The Geometric Substrate of Virtual Particles: Dirac Solutions and Dynamical Exchange in the Quaternionic I3 Space

This paper presents a novel geometric framework for interpreting virtual particlesand negative-energy states in quantum theory. We propose that the space of pureimaginary quaternions, I3, endowed with its inherent negative-definite metric ds2 =−(dx2 + dy2 + dz2), provides a natural habitat for off-shell phenomena. We derivethe form of the Dirac equation in I3 and show its free solutions are evanescent waves.Crucially, we extend this model by coupling the I3 sector to the standard R3 sectoracross a Planck-radius boundary. This coupling induces a dynamical exchange ofprobability amplitude between the sectors, governed by a sin2 law oscillating at theZitterbewegung frequency. Finally, we demonstrate that the positive energy densityof the quantum vacuum in I3, integrated over its inherent negative volume, resultsin a negative effective gravitational mass. This provides a geometric mechanismfor a short-range screening of the Newtonian potential and a natural, repulsivecosmological constant, offering a unified explanation for singularity resolution anddark energy.