In this report, we study the time evolution of a two-site Hubbard model when immersed in different environments by applying the Lindblad Master Equation. The equation was derived from microscopic dynamics and analysed in terms of symmetries and eigenvalues. Three environments were modelled; one which enabled the loss and gain of particles, one modelling continuous observation of the particle count and the final environment preserving particle-hole symmetry. The spectral gap of the Liouvillian, which determines the asymptotic decay rate, was numerically calculated for each environment with different parameter values (J, U, μ) for the Hubbard Hamiltonian. The ensuing time evolution was also simulated using a fourth-order Runge-Kutta scheme. For the third set of operators, a closure of the spectral gap was observed at U=2μ, as well as a non-decoherent steady state. This was attributed to the particle-hole symmetry of the Hubbard model. To test this hypothesis, Lindblad operators highlighting this symmetry were also simulated, showing expected results. An outlook for further study was formulated to address remaining questions concerning the interpretation of these results.
Paper 1 established quantum mechanics as the mathematics of partial closure within Atomic Continuum Ontology. This second paper develops the first major formal bridge: the Schrödinger equation as closure evolution. Rather than interpreting the wavefunction as a particle-cloud, ghostly object-wave, or hidden object-state, this paper interprets ψ as a phase-bearing partial-closure profile within the atomic continuum. The Schrödinger equation is then translated as the differential law by which partial closure remains dynamically admissible under Hamiltonian constraint. The Hamiltonian becomes a closure-admissibility operator; i expresses phase-rotation structure; ℏ functions as the quantum scale of closure-action; eigenstates become stable closure modes; eigenvalues become disclosed energy residues; and unitary evolution becomes phase-preserving closure flow. This framework preserves the standard quantum formalism while shifting its ontological reading from object motion to partial-closure dynamics. Measurement is not treated as ordinary Schrödinger evolution, but as a later reclosure transition, thereby preparing the transition to Paper 3.
We investigate relations between the group of symmetries of a subspace of the symmetric tensor product of a separable Hilbert space and a representation of the partition function of a quantum entangled family of particles associated with this subspace. Consider a system of
N noninteracting identical bosons. In general, the system is described by the
N-fold symmetric tensor product
E⊙N, where
E is the Hilbert space that describes the one-particle system. In some cases, the system can be described by some subspace of
E⊙N. We consider the case in which this subspace is the closure of the linear span of some subset of the eigenbasis of the Hamiltonian of the system. We describe the group of symmetries
S such that the partition function of the system is
S-symmetric.