Completely Discretized, Finite Quantum Mechanics

Emmanuel G. Floratos, I. Tsohantjis

Abstract Following the method of induced group representations of Wigner-Mackay, the explicit construction of all the unitary irreducible representations of the discrete finite Heisenberg-Weyl group $$HW_{2^{s}}$$ H W 2 s over the discrete phase space lattice $$\mathbb {Z}_{2^{s}}\otimes $$ Z 2 s ⊗ $$\mathbb {Z}_{2^{s}}$$ Z 2 s is presented. We explicitly determine their characters and their fusion rules. We discuss possible physical applications for finite quantum mechanics and quantum computation.

Robert W. Carroll

Stanley Gudder, Vladimir Naroditsky

Yohannes Yebabe Tesfay

Sympathetic, the topological structure of time is one of the fundamental and the most sophisticated research themes of modern physics. All the calculations of Newtonian mechanics are derived based on the assumption of absolute and independent time. Though, Einstein, in the SR, had thrown away the absolute and independent time conception of the Newtonian mechanics. Later, Einstein, in his GT, showed that gravitation is the curvature of four-dimensional spacetime. In this paper, the author proposed to test whether the metric time is continuous or discrete in a closed quantum dynamical systems considering a finite wave collapse. By applying the energy-time uncertainty quantum formalisation, the author gives a complete proof that the metric time is discrete. The proof of the theorem, therefore, gives strong evidence of the scheme of quantum gravity.

Fumiyoshi Tamura

This paper develops a quantum-generative perspective within Meaning-Based Theory (MBT) by re-examining the generative sequence from Self/Other formation to completed local space. MBT does not assume time, space, matter, particles, fields, or quantum structures as primitive entities. Instead, it begins from Meaning and investigates the order in which structures must become established. The central result of this paper is that a discrete periodic structure already appears before the establishment of Time. Continued self-reference generates the sequence Self₁ → Other₁ → Self₂ → Other₂ → … where Self₁ and Self₂ are distinct establishment events, and likewise Other₁ and Other₂ are distinct, while their role types recur: Type(Self₁) = Type(Self₂), Type(Other₁) = Type(Other₂). Thus, the most upstream form of periodicity in MBT is not motion repeating through time, but the re-establishment of the same role type across distinct establishment events. After Comparison, Distinction, and generative ordering become explicit, irreversible establishment precedence gives rise to Time. The pre-temporal role-type recurrence is then inherited as a time-indexed discrete recurrence: Time continues irreversibly while the same role types are re-established at later times. The paper further shows that four positions form the minimal finite positional structure containing both the first establishment and the first recurrence of both Self and Other. With A = Self₁,B = Other₁,C = Self₂,D = Other₂, the four-position structure contains the complete first finite realization of the alternating role-type recurrence. When the relations among these four positions are established, multiple relational actions become possible. If different orders of relational action produce different total results, relational interference arises. This leads to the finite I₄ closure structure, for which Q³ = −I, Q⁶ = I. A crucial distinction is established: the finite periodicity Q⁶ = I is not the origin of periodicity in MBT. Role-type recurrence already exists upstream at the Self/Other stage. The I₄ periodic structure is therefore interpreted as a downstream finite relational closure arising after the earlier discrete recurrence has been inherited through positional and relational structure. Because I₄ closure is also the completed local space in MBT, the resulting generative order is not completed space → quantum-like structure, but rather pre-temporal discrete recurrence→ time-indexed discrete recurrence→ four-position finite structure→ relational interference→ finite I₄ closure→ completed local space. This suggests that a candidate quantum-generative principle exists upstream of completed space within MBT. The paper does not identify Self/Other periodicity with the quantum of conventional quantum mechanics, nor does it identify I₄ with a particular physical quantum particle. The proposed identification of I₄ as a minimal quantum-closure candidate therefore remains a research hypothesis rather than an established theorem. The main result is instead structural: without assuming space, particles, matter, or conventional quantum theory, MBT generates a continuous logical sequence involving discreteness, recurrence of state type, relational interference, finite closure, and finally completed local space. A remaining problem is to derive the precise generative correspondence between the upstream two-role recurrence and the six-action finite closure Q⁶ = I. Establishing this correspondence is left for subsequent work.