
Raymond AschheimQuantum Gravity Research · Theory
Raymond Aschheim
Dipl. Ing. CentraleSupelec, GIF, FR
About
69
Publications
16,759
Reads
How we measure 'reads'
A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Learn more
355
Citations
Publications
Publications (69)
Background an objectives
Our recent work has focused on the application of infinite group theory and related algebraic geometric tools in the context of transcription factors and microRNAs. We were able to differentiate between “healthy” nucleotide sequences and disrupted sequences that may be associated with various diseases. In this paper, we ext...
We present three new coset manifolds named Dixon-Rosenfeld lines that are similar to Rosenfeld projective lines except over the Dixon algebra $$\mathbb {C}\otimes \mathbb {H}\otimes \mathbb {O}$$ C ⊗ H ⊗ O . Three different Lie groups are found as isometry groups of these coset manifolds using Tits’ formula. We demonstrate how Standard Model intera...
In his study on the geometry of Lie groups, Rosenfeld postulated a strict relation between all real forms of exceptional Lie groups and the isometries of projective and hyperbolic spaces over the (rank-2) tensor product of Hurwitz algebras taken with appropriate conjugations. Unfortunately, the procedure carried out by Rosenfeld was not rigorous, s...
Our recent work has focused on the application of infinite group theory and related algebraic geometric tools in the context of transcription factors and microRNAs. We were able to differentiate between “healthy" nucleotide sequences and disrupted sequences that may be associated with various diseases. In this paper, we extend our efforts to the st...
In this paper we present a general setting for aperiodic Jordan algebras arising from icosahedral quasicrystals that are obtainable as model sets of a cut-and-project scheme with a convex acceptance window. In these hypothesis, we show the existence of an aperiodic Jordan algebra structure whose generators are in one-to-one correspondence with elem...
We present three new coset manifolds named Dixon-Rosenfeld lines that are similar to Rosenfeld projective lines except over the Dixon algebra $\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$. Three different Lie groups are found as isometry groups of these coset manifolds using Tits' formula. We demonstrate how Standard Model interactions with the Di...
Aperiodic algebras are infinite dimensional algebras with generators corresponding to an element of the aperiodic set. These algebras proved to be an useful tool in studying elementary excitations that can propagate in multilayered structures and in the construction of some integrable models in quantum mechanics. Starting from the works of Patera a...
Aperiodic algebras are infinite dimensional algebras with generators corresponding to an element of the aperiodic set. These algebras proved to be an useful tool in studying elementary excitations that can propagate in multilayered structures and in the construction of some integrable models in quantum mechanics. Starting from the works of Patera a...
In his study on the geometry of Lie groups, Rosenfeld postulated a strict relation between all real forms of exceptional Lie groups and the isometries of projective and hyperbolic spaces over the (rank-2) tensor product of Hurwitz algebras taken with appropriate conjugations. Unfortunately, the procedure carried out by Rosenfeld was not rigorous, s...
Taking a DNA sequence, a word with letters/bases A, T, G and C, as the relation between the generators of an infinite group π, one can discriminate between two important families: (i) the cardinality structure for conjugacy classes of subgroups of π is that of a free group on one to four bases, and the DNA word, viewed as a substitution sequence, i...
Citation: Planat, M.; Amaral, M.M.; Fang, F.; Chester, D.; Aschheim, R.; Irwin, K. DNA sequence and structure under the prism of group theory and algebraic surfaces. Int. J. Mol. Sci. 2022, 1, 0. https://doi.org/
Taking a DNA-sequence, a word with letters/bases A, T, G and C, as the relation between the generators of an infinite group $\pi$, one can discriminate two important families: (i) the cardinality structure for conjugacy classes of subgroups of $\pi$ is that of a free group on $1$ to $4$ bases and the DNA word, viewed as a substitution sequence, is...
Motivated by the recent interest in Lie algebraic and geometric structures arising from tensor products of division algebras and their relevance to high energy theoretical physics, we analyze generalized bioctonionic projective and hyperbolic planes. After giving a Veronese representation of the complexification of the Cayley plane [Formula: see te...
It is shown that the representation theory of some finitely presented groups thanks to
their SL2(C) character variety is related to algebraic surfaces. We make use of the Enriques–Kodaira classification of algebraic surfaces and related topological tools to make such surfaces explicit. We study the connection of SL2(C) character varieties to topolo...
It is shown that the representation theory of some finitely presented groups thanks to their $SL_2(\mathbb{C})$ character variety is related to algebraic surfaces. We make use of the Enriques-Kodaira classification of algebraic surfaces and the related topological tools to make such surfaces explicit. We study the connection of $SL_2(\mathbb{C})$ c...
