Field operator

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 $P(\varphi)_2$-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)
[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