*2.2. Local Net*

Let *M* be a manifold or a (locally finite) graph. We suppose that *M* describes the space-time or the space under consideration. R denotes the set of bounded regions of *M*, which satisfies ∪R = *M*. *M* ∈ R is assumed when *M* is bounded.

**Definition 1** (local net)**.** *A family* {A(O)}O∈R *of C*∗*-algebras is called a local net on M if it satisfies the following conditions:*

(*i*) *For every inclusion* O1 ⊂ O<sup>2</sup>*, we have* A(O1) ⊂ A(O2)*.*

(*ii*) *For any mutually causally separated (spatial) regions* O1 *and* O<sup>2</sup>*,*

$$[\mathcal{A}(\mathcal{O}\_1), \mathcal{A}(\mathcal{O}\_2)] = \{AB - BA | A \in \mathcal{A}(\mathcal{O}\_1), B \in \mathcal{A}(\mathcal{O}\_2)\} = \{0\}.\tag{2}$$

For every local net {A(O)}O∈R on *M*, there exists a C∗-algebra

$$\mathcal{A} = \overline{\bigcup\_{\mathcal{O} \in \mathcal{R}} \mathcal{A}(\mathcal{O})}^{\|\cdot\|}, \tag{3}$$

called the global algebra of {A(O)}O∈R. If *M* is bounded, then A = A(*M*) since *M* ∈ R and O ⊂ *M* for all O∈R. When a group *G* acts on R as a symmetry, we assume the covariance condition for {A(O)}O∈R: there exists an automorphic action *α* of *G* on A such that

$$\mathfrak{a}\_{\mathcal{X}}(\mathcal{A}(\mathcal{O})) = \mathcal{A}(\mathcal{g}\mathcal{O}) \tag{4}$$

for all *g* ∈ *G* and O∈R, where *g*O = {*gx*|*x* ∈ O}.

To describe the statistical aspect of quantum fields by a local net {A(O)}O∈R, states on the global algebra A or "local states" [23] are used.