Transcription factors (TFs) are proteins that recognize specific DNA fragments in order to decode the genome and ensure its optimal functioning. TFs work at the local and global scales by specifying cell type, cell growth and death, cell migration, organization and timely tasks. We investigate the structure of DNA-binding motifs with the theory of...
In this work we present a useful way to introduce the octonionic projective and hyperbolic plane through the use of Veronese vectors. Then we focus on their relation with the exceptional Jordan algebra and show that the Veronese vectors are the rank-one elements of the algebra. We then study groups of motions over the octonionic plane recovering al...
Motivated by the recent interest in Lie algebraic and geometric structures arising from tensor products of division algebras and their relevance to high energy theoretical physics, we analyze generalized bioctonionic projective and hyperbolic planes. After giving a Veronese representation of the complexification of the Cayley plane $\mathbb{O}P_{\m...
Transcription factors (TFs) are proteins that recognize specific DNA fragments in order to decode the genome and ensure its optimal functioning. TFs work at the local and global scales by specifying cell type, cell growth and death, cell migration, organization and timely tasks. We investigate the structure of DNA-binding motifs with the theory of...
We consider quantum transition amplitudes, partition functions and observables for 3D spin foam models within SU(2) quantum group deformation symmetry, where the deformation parameter is postulated to be a complex fifth root of unity. By considering fermionic cycles through the foam we couple this SU(2) quantum group with the same deformation of SU...
In this work we present a useful way to introduce the octonionic projective and hyperbolic plane $\mathbb{O}P^{2}$ through the use of Veronese vectors. Then we focus on their relation with the exceptional Jordan algebra $\mathfrak{J}_{3}^{\mathbb{O}}$ and show that the Veronese vectors are the rank-one elements of the algebra. We then study groups...
Transcription factors (TFs) are proteins that recognize specific DNA fragments in order to decode the genome and ensure its optimal functioning. TFs work at the local and global scales by specifying cell type, cell growth and death, cell migration, organization and timely tasks. We investigate the structure of DNA-binding motifs with the theory of...
We consider partition functions, in the form of state sums, and associated probabilistic measures for aperiodic substrates described by model sets and their associated tiling spaces. We propose model set tiling spaces as microscopic models for small scales in the context of quantum gravity. Model sets possess special self-similarity properties that...
Citation: Planat, M.; Aschheim, R.; Amaral, M.M.; Fang, F.; Irwin, K. Graph Coverings for Investigating Non Local Structures in Proteins, Music and Poems. Sci 2021, 3, 39. Abstract: We explore the structural similarities in three different languages, first in the protein language whose primary letters are the amino acids, second in the musical lang...
It is shown how the secondary structure of proteins, musical forms and verses of poems are approximately ruled by universal laws relying on graph coverings. In this direction, one explores the group structure of a variant of the SARS-Cov-2 spike protein and the group structure of apolipoprotein-H, passing from the primary code with amino acids to t...
It is shown how the secondary structure of proteins, musical forms and verses of poems are approximately ruled by universal laws relying on graph coverings. In this direction, one explores the group structure of a variant of the SARS-Cov-2 spike protein and the group structure of apolipoprotein-H, passing from the primary code with amino acids to t...
Every protein consists of a linear sequence over an alphabet of 20 letters/amino acids. The sequence unfolds in the 3-dimensional space through secondary (local foldings), tertiary (bonds) and quaternary (disjoint multiple) structures. The mere existence of the genetic code for the 20 letters of the linear chain could be predicted with the (informa...
Every protein consists of a linear sequence over an alphabet of $20$ letters/amino acids. The sequence unfolds in the $3$-dimensional space through secondary (local foldings), tertiary (bonds) and quaternary (disjoint multiple) structures. The mere existence of the genetic code for the $20$ letters of the linear chain could be predicted with the (i...
The Kummer surface was constructed in 1864. It corresponds to the desingularization of the quotient of a 4-torus by 16 complex double points. Kummer surface is known to play a role in some models of quantum gravity. Following our recent model of the DNA genetic code based on the irreducible characters of the finite group G5:=(240,105)≅Z5⋊2O (with 2...
The Kummer surface was constructed in 1864. It corresponds to the desingularisation of 1 the quotient of a 4-torus by 16 complex double points. Kummer surface is kwown to play a role in 2 some models of quantum gravity. Following our recent model of the DNA genetic code based on the 3 irreducible characters of the finite group G 5 := (240, 105) ∼ =...
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=Z5⋊H where H=Z2.S4≅2O is isomorphic to the binary octahedral group 2O and S4 is the symmetric group on four letters/bases. The second group has signature G=Z5...
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 lette...
