Scientific Legacy of Professor Zbigniew Oziewicz

The following sections are included:

• Advisory Board
• Preface
• Contents

Professor Zbigniew Oziewicz was a restless scientific researcher in physics and abstract mathematics, always looking for new ideas to help understand the physical world, starting from fundamental mathematical formalism. His work touched many research fields, leading him to collaborate with many people in different parts of the world. In this paper, we present biographical notes. We focus on topics that professor Oziewicz touched on during his almost 30 years at the Faculty of Higher Studies Cuautitlan of the National Autonomous University of Mexico, where some of us were working with him and learning from his ideas.

This work is based on the key-note presentation at the International Conference “Applied Category Theory Graph-Operad-Logic. Professor Zbigniew Oziewicz in Memoriam” which was subtitled as “Remembering 25 years of activities with Professor Zbigniew Oziewicz in Mexico”. The present exposition is dedicated to the last part of the invited lecture related to the proposal of a joint project and the alternative theory suggested by Professor Oziewicz. Here we give a brief account on how a set with a binary operation, called quasigroup, is related to symmetric spaces of E. Cartan as the underlying geometry for Einstein’s composition of velocities. We shall also give a brief review on the relations between homogeneous reductive spaces and the smooth loop theory conceived by Lev Sabinin.

This work has the purpose of presenting a synthesis of the philosophical thoughts and applications as well as practice of Professor Zbigniew Oziewicz in terms of the didactic strategies developed by him with the support of the Moodle virtual classroom. This environment facilitates promoting interest in science in the students. We demonstrate the benefits that this technology may have in favor of the development of critical thinking, learning, and research. Some experiences and academic dynamics developed by Professor Oziewicz using the virtual classrooms of UNAM are synthesized. It is stated that the virtual classroom is an excellent pedagogical tool useful in teaching basic concepts of mathematics, philosophy, epistemology, physics, and the vision of the universe.

The chapter explains a basic construction producing Clifford algebras inductively, starting with a base algebra A that is associative and has an involution. The basic construction always produces associative algebras and can be iterated indefinitely. The basic construction is generalized to a group theoretic construction where a group G acts on the algebra A. This group theoretic construction generates Clifford algebras and matrix algebras. The chapter then studies applications of this algebra to mathematical and physical examples.

The exponential of an endomorphism with a minimal polynomial is said to be a group-polynomial or an Euler polynomial. The coefficients of the group-polynomial (elements of a ring of scalars) are functions of traces of powers of the endomorphism, or equivalently, functions of the roots. These functions are explicitly deduced for a small number of group-polynomials. It is noted that the exponential of the root is the root of the exponential. Applications are given of this concept to isometries from reflections and bivectors, and to the Lorentz pre-factor in relativity. The historical relationship to the works of other authors is detailed.

A vector quaternion Clifford algebra, based on fundamental symmetries, is applied to produce a nilpotent form of the Dirac equation. Many significant results follow automatically, including Pauli exclusion, antisymmetric fermion wavefunctions, the combined symmetries of charge conjugation, parity and time reversal, CPT symmetry, and the structures and properties of bosons and baryons. Calculations become streamlined and problematic renormalizations are removed. The basis of relativistic quantum mechanics is reduced from applying an equation to defining an operator. The algebraic structure also appears to indicate the fermionic states that are available within the Standard Model.

The algebra of the relativistic composition of velocities is shown to be isomorphic to an algebraic loop defined on division algebras. This makes calculations in special relativity effortless, straightforward, and restores the elegant appearance of Poincaré formula. Next, we present a remarkably simple geometric construction for relativistic composition of velocities. The proof is based on the pseudo-unitary representation of the Lorentz group.

The mass–shell equation is treated as the characteristic polynomial of the second-order ordinary differential equation describing the damped oscillator model with periodic or aperiodic oscillations. The fundamental solutions of the differential equation are connected with the momenta of the relativistic particle via solutions of the related Riccati equation. In accordance with two kinds of oscillations, periodic or aperiodic, two kinds of rapidity-type parameters are introduced. These parameters are interpreted as the length of the geodesic lines on the planes with elliptic or hyperbolic metrics, correspondingly. The set of equations based on the trigonometry is unified by taking advantage from the Jacobi–Glaisher transformation within the theory of elliptic functions.

