Quantum Gravity Research

We introduce countably infinite series of finite dimensional generalizations of the exceptional Lie algebras: in fact, each exceptional Lie algebra (but g2) is the first element of an infinite series of finite dimensional algebras, which we name Magic Star algebras. All these algebras (but the first elements of the infinite series) are not Lie algebras, but nevertheless they have remarkable similarities with many characterizing features of the exceptional Lie algebras; they also enjoy a kind of periodicity (inherited by Bott periodicity), which we name Exceptional Periodicity. We analyze the graded algebraic structures arising in a certain projection (named Magic Star projection) of the generalized root systems pertaining to Magic Star algebras, and we highlight the occurrence of a class of rank-3, Hermitian matrix (special Vinberg T)-algebras (which we call H algebras) on each vertex of such a projection. We then focus on the Magic Star algebra f4(n), which generalizes the non-simply laced exceptional Lie algebra f4, and deserves a treatment apart. Finally, we compute the Lie algebra of the inner derivations of the H algebras, pointing out the enhancements occurring for each first element of the series of Magic Star algebras, thus retrieving the result known for the derivations of cubic simple Jordan algebras.

This paper is part of a series that describes the Fibonacci icosagrid quasicrystal (FIG) and its relation to the E8 root lattice. The FIG was originally constructed to represent the intersection points of an icosahedrally symmetric collection of planar grids in three dimensions, with the grid spacing of each following a Fibonacci chain. It was found to be closely related to a five-fold compound of 3D sections taken from the 4D Elser–Sloane quasicrystal (ESQC), which is derived via a cut-and-project process from E8. More recently, a direct cut-and-project from E8 has been found which yields the FIG (presented in another paper of this series). The present paper focuses not on the full quasicrystal, but on the relationship between the root polytope of E8 (Gosset’s 421 polytope) and the core polyhedron generated in the FIG, a compound of 20 tetrahedra referred to simply as a 20-Group. In particular, the H3 symmetry of the FIG can be seen as a five-fold or “golden” composition of tetrahedral symmetry (referring to the characteristic appearance of the golden ratio). This is shown to mirror a connection between tetrahedral and five-fold symmetries present in the 421. Indeed, the rotations that connect tetrahedra contained within the 421 are shown to induce, in a certain natural way, the tetrahedron orientations in the 20-Group.

Revealing the time structure of physical or biological objects is usually performed thanks to the tools of signal processing such as the fast Fourier transform, Ramanujan sum signal processing, and many other techniques. For space-time topological objects in physics and biology, we propose a type of algebraic processing based on schemes in which the discrimination of singularities within objects is based on the space-time-spin group SL(2,C). Such topological objects possess an homotopy structure encoded in their fundamental group, and the related SL(2,C) multivariate polynomial character variety contains a plethora of singularities somehow analogous to the frequency spectrum in time structures. Our approach is applied to a model of quantum computing based on an Akbulut cork in exotic R4, to an hyperbolic model of topological quantum computing based on magic states and to microRNAs in genetics. Such diverse topics reveal the manifold of possibilities of using the concept of a scheme spectrum.

We recently proposed that topological quantum computing might be based on $SL(2,\mathbb{C})$ representations of the fundamental group $\pi_1(S^3\setminus K)$ for the complement of a link $K$ in the $3$-sphere. The restriction to links whose associated $SL(2,\mathbb{C})$ character variety $\mathcal{V}$ contains a Fricke surface $\kappa_d=xyz -x^2-y^2-z^2+d$ is desirable due to the connection of Fricke spaces to elementary topology. Taking $K$ as the Hopf link $L2a1$, one of the three arithmetic two-bridge links [the Whitehead link $5_1^2$, the Berge link $6_2^2$, the double-eight link $6_3^2$] or the link $7_3^2$, the $\mathcal{V}$ for those links contains the reducible component $\kappa_4$, the so-called Cayley cubic. In addition, the $\mathcal{V}$ for the later two links contains the irreducible component $\kappa_3$, or $\kappa_2$, respectively. Taking $\rho$ to be a representation with character $\kappa_d$ ($d<4$), with $|x|,|y|,|z| \le 2$, then $\rho(\pi_1)$ fixes a unique point in the hyperbolic space $\mathcal{H}_3$ and is conjugate to a $SU(2)$ representation (a qubit). Even though details on the physical implementation remain open, more generally, we show that topological quantum computing may be developed from the point of view of three-bridge links, the topology of the $4$-punctured sphere and Painlev\'e VI equation. The $0$-surgery on the $3$ circles of the Borromean rings L6a4 is taken as an example.

