#### Peter Morgan

Gold Member

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The completeness of the algebra of operators generated by a Wightman operator-valued distribution ##\hat\phi(x)## means that we can construct the vacuum projection operator ##|0\rangle\langle 0|## (the spectrum is ##\{0,1\}##, so this is in ##\mathcal{B(H)}##, although I've never seen an explicit construction). This is a projection but it's definitely not a local operator.The Reeh-Schlieder theorem says that you can reach all states by repeated application of local operators to the vacuum. However, it doesn't say anything about locality, because there is no physical process that is modeled by an application of a local operator to the state. All physical processes are modeled by unitary evolution or projection and both these operations respect locality. The Reeh-Schlieder theorem is just a mathematical fact about the cyclicity of the vacuum state with respect to local algebras.

We also know, however, that no operator in a local algebra ##\mathcal{A(O)}## annihilates any given state with bounded energy (Haag LQP, Ch. II, Theorem 5.3.2), which the vacuum projection operator does, so although Reeh-Schlieder proves that all

*states*in the Hilbert space ##\mathcal{H}## can be approximated by the action of a local algebra ##\mathcal{A(O)}## on the vacuum state, not all

*operators*in ##\mathcal{B(H)}## can be approximated by operators in ##\mathcal{A(O)}##.

I think your claim, @rubi, that "physical processes are modeled by unitary evolution or projection" would be better stated as "physical processes are modeled by unitary evolution or local projections".

A separate difficulty, however, is that unitary evolution is not local insofar as the generators of time-like translations do not commute with ##\mathcal{A(O)}##; one uses in quantum field theory not advanced or retarded propagators, but instead the time-ordered Wightman function of the free scalar field (where we can state something concrete) is the Feynman propagator, a Greens function corresponding to different boundary conditions, so that it does not respect locality quite so much (which comes down to analyticity again). Of course there are cases when boundary conditions are such that classical physics also uses noncausal propagators, so this is not anything new to QFT.