How to Describe the Space-Time Structure with Nets of C*-Algebras

Abstract

The major subject of algebraic quantum fieldtheory is the study of nets of local C*-algebras, i.e.,maps $\mathcal{O}$ ↦ $A$($\mathcal{O}$) assigning to each open,relatively compact region of space-time (M, g) aC*-algebra $A$($\mathcal{O}$), whose self-adjoint elements describe localobservables measurable in the region $\mathcal{O}$. A question discussed recently in a number ofpapers is how much information about the geometricstructure of the underlying space-time (M, g) is encoded in the algebraicstructure of the net $\mathcal{O}$ ↦$A$($\mathcal{O}$). Followingthese ideas, it is demonstrated in this paper howspace-time-related concepts like causality and observerscan be described in a purely algebraic way, i.e., using only thelocal algebras $A$($\mathcal{O}$).These results are then used to show how the space-time(M, g) can be reconstructed from the set $L$ _loc := {$A$($\mathcal{O}$)|$\mathcal{O}$ ⊂M open, $\overline {\mathcal{O}}$ compact} of local algebras.