# Mass operator

*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