A Hilbert Space Model for a Coupled Universe: Self-Adjoint Extension, Mirror Symmetry, and Dimensional Constraint
Authors/Creators
Description
We propose a Hilbert space model of the universe based on a coupled Hamiltonian H = HA ⊗
HB where HA = L2(R+) models the observable sector and HB = Ck is a finite-dimensional
auxiliary sector. Using the theory of self-adjoint extensions (boundary triples and Krein’s formula),
we show that physical stability and a fundamental anti-linear symmetry analogous to the functional
equation uniquely determine the extension, confining the observable spectrum to a “critical line.”
Furthermore, we introduce a vacuum energy condition based on the analytic continuation of the
Riemann zeta function, ζ(−1) = −1/12, which fixes the auxiliary dimension k = 12. This suggests
a link between spectral theory, quantum vacuum structure, and emergent dimensionality in physics.