Zbigniew Oziewicz was a pioneer of the four-dimensional space–time approach to covariant relative velocities. In the 1980s (according to private correspondence), he discovered two types of four-dimensional relative velocities: binary and ternary, along with the rules for adding them. They were first published in conference materials in 2004, and the second time in a peer-reviewed journal in 2007. These physically logical and mathematically precise concepts are so subtle that Oziewicz’s numerous preprints have yet to receive the recognition they deserve.

This chapter was planned to be a mere review, but a thorough review of the little-known results was made in an original synthetic manner with numerous generalizations. The chapter presents the Oziewicz–Świerk–Bolóos (and Matolcsi or Bini–Carini–Jantzen) binary relative velocity and the Oziewicz–Ungar–Dragan (also Celakoska–Chakmakov–Petrushevski on the basis of Urbantke, as well as Wyk) canonical ternary relative velocity. The Einstein–Oziewicz and Einstein–Minkowski relative velocities, which are a four-dimensional proper generalizations of Einstein relative velocity, also have a ternary character (or pseudobinary). This also applies to Oziewicz–Minkowski relative velocity, which is a time-like equivalent (generalization) of the canonical ternary velocity. It turns out that the Lorentz transformation of velocity itself is already ternary anchored. This fact is explicitly revealed by the manifestly covariant Lorentz transformation of velocity, which is the main tool of this chapter.

I propose we make use of the formal concept analysis (FCA) methods to analyze the internal structure of the fundamentals of physics from the set semantic point of view. Experiments for this proposal are presented with interactive concept lattices for more than 50 different theories of space–time and gravitation that can be explored with a macroscope. Formal contexts are built according to recognized formal classifications and reconstructions, most of them under the Lagrangian formalism, coordinate independent formulation in four-dimensional manifolds. Attributes represent groups of axioms and are color-coded according to their function. Clicking on model concepts will show their super-concepts (foundations) and subconcepts (specialized models). Such interpretations are based on Balzer, Moulines, and Sneed’s structuralist semantic set theoretic school of philosophy of science. The interactive network visualizations allow the user to appreciate the hierarchical concept relations between theories, to identify groups of theories, and some patterns on theory evolution. The Space–Time Theories Exploratorium is a useful tool for science communicators, educators, students, and public interested in the foundations of physics. It is also a useful map for researchers studying alternative theories of gravitation as it allows them to get a sense of where they stand in a sea of theories.

Zbigniew Oziewicz’s notice in 2003 that Lorentz transformations do not meet the requirements of a group prompted me to look at Special Relativity from a new perspective, in particular to search for the genesis of the concept of Relativity. In this report, I discuss Classical Relativity, which originates from Aristotle’s Physik and includes the concepts of Copernicus, Galileo, Romer, Bradley and Doppler: Quantum Relativity, with its well-balanced particle–wave concepts of Newton, Bradley, Doppler, Umow, Bartoli, Lebedew, and Planck, and the Special Relativity, which, with its a priori imposed Lorentz factor, does not agree either with Classical or Quantum Relativity.

In this chapter, we revise the concept of non-commutative vector fields introduced previously in Refs. 1 and 2, extending the framework, adding new results, and clarifying the old ones. Using appropriate algebraic tools, certain shortcomings in the previous considerations are filled and made more precise. We focus on the correspondence between so-called Cartan pairs and first-order differentials. The case of free bimodules admitting more friendly “coordinate description” and their braiding is considered in more detail. Bimodules of right/left universal vector fields are explicitly constructed.

Many scientific topics formed part of Professor Zbigniew Oziewicz’s research, all of which he treated with a strong mathematical formalism. In this chapter, we present some elements of his algebraic approach to differential calculus in the context of classical thermodynamics. In this theory, concepts such as temperature, pressure, entropy among others are identified as scalar fields. The derivations are contravariant vectors called processes on the scalar fields, these are free-coordinates. Relevant processes are called equilibrium processes, examples of which are isothermal, isobaric, etc., which contrasts with the standard interpretation of the process, as a change in the thermodynamic system. Another essential element in this theory is the differential forms, the one-differential forms are identified with wave fronts. Thus appears the concept of implicit duality, vector–covector, particle–wave. This allows the definition of creation and annihilation operators that are elements for the construction of the graded algebra.

We give a survey on our recent results on the theory of Binary Lie algebras (BL-algebras). We are studying properties of the BL-algebras and Malcev algebras with the identities J(x, y, zt) = 0.

