On the level order for Dirac operators

Dehai Bao, Z. Y. Zhu

Greg Partin

This paper proposes a new interpretation of one of the foundational equations of modern physics. Rather than treating the Dirac equation as the microscopic engine of the universe, it argues that the Dirac spinor is the observer-level readout of a deeper second-order computational substrate. Building on previous Lattice Field Medium (LFM) research establishing the emergence of a stable electron from the governing substrate equations, this work asks the next logical question: once an electron has formed, how does the familiar Dirac equation emerge? The paper introduces three new concepts: • Observer Compression Theorem — demonstrating that observers embedded within a local causal computational substrate cannot directly access the complete microscopic state and must necessarily describe physics through a compressed observer-accessible quotient. • Readout Operator — an explicit mathematical mapping from the full two-frame LFM lattice state to the four-component Dirac spinor by extracting observer-accessible quantities including momentum, effective mass, spin, helicity, phase, and particle/antiparticle branch information. • Minimal Readout Theorem — proving that any observer-level description reproducing one-electron Dirac observables must recover the same invariant physical quantities, establishing the minimal information required for relativistic electron dynamics. The paper further demonstrates that repeated application of the Readout Operator to the evolving second-order substrate reproduces Dirac evolution in the local continuum limit while preserving the standard predictive success of relativistic quantum mechanics. Numerical readout diagnostics validate the construction through controlled benchmark tests, including near-machine-precision one-step Dirac fidelity, exact branch recovery, spin-precession agreement, and information compression exceeding four orders of magnitude while maintaining high reconstruction fidelity. This work does not replace the Dirac equation. Instead, it proposes a deeper computational interpretation in which Dirac theory remains the correct observer-level description while emerging naturally from a richer second-order substrate. More broadly, it introduces an information-theoretic hierarchy separating microscopic computation from observer-accessible physics, providing a new framework for understanding why first-order quantum equations arise from deeper computational dynamics.

K Shimizu

We show an equivalence between Dirac quantization and the reduced phase space quantization. The equivalence of the both quantization methods determines the operator ordering of the Hamiltonian. Some examples of the operator ordering are shown in simple models. I.

C. C. Lin

K. Shimizu

We demonstrate an equivalence between Dirac quantization and reduced phase space quantization. The equivalence of the two quantization methods determines the operator ordering of the Hamiltonian. Some examples of the operator ordering are shown in simple models.