Categorial Subsystem Independence as Morphism Co-possibility

Abstract

This paper formulates a notion of independence of subobjects of an object in a general (i.e., not necessarily concrete) category. Subobject independence is the categorial generalization of what is known as subsystem independence in the context of algebraic relativistic quantum field theory. The content of subobject independence formulated in this paper is morphism co-possibility: two subobjects of an object will be defined to be independent if any two morphisms on the two subobjects of an object are jointly implementable by a single morphism on the larger object. The paper investigates features of subobject independence in general, and subobject independence in the category of C * -algebras with respect to operations (completely positive unit preserving linear maps on C * -algebras) as morphisms is suggested as a natural subsystem independence axiom to express relativistic locality of the covariant functor in the categorial approach to quantum field theory. MotivationSubsystem independence is a crucial notion in the specific axiomatic approach to (relativistic) quantum field theory known as "Local Quantum Physics" (also called "Algebraic Quantum Field Theory"). This approach to quantum field theory was initiated by Haag and Kastler [19], and since its inception it has developed into a rich field. (For monographic summaries see [1,16,20]; for compact, more recent reviews we refer to [9,10,36]; the papers [13,17,18] recall some episodes in the history of this approach.) The key element in the approach is the implementation of locality, and "The locality concept is abstractly encoded in a notion of independence of subsystems. . ." [5]. It turns out that independence of subsystems of a larger system can be specified in a number of nonequivalent ways: Summers' 1990 paper [34] gives a review of the rich hierarchy of independence notions; for a non-technical review of subsystem independence concepts that include more recent developments as well, see [35].