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 = Z5 ⋊ H where H = Z2.S4 ∼ = 2O is isomorphic to the binary octahedral group 2O and S4 is the symmetric group on four letters/bases. The second group has sign...
A popular account of the mixing patterns for the three generations of quarks and leptons is through the characters κ of a finite group G. Here, we introduce a d-dimensional Hilbert space with d = cc(G), the number of conjugacy classes of G. Groups under consideration should follow two rules, (a) the character table contains both two-and three-dimen...
The authors previously found a model of universal quantum computation by making use of the coset structure of subgroups of a free group G with relations. A valid subgroup H of index d in G leads to a 'magic' state |ψ in d-dimensional Hilbert space that encodes a minimal informationally complete quantum measurement (or MIC), possibly carrying a fini...
A popular account of the mixing patterns for the three generations of quarks and leptons is through the characters $\kappa$ of a finite group $G$. Here we introduce a $d$-dimensional Hilbert space with $d=cc(G)$, the number of conjugacy classes of $G$. Groups under consideration should follow two rules, (a) the character table contains both two- an...
The authors previously found a model of universal quantum computation by making use of the coset structure of subgroups of a free group G with relations. A valid subgroup H of index d in G leads to a 'magic' state |ψ in d-dimensional Hilbert space that encodes a minimal informationally complete quantum measurement (or MIC), possibly carrying a fini...
The authors previously found a model of universal quantum computation by making use of the coset structure of subgroups of a free group G with relations. A valid subgroup H of index d in G leads to a ‘magic’ state ∣∣ψ⟩ in d-dimensional Hilbert space that encodes a minimal informationally complete quantum measurement (or MIC), possibly carrying a fi...
Let H be a nontrivial subgroup of index d of a free group G and N be the normal closure of H in G. The coset organization in a subgroup H of G provides a group P of permutation gates whose common eigenstates are either stabilizer states of the Pauli group or magic states for universal quantum computing. A subset of magic states consists of states a...
Let $H$ be a non trivial subgroup of index $d$ of a free group $G$ and $N$ the normal closure of $H$ in $G$. The coset organization in a subgroup $H$ of $G$ provides a group $P$ of permutation gates whose common eigenstates are either stabilizer states of the Pauli group or magic states for universal quantum computing. A subset of magic states cons...
Based on the Cayley-Dickson process, a sequence of multidimensional structured natural numbers (infinions) creates a path from quantum information to quantum gravity. Octonionic structure, exceptional Jordan algebra, and E 8 Lie algebra are encoded on a graph with E 9 connectivity, decorated by integral matrices. With the magic star, a toy model fo...
We consider quantum transition amplitudes, partition function and observables for 3D spin foam model with the SU (2) quantum group deformation symmetry where the deformation parameter is a complex fifth root of unity. By considering fermionic cycles through the foam we couple this SU (2) quantum group with the same deformation of SU (3) and G 2 so...
The fundamental group $\pi_1(L)$ of a knot or link $L$ may be used to generate magic states appropriate for performing universal quantum computation and simultaneously for retrieving complete information about the processed quantum states. In this paper, one defines braids whose closure is the $L$ of such a quantum computer model and computes their...
The fundamental group π1(L) of a knot or link L may be used to generate magic states appropriate for performing universal quantum computation and simultaneously for retrieving complete information about the processed quantum states. In this paper, one defines braids whose closure is the L of such a quantum computer model and computes their Seifert...
A single qubit may be represented on the Bloch sphere or similarly on the 3-sphere S 3 . Our goal is to dress this correspondence by converting the language of universal quantum computing (UQC) to that of 3-manifolds. A magic state and the Pauli group acting on it define a model of UQC as a positive operator-valued measure (POVM) that one recognize...
A single qubit may be represented on the Bloch sphere or similarly on the 3-sphere S3. Our goal is to dress this correspondence by converting the language of universal quantum computing (UQC) to that of 3-manifolds. A magic state and the Pauli group acting on it define a model of UQC as a positive operator-valued measure (POVM) that one recognizes...
A single qubit may be represented on the Bloch sphere or similarly on the $3$-sphere $S^3$. Our goal is to dress this correspondence by converting the language of universal quantum computing (uqc) to that of $3$-manifolds. A magic state and the Pauli group acting on it define a model of uqc as a POVM that one recognizes to be a $3$-manifold $M^3$....
It has been shown that non-stabilizer eigenstates of permutation gates are appropriate for allowing $d$-dimensional universal quantum computing (uqc) based on minimal informationally complete POVMs. The relevant quantum gates may be built from subgroups of finite index of the modular group $\Gamma=PSL(2,\mathbb{Z})$ [M. Planat, Entropy 20, 16 (2018...
