Field operator

A linear weakly-continuous mapping , , of the space of basic functions , , that take values in a finite-dimensional vector space , to the set of operators (generally speaking, unbounded) defined on a dense linear manifold of some Hilbert space . Here it is assumed that both in and in certain representations (in ) and (in ), , of the inhomogeneous Lorentz group act in such a way that the equation

(*)

holds, where

Depending on the representation (scalar, vector, spinor, etc.) in , the field is called, respectively, scalar, vector or spinor. A family of field operators together with representations and 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)

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