arXiv subjects

Niels Gresnigt

Publications and source records attributed to Niels Gresnigt.

9 recordsLinked to original sources

arXiv2026

Constraints for Physical Gauge Groups Coming from the Causal Action Principle

The constraints for the effective local gauge groups stemming from the causal action principle for causal fermion systems are reviewed. The constraint which are quadratic in the gauge potentials (coming from the so-called bilinear logarithmic terms) are shown to make a connection between the structures of Lie algebras and Clifford algebras. This connection is worked out systematically starting from general chiral potentials corresponding to the gauge group $\mathrm{U}(N) \times \mathrm{U}(N)$. The general results are illustrated in various examples.

math-ph↗

arXiv2026

Sedenions, Clifford Algebras, and Three Fermion Generations: A Focused Review

The existence of three fermion generations remains one of the unexplained structural features of the Standard Model. This article reviews an algebraic framework in which an intrinsic $S_3$ family symmetry relates three gauge-equivalent fermion sectors. The framework is motivated by division- and Clifford-algebra constructions in which the complex octonions and $\mathbb{C}\ell(6)$ organise the colour and electromagnetic quantum numbers of one generation. Continuing the Cayley--Dickson sequence, the sedenions provide an intrinsic $S_3$ automorphism structure, while their complexified left-multiplication operators generate an associative algebra isomorphic to $\mathbb{C}\ell(8)$. In the $\mathbb{C}\ell(8)$ formulation, the order-three family action generates three linearly independent fermion sectors while leaving a single $\mathfrak{su}(3)_C\oplus\mathfrak{u}(1)_{\mathrm{em}}$ gauge algebra. Extension to $\mathbb{C}\ell(10)$ incorporates a single $\mathfrak{su}(3)_C\oplus\mathfrak{su}(2)_L\oplus\mathfrak{u}(1)_Y$ gauge algebra, with each generation including a right-handed neutrino sterile under the SM gauge interactions. We place the construction in context by comparing it with representative algebraic and family-symmetry approaches to three generations, including triality-based proposals, and clarify why the family symmetry used here is distinct from standard $\operatorname{Spin}(8)$ triality despite the common appearance of $\mathbb{C}\ell(8)$ and $S_3$. The framework remains algebraic and representation theoretic rather than a complete dynamical theory; family-symmetry breaking, realistic fermion masses and mixing, and a dynamical account of the gauge and matter sectors remain open problems.

physics.gen-ph↗

arXiv2026

Higgs and Yukawa Structure in a Clifford Algebra Model with Three Generations and $S_3$ Family Symmetry

We construct the Higgs and Yukawa sectors as a structural completion of an algebraic three-generation model based on the complex Clifford algebra $\mathbb{C}\ell(10)$ with an intrinsic $S_3$ family symmetry. This addresses a common limitation of algebraic frameworks, in which Standard Model fermion multiplets and gauge symmetries may be described naturally, while the Higgs and Yukawa sectors remain less developed or absent. In the present framework, three algebraically distinguished fermion sectors are permuted by $S_3$, while the Standard Model gauge generators remain generation-independent. Higgs components are realised as right-action operators mapping weak-doublet fermion sectors into the corresponding weak-singlet sectors, and Yukawa coefficients are extracted using a Hilbert--Schmidt trace pairing. This yields two first-generation Higgs doublets with electroweak quantum numbers $(1,2,-1)$ and $(1,2,+1)$ under $SU(3)_C \times SU(2)_L \times U(1)_Y$, together with a Type-II-like separation between down-type and up-type Yukawa channels. Acting with the order-three family generator then generates a family-resolved Higgs sector organised into cyclic $S_3$ orbits. In the cyclically averaged Higgs limit, the Type-II-like Yukawa selection rule is preserved, while the generation-space Yukawa matrix is fixed algebraically and is non-diagonal in the algebraic generation basis. Under the usual implementation of electroweak symmetry breaking, the neutral Higgs couplings are aligned with the corresponding mass matrices, so tree-level flavour-changing neutral currents are not expected in this limit. The result is a constrained algebraic starting point for future $S_3$-breaking flavour phenomenology.

physics.gen-ph↗

arXiv2026

Electroweak Structure and Three Fermion Generations in Clifford Algebra with S3 Family Symmetry