It has been shown that non-stabilizer eigenstates of permutation gates are appropriate for allowing d-dimensional universal quantum computing (uqc) based on minimal informationally complete POVMs. The relevant quantum gates may be built from subgroups of finite index of the modular group Γ = P SL(2, Z) [M. Planat, Entropy 20, 16 (2018)] or more gen...
A single qubit may be represented on the Bloch sphere or similarly on the $3$-sphere $S^3$. Our goal is to dress this correspondence by converting the language of universal quantum computing (UQC) to that of $3$-manifolds. A magic state and the Pauli group acting on it define a model of UQC as a positive operator-valued measure (POVM) that one reco...
The goal of this work is to elaborate on new geometric methods of constructing exact and parametric quasiperiodic solutions for anamorphic cosmology models in modified gravity theories, MGTs, and general relativity, GR. There exist previously studied generic off-diagonal and diagonalizable cosmological metrics encoding gravitational and matter fiel...
We apply a discrete quantum walk from a quantum particle on a discrete quantum spacetime from loop quantum gravity and show that the related entanglement entropy drives an entropic force. We apply these concepts in a model where walker positions are topologically encoded on a spin network. Then, we discuss the role of the golden ratio in fundamenta...
We present the emergence of a root system in six dimensions from the tetrahedra of an icosahedral core known as the 20-group (20G) within the framework of Clifford’s geometric algebra. Consequently, we establish a connection between a three-dimensional icosahedral seed, a six-dimensional (6D) Dirichlet quantized host and a higher dimensional lattic...
We present the emergence of a root system in six dimensions from the tetrahedra of an icosahedral core known as the 20-group (20G) within the framework of Clifford's geometric algebra. Consequently, we establish a connection between a three dimensional icosahedral seed, a six dimensional Dirichlet quantized host and a higher dimensional lattice str...
The goal of this work on mathematical cosmology and geometric methods in modified gravity theories, MGTs, is to investigate Starobinsky-like inflation scenarios determined by gravitational and scalar field configurations mimicking quasicrystal, QC, like structures. Such spacetime aperiodic QCs are different from those discovered and studied in soli...
The goal of this work on mathematical cosmology and geometric methods in modified gravity theories, MGTs, is to investigate Starobinsky-like inflation scenarios determined by gravitational and scalar field configurations mimicking quasicrystal, QC, like structures. Such spacetime aperiodic QCs are different from those discovered and studied in soli...
The goal of this work is to elaborate on new geometric methods of constructing exact and parametric quasiperiodic solutions for anamorphic cosmology models in modified gravity theories, MGTs, and general relativity, GR. There exist previously studied generic off-diagonal and diagonalizable cosmological metrics encoding gravitational and matter fiel...
Quasicrystals are fractal due to their self similar property. In this paper, a new cycloidal fractal signature possessing the cardioid shape in the Mandelbrot set is presented in the Fourier space of a Fibonacci chain with two lengths, L and S, where L/S = φ. The corresponding pointwise dimension is 0.7. Various modifications, such as truncation fr...
Inspired by the Hilbert-Polya proposal to prove the Riemann Hypothesis we have studied the Schroedinger QM equation involving a highly non-trivial potential, and whose self-adjoint Hamiltonian operator has for its energy spectrum one which approaches the imaginary parts of the zeta zeroes only in the asymptotic (very large N) region. The ordinates...
We apply a discrete quantum walk from a quantum particle on a discrete quantum spacetime from loop quantum gravity and show that the related Entanglement Entropy can drive a entropic force. We apply this concepts to propose a model of a walker position topologically encoded on a spin network.
We explore discrete approaches in LQG where all fields, the gravitational
tetrad, and the matter and energy fields, are encoded implicitly in a graph
instead of being additional data. Our graph should therefore be richer than a
simple simplicial decomposition. It has to embed geometrical information and
the standard model. We start from Lisi's mode...
Among Cayley graphs on the symmetric group, the pancake graph is one as a viable interconnection scheme for parallel computers, which has been examined by a number of researchers. The pancake was proposed as alternatives to the hypercube for interconnecting processors in parallel computers. Some good and attractive properties of this interconnectio...
The new graphiton models described here are trivalent graphs which encode topologically binary information. They permit defining intrinsic discrete spaces which constitute supernode crystals. Besides encoding its own metric, the model supports disturbances due to fault tolerance through the redundancy of information in the paths of connection betwe...
Instantaneous, automatic reader for reading handwriting in real time, associating a pen equipped with a light source with two optical sensors measuring the position of the light source. The optical sensors are angle sensors of the CCD linear array type. The instantaneous measurement of the position of the pen on the paper allows measurement of the...