We report here that the Kepler problem for planetary motions, Lorentz transformations in relativity, and the Jordan algebras are all intimately related.

In this chapter, it is shown that a phase transition phenomenon takes place in a certain class of Leavitt path algebras. This result is attained by making use of some classical results of the theory of random simple graphs and by associating to each finite simple graph a Leavitt path algebra in such a way that there is a nice correspondence between the structural properties of the graph and those of the algebra.

Category theory approach to Khovanov homology perfectly fits into the broad idea which Zbyszek Oziewicz was developing: Applied Category Theory. Motivated by knot theory, we introduce a homology theory for small categories with functor coefficients. Under this general framework, different familiar homology theories such as group homology, poset homology, and Khovanov homology can be realized as homologies of small categories whose coefficients are specified functors.

The following sections are included:

• Dedication
• Introduction
• Variational Principles
• The Golden Ratio
• Beauty and Harmony
• Cycles and Waves
• Conclusion
• Acknowledgments
• References

In this work, some isomorphisms of categories are explored with the aim of getting a better understanding of Alexandroff spaces. Finite spaces are the first examples of this sort of topological spaces, so are Boolean groups and minimal models. There are some isomorphisms of categories involving categories of Alexandroff spaces, so dealing with them may be done from several interesting perspectives. Compactness reveals to be a very remarkable feature for function spaces based on Alexandroff ones. Some relevant aspects of the homotopy theory of Alexandroff spaces are developed, to prove a topological version of Arrow’s impossibility theorem.

In this chapter, parametric Hopf adjunctions are revised within the 2-category of adjunctions. In order to do so, the definition of adjoint objects is revised along with the definition of classical parametric adjunctions. Second, parametric Hopf monads are also studied. The chapter also explores the relation between these two concepts using the context of the 2-adjunction between the 2-category of adjunctions and the 2-category of monads. Finally, the characterization of liftings for parametric Hopf adjunctions is given.

The purpose of this chapter is to explore the relationship between states and observables. This will be done through a deep investigation of the variety of possibilities to derive one from the other, and by interpreting the different derivations as representations of the relations between the observer and the observed system. This analyses will naturally lead to the definition of physical states and epistemic states. We will realize a careful study of different definitions of state as points of a state space, and also as derived from the algebra of observables, as maximal ideals, characters, pure states, triples of irreducible representations with a Hilbert space and a cyclic element, and equivalence classes of irreducible representations, which are all equivalent in the classical (commutative) case. Once we have all these classically equivalent definitions, we will look at the corresponding ways to see the space described by the algebra M₂(ℂ). We will briefly argue that we can integrate the views from the Einstein–Bohr debate on the interpretation of quantum states using insights from the phenomenological tradition, that these states correspond to the relation between the observer and the observed system, and therefore are a suitable formalization of phenomenological states.

The presence of SU(n) or another Lie group symmetry of a physical system is its essential, frequently underutilized property. In many cases, it allows for finding analytical solutions to nonlinear differential equations describing this system. The power of the method is demonstrated on diversified examples from mathematical physics: Lie-group symmetries in finding solutions of generalized, multidimensional theory of gravity; analytical Dirac–equation solutions for a description of conducting polymers; stability of qubit states in quantum computers; spatial defects in condensed matter; reconstruction of a three-dimensional object from its two-dimensional tomographic image; significant improvement of numerical solutions’ stability for Euler equations. The next question after obtaining such a Lie group symmetric solution is: does a generalized solution with appropriate quantum group symmetry exist for the given physical system? If so, what is the physical meaning of the deformation parameter q introduced by such a solution?

In many cases, it can be identified. Any SU(n) solution is by its nature singular, assuming a perfect symmetry of the physical system discussed. Such a solution gives powerful insights into theoretical physics, yet the assumption may be too demanding for experimental applications. Deformation parameter q from a quantum group symmetry allows for a continuum of solutions, more applicable to experiments.

Discrete models in geometry and physics have been around for a while, but the past century brought new motivation for their rediscovery. Instead of the usual transition to continuity via the limit N → ∞ (resp. ħ → 0), we choose a rather geometric path through the rational realm, starting with a ℤ-module equipped with an inner product and the associated discrete Clifford algebra, whose Lipschitz semi-group yields via projection the Möbius transformations in the corresponding quotient. Thus, the projective parametrization used here emerges naturally in this context and bypasses the unit sphere and the exponential map, associated with the usual spin group approach, replacing them with the rational central projection and Cayley transform, respectively. Standard constructions like spinors and Hopf maps are rather straightforward in our description, while others, such as Euler-type decompositions, become non-trivial, but there are alternatives available. Using dual quaternions and infinitesimal analysis based on hyperrationals, we discuss kinematics as a prelude to a future study of differential geometry on ℚ-manifolds.