We find that the degeneracies and many peculiarities of the DNA genetic code may be described thanks to two closely related (fivefold symmetric) finite groups. The first group has signature $G=\mathbb{Z}_5 \rtimes H$ where $H=\mathbb{Z}_2 . S_4\cong 2O$ is isomorphic to the binary octahedral group $2O$ and $S_4$ is the symmetric group on four letters/bases. The second group has signature $G=\mathbb{Z}_5 \rtimes GL(2,3)$ and points out a threefold symmetry of base pairings. For those groups, the representations for the $22$ conjugacy classes of $G$ are in one-to-one correspondence with the multiplets encoding the proteinogenic amino acids. Additionally, most of the $22$ characters of $G$ attached to those representations are informationally complete. The biological meaning of these coincidences is discussed.

The hard problem of consciousness must be approached through the ontological lens of twentieth-century physics, which tells us that reality is information theoretic Wheeler (American Scientist, 74, 366–375, 1986; Wheeler (1990) and quantized at the level of Planck scale spacetime Snyder (Physical Review D, 67, 38–41, 1947). Through careful deduction, it becomes clear that information cannot exist without consciousness—the awareness of things. And to be aware is to hold the meaning of relationships of objects within consciousness—perceiving abstract objects, while enjoying degrees of freedom within the structuring of those relationships. This defines consciousness as language—(1) a set of objects and (2) an ordering scheme with (3) degrees of freedom used for (4) expressing meaning. And since even information at the Planck scale cannot exist without consciousness, we propose an entity called a “primitive unit of consciousness,” which acts as a mathematical operator in a quantized spacetime language. Quasicrystal mathematics based on E8 geometry Sadoc and Mosseri (Journal of Physics A: Mathematical and General, 26, 1789, 1993) seems to be a candidate for the language of reality, possessing several qualities corresponding to recent physical discoveries and various physically realistic unification models.

In this paper, we present the construction of several aggregates of tetrahedra. Each construction is obtained by performing rotations on an initial set of tetrahedra that either (1) contains gaps between adjacent tetrahedra, or (2) exhibits an aperiodic nature. Following this rotation, gaps of the former case are “closed” (in the sense that faces of adjacent tetrahedra are brought into contact to form a “face junction”), while translational and rotational symmetries are obtained in the latter case. In all cases, an angular displacement of β = arccos 3 ϕ − 1 / 4 (or a closely related angle), where ϕ = 1 + 5 / 2 is the golden ratio, is observed between faces of a junction. Additionally, the overall number of plane classes, defined as the number of distinct facial orientations in the collection of tetrahedra, is reduced following the transformation. Finally, we present several “curiosities” involving the structures discussed here with the goal of inspiring the reader’s interest in constructions of this nature and their attending, interesting properties.

The Boerdijk–Coxeter helix is a helical structure of tetrahedra which possesses no non-trivial translational or rotational symmetries. In this document, we develop a procedure by which this structure is modified to obtain both translational and rotational (upon projection) symmetries along/about its central axis. We show by construction that a helix can be obtained whose shortest period is any whole number of tetrahedra greater than one except six, while a period of six necessarily entails a shorter period. We give explicit examples of two particular forms related to the pentagonal and icosahedral aggregates of tetrahedra as well as Buckminster Fuller’s “jitterbug transformation”.