# Mass operator

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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)