Classification of Subsystems for Local Nets¶with Trivial Superselection Structure

Abstract

Let F be a local net of von Neumann algebras in four spacetime dimensions satisfying certain natural structural assumptions. We prove that if F has trivial superselection structure then every covariant, Haag-dual subsystem B is of the form F G 1 ⊗ I for a suitable decomposition F = F 1 ⊗ F 2 and a compact group action. Then we discuss some application of our result, including free field models and certain theories with at most countably many sectors.

Scaling Limit for Subsystems and Doplicher–Roberts Reconstruction

“…Below, we present some examples which corroborate the natural conjecture that, at least in typical cases, we have an equality in the last column of diagram (10). In subsection 4.3, we outline a strategy for proving that F(B 0,ι ) = F(A 0,ι ).…”

“…For completeness we also analyse the relations between the different gauge groups that arise in diagram (10). According to [14, sec.…”

“…Until now we have employed the minimal set of assumptions on the scaling limit nets which allow us to make sense of the elements in diagram (10). In order to proceed further in the discussion of its properties, it is useful at this point to apply the general machinery that has recently become available in the theory of subsystems, which requires some rather natural additional assumptions on the scaling limit nets, see definition 4.6.…”

“…It has been very successful in the mathematical description of superselection sectors and of the global gauge group of a given QFT [17]. Also, the mathematical tools that are available in this setting are well suited for providing a detailed analysis of subsystems, an issue that is central in order to obtain an intrinsic description of the observable system one starts with [12,10,11]. In another direction, the recently proposed algebraic approach to the renormalization group [8] (see also section 2) has opened the possibility of studying the short distance limit in the local quantum physics framework, and has started to convey new insight into our understanding of physically relevant issues such as confinement of colour charges and renormalization of pointlike fields [3,14,1].…”

“…Some functoriality aspects of the reconstruction have been investigated in [12], and a classification result for subsystems of F(A) has been obtained in [10,11].…”