Fractals and Chaos Related to Ising–onsager–zhang Lattices Versus the Jordan–von Neumann–wigner Procedures: Quaternary Approach

Abstract

The paper is inspired by a spectral decomposition and fractal eigenvectors for a class of piecewise linear maps due to Tasaki et al. [1994] and by an ad hoc explicit derivation of the Heisenberg uncertainty relation based on a Peano–Hilbert planar curve, due to El Nashie [1994]. It is also inspired by an elegant generalization by Zhang [2008] of the exact solution by Onsager [1944] to the problem of description of the Ising lattices [Ising, 1925]. This generalization involves, in particular, opening the knots by a rotation in a higher dimensional space and studying important commutators in the corresponding algebra. The investigations of Onsager and Zhang, involving quaternion matrices of order being a power of two, can be reformulated with the use of the "quaternionic" sequence of Jordan algebras implied by the fundamental paper of Jordan et al. [1934]. It is closely related to Heisenberg's approach to quantum theories, as summarized by him in his essay dedicated to Bohr on the occasion of Bohr's seventieth birthday (1955). We show that the Jordan structures are closely related to some types of fractals, in particular, fractals of the algebraic structure. Our study includes fractal renormalization and the renormalized Dirac operator, meromorphic Schauder basis and hyperfunctions on fractal boundaries, and a final discussion.

Clifford algebra approach of 3D Ising model

“…As pointed out in ref. [20][21][22], one of the present authors (ZDZ) gave quaternion-based 3D (quantum) models of order-disorder transition for simple orthorhombic Ising lattices, based on Jordan-von Neumann-Wigner procedure [14]. Two matrices A and B, which are in Jordan algebra [12][13][14]., one has:…”

Fractals and Chaos Related to Ising-Onsager-Zhang Lattices vs. The Jordan-von Neumann-Wigner Procedures. Ternary Approach

“…As in the case of quaternary approach [1] our results can modify and simplify the Onsager-Zhang procedure of analyzing simple orthorhombic Ising lattices [2] in terms of fractal renormalization and the renormalized Dirac operator as well as hyperfunctions on fractal boundaries. In contrast to [1] the ternary approach is more specific but leads to stronger results, in particular for ternary alloys without vacancies and binary alloys with vacancies.…”

“…In contrast to [1] the ternary approach is more specific but leads to stronger results, in particular for ternary alloys without vacancies and binary alloys with vacancies. Also, we can specify the Zhang idea of decomposing the knots related to lattice vertices by deformations on formal knot sequences arising from Reidemeister moves [7].…”

“…Firstly, the natural appearance of the new multiplication (4) instead of the usual matrix multiplication replaces in an elegant way the desire of Onsager [2], Zhang [3], and Staples [5] to look for commutative subalgebras of the algebra constructed and for their combinatorial properties. Secondly, thanks to the main theorem of [1] the results of P. Jordan found a very impressive representation in the mathematical scheme of quantum mechanics.…”

“…This simplifies a recent paper [1] involving two of us, concentrated on a parallel quaternary approach.…”