Prolate and oblate spheroidal coordinates have found many recent applications in theoretical physics, physical chemistry, planetary science, and engineering in making precise calculations and predictions where the limiting case of spherical coordinates is not adequate. A prolate ellipsoid is generated by rotating an ellipse about its principal axis, whereas an oblate ellipsoid is generated by rotating an ellipse about its non-principal axis. Prolate and oblate coordinates are introduced in a unified way in terms of spheroidal quaternions I_p, J_p, K_p having the unusual properties that dI_p = J_p, dJ_p = −I_p, and dK_p = 0. We explore the properties of these quaternions, and their relationship to prolate and oblate coordinates in the geometric (Clifford) algebra of Euclidean space. In addition, properties of the Euclidean and conformal Lie groups and algebras, and basic symmetries of the Laplace and Helmholtz equation, are considered.

This chapter undertakes a special affine Fourier transformation generalization of the quaterion domain Fourier transform (QDFT) that can process signals with both range and domains being subsets of quaternions ℍ. The resulting special affine quaternion domain Fourier transform includes a.o. the QDFT itself, and for quaternion domain functions a linear canonical transform, a fractional Fourier transform, a lense transform, a free-space propagation transform, and a magnification transform. We establish the following properties of the SAQDFT: computation in terms of the QDFT, quaternion coefficient linearity, the quaternionic frequency magnitude limit |ω| → 0, continuity, shift and modulation, inversion, Rayleigh (Parseval) energy preservation, and partial and directional derivatives. Finally, we establish the directional uncertainty principle for the SAQDFT, as well as the (direction-independent) uncertainty principle.

A “program” for development in knot theory is formulated, in which it is postulated existence of a “mother of skein algebras”, which means a variety of representations of the fundamental group in the algebra of all 2 × 2 matrices, generalized Chebyshev polynomials would be traces of generalized Jones-Wenzl idempotents, and perhaps this could be generalized to matrices of higher dimension, etc. The chapter is a certain justification for the above “program”. Among the results are the generalization of the formula for multiplication of Chebyshev polynomials to generalized Chebyshev polynomials that justifies categorification of the Chebyshev polynomials and points to representations of the Mat(2, ℂ) as possible step towards looking for generalized Jones-Wenzl idempotents. Then, looking for more general Frobenius algebras than the one used in defining the Khovanov homology, with relaxed commutativity condition, the Frobenius algebra of deRham cohomology of a 2D torus is explicitly calculated.

Both algebraic equations $Det(\widehat{p}-m)=0$ and $Det(\widehat{p}+m)=0$ for u− and v− 4-spinors have solutions with $p_0=\pm E_p=\pm \sqrt{p^2+m^2}$. The same is true for higher-spin equations (or they may even have more complicated dispersion relations). Meanwhile, every book considers the equality p₀ = E_p for both u- and v-spinors of the (1/2, 0) ⨁ (0, 1/2)) representation only, thus applying the Dirac–Feynman–Stueckelberg procedure for elimination of negative-energy solutions. The recent Ziino works (and, independently, the articles of several other authors) show that the Fock space can be doubled. We re-consider this possibility on the quantum-field level for both s = 1/2 and higher spin particles. Some parts have also been presented at the XIII DGFM SMF Workshop, November 4–8, 2019. Leon, Gto., México.

The purpose of this work is to present a non-commutative geometrical version of the Yang–Mills theory and the Yang–Mills–Scalar–Matter theory by constructing a concrete example using M. Durdevich’s theory of quantum principal bundles.

