Analytical properties of standard subspaces and reflection positivity in AQFT

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For the beta-strip, we see that the one-parameter group of multiplication acts as a unitary on its lower boundary. However, its action is not unitary on the upper boundary since an exponential factor depending on beta appears. On the other hand, we find a realization of the Hardy space of the beta-strip as a graph of a translation operator R on the imaginary axis, which is invariant under the above non-unitary one-parameter group. Such multiplication on the lower boundary and the translation operator R on the imaginary axis verify canonical commutation relations, which produce a normal form result: for a Hilbert space equipped with a unitary one parameter group and a self-adjoint operator which satisfy a CCR produce a realization of the Hilbert space as L2-functions, the one-parameter group as translations and the self-adjoint operator as multiplication. Similarly, we show that for every positive self-adjoint operator on a Hilbert space, its graph is a standard subspace in a double copy of the original Hilbert space endowed with a suitable complex structure.In the framework of AQFT, we consider full conformal field theories in 2-dimensions whose chiral components admit representation categories with enough irreducible automorphisms.We construct the Wightman fields and their Hilbert structure for such theories using a suitable combination of primary fields for each chiral component. We also build the 2-dimensional Haag-Kastler net that corresponds to the Wightman fields. We worked out our construction when the two chiral components are given by the U(1)-current.