By Halvorson, Mueger.

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But the key is writing it down in a fashion that suggests a particle interpretation. e. the state space for a single particle. Now depending on whether our particle is a Boson or a Fermion, the state space of a pair of these particles is either Es (H ⊗ H) or Ea (H ⊗ H), where Es is the projection onto the vectors invariant under the permutation ΣH,H on H ⊗ H, and Ea is the projection onto vectors that change signs under ΣH,H . For 32 present purposes, we ignore these differences, and simply use H ⊗ H to denote one possibility or the other.

Then for each point x ∈ M , R(O) = CI. {O∈K : x∈O} Proof. Fix x ∈ M , and fix a double cone x ∈ O. Choose an arbitrary operator, denoted by A(x), in R(O). 12. 12. It then follows that there is a c ∈ C such that A(x) = cI. Since x was an arbitrary element of M , the result is proven. 2 Go theorems Why should we care if Φ(x) cannot be taken to denote any non-trivial operator on Hilbert space? Does this have any implications for the interpretation of QFT? After all, for any neighborhood O of x, we can find a test-function f that is supported in O, and we can replace the non-denoting term “Φ(x)” with the denoting term “Φ(f )”.

9 Definition. g. regions in some manifold ordered by inclusion). Let O → A(O) and O → B(O) be nets of C ∗ -algebras over K. A net morphism α : A → B is a natural transformation between the functors. That is, α consists of a collection of morphisms αO : A(O) → B(O) : O ∈ K , that is natural in O. 10 Fact. Net automorphisms correspond to automorphisms of the quasilocal algebra that leave each local subalgebra globally invariant. To state this precisely, let A denote the functor from K into C∗ , and let B denote the inductive limit of A.