Complex-Mass Representations of the Poincar\'e Group via Exceptional Structures and Octonionic Intertwiners

The Wigner classification of unitary irreducible representations of the Poincaré
group restricts the mass-squared parameter to real values m2 ∈ R, a consequence
of self-adjointness of the translation generators. We propose a systematic general-
ization to complex mass m2 ∈ C by embedding the problem in the representation
theory of exceptional Lie groups. Building on established approaches to complex
mass, Krein spaces, rigged Hilbert spaces, complexified groups, categorical meth-
ods, pseudo-Hermitian deformations, analytic continuation, and quantum group
deformations, and on the octonionic twistor program that realizes E6 , E7 , and
E8 as symmetries of generalized twistor spaces, we identify six new directions for
synthesis.
The central results are: (H) octonionic mass complexification via the rank
stratification of the Albert algebra J3 (O), where the cubic determinant replaces
the quadratic Casimir; (I) a Freudenthal–Krein duality that pairs complex masses
through the E7 symplectic structure on the 56-dimensional Freudenthal triple sys-
tem; (J) modular resonances in E8 -structured type III1 von Neumann factors, where
decay widths are constrained by the Coxeter number h = 30; (K) identification of
the infinity twistor as the pseudo-Hermiticity operator, linking spontaneous PT -
breaking to particle instability; (L) slice-regular mass parameters over the octonions
with G2 -holonomy orbits; and (M) an exceptional Gamow–Wigner classification us-
ing orbits of E6 (C) on the complexified Albert algebra, with F4 (C) as the enriched
little group for resonances.
These directions converge on a unified physical picture: an unstable particle is
a topological defect in the modular flow of the E8 root lattice, instability originates
from non-associativity of the octonions, and three fermion generations arise from
the cubic monodromy of the Albert determinant. We formulate precise conjectures
and identify nine testable structural predictions.