Intensive Observables in Quantum Theory

Abstract

The notion of strictly intensive observables is introduced in a theory of local observables (such as that of R. Haag). If O1, O2 are two disjoint regions, then the value of a strictly intensive observable in O1 ∪ O2 is the sum of its values in O1 and O2. It is shown that energy-momentum can never be strictly intensive. This result is used to prove that the algebras of observables is not of Type I for some regions. By analogy with the energy-momentum tensor density for the free field, the definition is weakened, so that an intensive observable in a region O is only ``approximately'' in a(O). This leads to the introduction of germs of intensive observables. It is proved that the unitary intensive operators form a sheaf 𝔉 of groups, and the Hermitian intensive operators form a sheaf 𝔊 of Abelian groups with operators in 𝔉, and on which the inner derivative Y → i(XY − YX) is defined.