Algebras of observables of the free Dirac field

A net of algebras of local observables of the free Dirac field satisfying the Haag–Araki axioms is constructed and investigated. It is shown that because of the $C$-number nature of the commutation relations the model also satisfies the axiom of weak additivity, in contrast to fermion systems of general form. A new representation for a spinor field is constructed that is unitarily equivalent to the usual one and taken as a basis for constructing a net of algebras of observables of threedimensional regions on the $t=0$ hyperplane of Minkowski space. It is shown that the algebra of observables of a three-dimensional region $B$ coincides with that of the four-dimensional double cone $C(B)$ with base $B$. This correspondence is used to prove structure theorems for the algebras of observables of the regions $C(B)$ and $C(B)'$. The methods of Tomita–Takesaki theory are used to show that these algebras are type III factors after restriction to coherent superselection sections and satisfy the duality condition.