By Halvorson, Mueger.

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21 apply to algebras R(O1 ) and R(O2 ) when O1 and O2 are strongly spacelike separated. Notes: For a comprehensive review of pre-1990 results on independence of local algebras in AQFT, see [Summers, 1990]. For some more recent results, see [Summers, 1997; Florig and Summers, 1997; R´edei, 1998; Halvorson and Clifton, 2000; Summers and Buchholz, 2005]. 4 Intrinsically entangled states According to Clifton and Halvorson [2001b], the type III property of local algebras in AQFT shows that it is impossible to disentangle local systems from their environment.

A mapping O → S(O) from double cones in Minkowski spacetime to subspaces of S. Then, if certain constraints are satisfied, the pair of mappings O → S(O) → A(O) ≡ C ∗ {W (f ) : f ∈ S(O)}, can be composed to give a net of C ∗ -algebras over Minkowski spacetime. ) Now if we are given some dynamics on S, then we can — again, if certain criteria are satisfied — define a corresponding dynamical automorphism group αt on A[S, σ]. There is then a unique dynamically stable pure state ω0 of A[S, σ], and we consider the GNS representation (H, π) of A[S, σ] induced by ω0 .

155]), which we denote by φ(x) = a− (x) + ia+ (x), (9) ia− (x). (10) π(x) = a+ (x) − It then follows that φ(x) and π(x) are self-adjoint, and on a dense domain D in F(H), we have [π(x), φ(x )] = i δx , δx = i δ0 (x − x ), (11) 51 where now δ0 is a completely legitimate mathematical object — viz. the probability measure supported on {0}. Consider the (discontinuous) representation x → V (x) of the translation group on l2 (R) defined on the basis elements {δy : y ∈ R} by V (x)δy = δy−x . e. Γ maps a unitary operator V on the single particle space H to the corresponding operator I ⊕ V ⊕ (V ⊗ V ) ⊕ · · · , on F(H).