In this chapter, (1) We review historical and recent verifications of the Einstein–Podolsky–Rosen (EPR) argument that Quantum mechanics is not a complete theory supporting the completion of quantum into hadronic mechanics (hm). (2) We review the recently proposed completion of the conventional quantum entanglement of point-like particles without known interactions into the novel EPR entanglement of extended particles under non-Hamiltonian interactions quantitatively represented by the novel iso-mathematics of hadronic mechanics. (3) We review and extend the representation via hadronic mechanics achieved in preceding works of all characteristics of muons, including their anomalous magnetic moment, under the assumption that the physical constituents of the muons are ordinary electrons and positrons released free in the spontaneous decay with lowest mode μ^± → e⁻, e^±, e⁻, resulting in the structure model ${\mu }^{\pm }={\left(e_{\downarrow }^{-},e_{\uparrow }^{\pm },e_{\downarrow }^{-}\right)}_{hm}$ in which the annihilation of the electron–positron pair is confirmed by the muon decay mode μ^± → e^±+2γ and explain the instability of the muons. (4) We review and expand preceding works by various authors on the apparent unsettled character of recent measurements of the anomalous magnetic moment of the muons in view of internal non-local and time-irreversible effects. (5) Thanks to the preceding studies, we propose, apparently for the first time, the completion of conventional quantum computers based on 20th century applied mathematics essentially describing isolated components, into the broader EPR computers based on iso-mathematics for the description of components in continuous and instantaneous communication, and point out their advantages for increased computational speed, better cybersecurity, and increased efficiency.

The Standard Model (SM) of particle physics is a quantum field theory containing the internal symmetries of the group SU(3)_C × SU(2)_L × U(1)_Y. The theory contains a set of fundamental particles: leptons, quarks, gauge bosons, and Higgs bosons. The model describes the fundamental structure of matter considering elementary particles as irreducible entities governed by three of the four known interactions (does not include gravity). Currently, almost all the experimental tests of the three forces described by the model are in agreement with its predictions. However, several problems have not been solved within the framework of the Model Standard, such as neutrino oscillations, dark matter, among others. In this work, there will be a revision of some aspects of the SM, mainly in the electroweak sector.

We consider high-dimensional quantum states which generalize multivalued logics. Through maximally entangled states, a central part is able to receive 2(n−1) classical bits from n corresponding parts by one two-dimensional qubit from each of them. As a second generalization, we introduce a multiparty protocol for this kind of superdense coding using quantum systems with k quantum levels.

Isomorphism is one of the most important equivalence relations between two structures of the same kind. We present some recent results on the maximal complexity of the isomorphism problem for computable structures from a class K closed under isomorphism. Formally, the computable isomorphism problem for K is the set of pairs of computable indices for computable structures in K, which are isomorphic. For many familiar classes, the computable isomorphism problem is complete at some level of arithmetical hierarchy, such as for vector spaces over a fixed infinite computable field or for algebraically closed fields of a fixed characteristic, as established by Calvert. On the other hand, there are classes of structures with ${\text{Σ}}_1^1$-complete computable isomorphism problem, which is of maximal complexity. These classes contain isomorphic computable structures that are not even hyperarithmetically isomorphic. For example, Friedman and Stanley established that fields of a fixed characteristic have ${\text{Σ}}_1^1$-complete computable isomorphism problem. Recent examples of such classes include 2-step nilpotent groups, distributive lattices, and nilpotent rings. The method we use to establish ${\text{Σ}}_1^1$-completeness is based on a uniform effective interpretation of computable structures from a certain class into computable structures in K.

We present detailed calculations for a one-parameter family of quantum spaces, which quantize the classical Poincaré model of the hyperbolic plane, in such a way that all classical symmetries are preserved. Using the theory of quantum principal bundles, it is then possible to naturally define and study quantum counterparts of all classical hyperbolic Riemann surfaces. The one-parametric family includes, as its two extreme cases, the classical hyperbolic plane as well as the quantum space of the standard Toeplitz extension. We find a closed expression for the generating function and compute the hyperbolic metric as a natural part of the quantum structure, arising from the stochastic points interpreted as coherent states and considering their transition probabilities.

The purpose of this chapter is to develop gravitational theory using spacetime geometric algebra. We invite readers to familiarize themselves with the concepts used in this powerful mathematical tool, reading the materials written by Hestenes and Sobczyk. Using the dichotomy between the concepts absolute and relative given by Oziewicz, we will show how the linear gravitational equations obtained from different approaches, namely, linearized Einstein field equations (from tensor analysis), the Heaviside gravitational equations (from Gibbs’s vector calculus), gravitational equations obtained using quaternions by Arbab, the linear equations from Logunov’s relativistic theory of gravitation (all of them, cited in what follows) can be summarized into a single equation, instead of four. Another advantage of using geometric algebra is that all equations are basis free, which means that the system of gravitational equations and its solutions are independent of reference frames.

The following section is included:

• Index