 AQFT takes so-called nets of algebras as basic for the mathematical description of quantum systems, i.e. the mapping \(\mathcal{O}\rightarrow\mathcal{A}(\mathcal{O})\) from finite space-time regions to algebras of local observables. The insight behind this apporoach is that the net structure of algebras, i.e. the very way how algebras of local observables are linked to space-time regions, supplies observables with physical significance. In this rather abstract setting, physical states are identified as positive, linear, normalized functionals which map elements of local algebras to real numbers. States can thus be understood as assignments of expectation values to observables. Via the so-called Gelfand-Neumark-Segal construction, one can recover the concrete Hilbert space representations in the conventional formalism. Thus, “all the Hilbert spaces we will ever need are hidden inside the algebra itself” (Halvorson &amp; Müger 2007, section 7).