interaction with the particle's own field and other fields, then $  M ( x) $
 +
is called the mass operator. The mass operator is an integral operator with kernel $  M ( x , x  ^  \prime  ) $:
  
<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/m/m062/m062600/m06260011.png" /></td> </tr></table>
+
$$
 +
M ( x)  = \Psi ( x)  = \
 +
\int\limits M ( x , x  ^  \prime  ) \Psi ( x  ^  \prime  )  d x  ^  \prime  .
 +
$$
  
The mass operator is closely related to the one-particle [[Green function|Green function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062600/m06260012.png" />, which is a solution of an equation similar to (*) but with a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062600/m06260013.png" />-function source on the right-hand side:
+
The mass operator is closely related to the one-particle [[Green function|Green function]] $  G ( x , x  ^  \prime  ) $,  
 +
which is a solution of an equation similar to (*) but with a $  \delta $-
 +
function source on the right-hand side:
  
<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/m/m062/m062600/m06260014.png" /></td> </tr></table>
+
$$
 +
[ L ( x) + M ( x) ] G ( x , x  ^  \prime  )  = \delta ( x - x  ^  \prime  ) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062600/m06260015.png" /> is the four-dimensional [[Delta-function|delta-function]].
+
where $  \delta ( x - x  ^  \prime  ) $
 +
is the four-dimensional [[Delta-function|delta-function]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</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">[2]</TD> <TD valign="top">  A.A. Abrikosov,  L.P. Gor'kov,  I.E. Dzyaloshinskii,  "Methods of quantum field theory in statistical physics" , Prentice-Hall  (1963)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</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">[2]</TD> <TD valign="top">  A.A. Abrikosov,  L.P. Gor'kov,  I.E. Dzyaloshinskii,  "Methods of quantum field theory in statistical physics" , Prentice-Hall  (1963)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
The concept of a  "mass operator"  can only be given some sense in the context of quantum field perturbation theory, and plays a minor role in that context.
 
The concept of a  "mass operator"  can only be given some sense in the context of quantum field perturbation theory, and plays a minor role in that context.

Latest revision as of 07:59, 6 June 2020


operator of mass

The operator taking account of the interaction of a particle with its own field and other fields. Let the state of a system be described by the quantity

$$ \Psi ( x) = \Psi _ {0} ( x) \psi ( x) , $$

where $ \psi ( x) $ is the field operator acting on the wave function $ \Psi _ {0} $( the state vector) and $ x $ is a four-dimensional coordinate vector. If $ \Psi ( x) $ satisfies the equation

$$ \tag{* } [ L ( x) + M ( x) ] \Psi ( x) = 0 , $$

where the operator $ L ( x) $ corresponds to a free particle and $ M ( x) $ accounts for its interaction with the particle's own field and other fields, then $ M ( x) $ is called the mass operator. The mass operator is an integral operator with kernel $ M ( x , x ^ \prime ) $:

$$ M ( x) = \Psi ( x) = \ \int\limits M ( x , x ^ \prime ) \Psi ( x ^ \prime ) d x ^ \prime . $$

The mass operator is closely related to the one-particle Green function $ G ( x , x ^ \prime ) $, which is a solution of an equation similar to (*) but with a $ \delta $- function source on the right-hand side:

$$ [ L ( x) + M ( x) ] G ( x , x ^ \prime ) = \delta ( x - x ^ \prime ) , $$

where $ \delta ( x - x ^ \prime ) $ is the four-dimensional delta-function.

References

[1] N.N. Bogolyubov, D.V. Shirkov, "Introduction to the theory of quantized fields" , Interscience (1959) (Translated from Russian)
[2] A.A. Abrikosov, L.P. Gor'kov, I.E. Dzyaloshinskii, "Methods of quantum field theory in statistical physics" , Prentice-Hall (1963) (Translated from Russian)

Comments

The concept of a "mass operator" can only be given some sense in the context of quantum field perturbation theory, and plays a minor role in that context.

How to Cite This Entry:
Mass operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mass_operator&oldid=47782
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article