Differential forms of fundamental completeness relations in quantum mechanics

Hong-Yi Fan, Shen-xi Yu

T. E. Phipps

lichao

This paper presents a complete emergent theory on the origin of symmetry and quan-tum mechanics. Starting from first principles, we construct a mathematical model basedon time-scale dynamics, demonstrating that gauge symmetry and quantum structures canspontaneously emerge from classical optimization constraints. The core discovery is a fun-damental uncertainty relation: S × A ≥ Cd, where S represents structural stability, Arepresents update activity, and Cd is a dimension-dependent constant (Cd ∝ d1.435). Thisconstraint forces systems to differentiate into specialized entities with different time scales,forming τ -clusters. Within high time-scale (frozen) clusters, interaction matrices sponta-neously evolve into antisymmetric forms (Rij ≈ −RTji), naturally constituting the generatorsof Lie algebras. Through complete numerical simulations, we verify that from these emer-gent antisymmetric matrices, one can construct Hermitian Hamiltonians, observe discreteenergy spectra, and establish approximate commutation relations. This indicates that themathematical structures of quantum mechanics are not fundamental assumptions but rathermathematical representations of optimal steady states achieved by complex systems throughtime-scale coordination under the fundamental ”stability-activity” contradiction. This paperprovides complete Python implementations, making all results independently reproducible.Keywords: time scale, emergent symmetry, quantum origin, Lie algebra, uncertaintyrelation, τ -clustering

Jeffrey Bub

Fan Hong-Yi, Liu NaiLe

On the basis of the common eigenvectors 〈η| and 〈ξ|, which respectively are eigenvectors of two particles' relative position and total momentum as well as center-of-mass coordinate and relative momentum, we find that there exist differential forms of completeness relation. The applications of the new completeness relation are pointed out. Throughout our discussion the technique of differentiation within an ordered product of operators is fully employed.