On the Locality Ideal In the Algebra of Test Functions for Quantum Fields

Abstract

Some basic properties of the locality ideal in Borchers's tensor algebra are established. It is shown that the ideal is a prime ideal and that the corresponding quotient algebra has a faithful Hilbert space representation. A topology is determined for which the positive cone in the quotient algebra is normal, and it is shown that every w-point distribution satisfying the locality condition is a linear combination of positive functional which also satisfy that condition. §L IntroductionThe locality ideal in Borchers 9 tensor algebra [1, 2] is the two-sided ideal generated by commutators of test functions with space-like separated supports. Its importance comes from the fact that quantum fields satisfying the requirement of local commutativity can be regarded as Hilbert space representations of the tensor algebra annihilating this ideal. Equivalently, such fields define representations of the corresponding quotient algebra. Among other things it will be shown that the states on this algebra separate points, and consequently that it has a faithful Hilbert space representation. For the tensor algebra itself this was first shown in [3]. The question whether this is also true for the quotient algebra was posed in [2], but has remained unsettled till now.The present paper is a sequel to [4, 5 3 6], and we refer to these papers and also to [2] for definitions and further references. Borchers 9 tensor algebra will be denoted by ^; it is the completed tensor algebra over the Schwartz space tf (R d ). The locality ideal 3* c is generated by elements of the form f®g - §®f with /, g^ff(R d ) satisfying the condition f(x)g(y)=Q whenever (x-y) e R d is not space-like. The method used for the investigation of 3 f e and the

Deformations of Quantum Field Theories and Integrable Models

“…(2.4) This net becomes local, i.e., subalgebras S (O 1 ), S (O 2 ) ⊂ S associated with spacelike separated regions O 1 ⊂ O ′ 2 commute, after dividing by the so-called locality ideal [Bor62,Yng84], the two-sided ideal L ⊂ S generated by all commutators f 1 ⊗ g 1 − g 1 ⊗ f 1 , with f 1 , g 1 ∈ S 1 having spacelike separated supports.…”

An algebraic formulation of the locality principle in renormalisation

“…Separation of supports, which in QFT reflects independence of events, also arises in the early algebraic study of locality, in terms of locality ideals, initiated by the work of H.-J. Borchers [2,23]. Here a locality ideal is defined to be the two-sided ideal generated by commutators of test functions with space-like separated supports.…”