Statement
A group-specific spin-off of Theme 2 of Goto et al. (Problems 3.1/3.4, dynamical-observer-backreaction-and-algebra): can the group’s Bell-CHSH and relative-entropy computations be redone on the type II “observer’s algebra” obtained by adjoining a bounded-below observer Hamiltonian via the crossed product (CLPW/CPW), rather than on the bare type III₁ algebra?
Why It Matters
A type II algebra has a trace and well-defined density matrices, so a Bell-CHSH or relative-entropy computation there has a genuinely operational observer attached — potentially sidestepping the impossible-measurements worry that shadows sharp measurements in relativistic QFT, and giving the group’s numbers a cleaner information-theoretic reading. It ties the group’s flat-space toolkit to the observer program the paper’s Theme 2 pushes.
What We Know
- CLPW: adjoining an observer with a bounded-below Hamiltonian turns the (de Sitter / horizon) algebra into type II₁ with a finite maximal-entropy state; the AQFT course develops the free-field analog (Sem II).
- The group already computes Bell-CHSH on the TFD and vacuum wedge algebras (course Sem II Wk 8).
- What changes when the dressed (crossed-product) observables replace the bare ones is not yet worked out.
Possible Approaches
- Construct dressed cosine-Weyl observables on the crossed-product algebra and recompute the CHSH correlator in the dressed vacuum; compare with the bare-algebra Tsirelson approach.
- Compute a relative entropy between dressed coherent states, using the type II trace, and compare with the type III₁ Araki–Uhlmann result.