The SR-selection conjecture for Haag–Kastler nets

Let $\{\mathcal{A}(\mathcal{O})\}$ be a Haag–Kastler net on a spacetime $\mathcal{M}$ satisfying axioms (HK1)–(HK4), with (HK2) interpreted as a sharper-than-equal-time commutation requirement. Assume non-trivial time-translation dynamics, a non-abelian observable algebra, and canonical commutation relations whose sole quantum-deformation scale is $\hbar$, in either the $(\hat{\phi},\hat{\pi})$ form or the creation–annihilation form. Let $G$ denote the kinematic group in (HK3). SR-selection conjecture. The group $G$ cannot be the Galilei group; it must support spacelike separation through a metric structure, and it must have a finite maximum invariant signaling speed. The conjecture is motivated by the claimed incompatibility between sharper-than-equal-time locality, canonical quantum commutation relations, and Galilean kinematics. A related no-go theorem is established under narrower hypotheses, including Fock representation or suitable Bargmann-charge regularity, but the full conjecture for arbitrary CCR realizations remains open.