We construct an explicit algebraic realisation of three fermion generations within a single Clifford algebra, transforming under the full Standard Model $SU(3)_C\times SU(2)_L\times U(1)_Y$ gauge group, in which an intrinsic $S_3$ family symmetry permutes three algebraically distinguished but gauge-equivalent fermion sectors without replicating the gauge bosons. Fermionic states are represented by minimal left ideals of the complex Clifford algebra $\mathbb{C}\ell(10)$, while the three-generation structure arises from an embedded discrete $S_3$ symmetry acting on the space of algebraic spinors. The Standard Model gauge generators are identified as elements commuting with this $S_3$ action and act on physical states via the adjoint (commutator) action. The resulting spectrum reproduces the correct Standard Model quantum numbers for three linearly independent generations of fermions.

physics.gen-ph↗

arXiv2024

Algebraic realisation of three fermion generations with $S_3$ family and unbroken gauge symmetry from $\mathbb{C}\ell(8)$

Building on previous work, we extend an algebraic realisation of three fermion generations within the complex Clifford algebra $\mathbb{C}\ell(8)$ by incorporating a $U(1)_{em}$ gauge symmetry. The algebra $\mathbb{C}\ell(8)$ corresponds to the algebra of complex linear maps from the (complexification of the) Cayley-Dickson algebra of sedenions, $\mathbb{S}$, to itself. Previous work represented three generations of fermions with $SU(3)_C$ colour symmetry permuted by an $S_3$ symmetry of order-three, but failed to include a $U(1)$ generator that assigns the correct electric charge to all states. Furthermore, the three generations suffered from a degree of linear dependence between states. By generalising the embedding of the discrete group $S_3$, corresponding to automorphisms of $\mathbb{S}$, into $\mathbb{C}\ell(8)$, we include an $S_3$-invariant $U(1)$ that correctly assigns electric charge. First-generation states are represented in terms of two even $\mathbb{C}\ell(8)$ semi-spinors, obtained from two minimal left ideals, related to each other via the order-two $S_3$ symmetry. The remaining two generations are obtained by applying the $S_3$ symmetry of order-three to the first generation. In this model, the gauge symmetries, $SU(3)_C\times U(1)_{em}$, are $S_3$-invariant and preserve the semi-spinors. As a result of the generalised embedding of the $S_3$ automorphisms of $\mathbb{S}$ into $\mathbb{C}\ell(8)$, the three generations are now linearly independent.

hep-th↗

arXiv2022

On the dynamical emergence of $SU_q(2)$ from the regularization of $2+1D$ gravity with cosmological constant

The quantization of the reduced phase-space of the Einstein-Hilbert action for gravity in $2+1D$ has been shown to bring about the emergence, at the quantum level, of a topological quantum field theory endowed with an $SU_q(2)$ quantum group symmetry structure. We hereby tackle the same problem, but start from the kinematical $SU(2)$ (quantum) Hilbert space of the theory of $2+1D$ gravity with non-zero cosmological constant in the Palatini formalism, and subsequently impose the constraints. We hence show the dynamical emergence of the $SU_q(2)$ quantum group at the quantum level within the spin-foam framework. The regularized curvature constraint is responsible for the effective representations of $SU_q(2)$ that are recovered for any Wilson loop evaluated at the $SU(2)$ group element that encodes the discretization of the space-time curvature induced by the cosmological constant. The extension to the spin-network basis, and consequently to any transition amplitude between its generic states, enables us to derive in full generality the recoupling theory of $SU_q(2)$. We provide constructive examples for the scalar product of two loop states and spin-networks encoding trivalent vertices. We further comment on the diffeomorphism symmetry generated by the implementation of the curvature constraint, and finally derive explicitly the partition function amplitude of the Turaev-Viro model.

gr-qc↗

arXiv2021

Braided matter interactions in quantum gravity via 1-handle attachment

In a topological description of elementary matter proposed by Bilson-Thompson, the leptons and quarks of a single generation, together with the electroweak gauge bosons, are represented as elements of the framed braid group of three ribbons. By identifying these braids with emergent topological excitations of ribbon networks, it has been possible to encode this braid model into the framework of quantum geometry provided by loop quantum gravity. In the case of trivalent networks, it has not been possible to generate particle interactions, because the braids correspond to noiseless subsystems, meaning they commute with the evolution algebra generated by the local Pachner moves. In the case of tetravalent networks, interactions are only possible when the model's original simplicity, in which interactions take place via the composition of braids, is sacrificed. We demonstrate that it possible to preserve both the original classification of fermions, as well as their interaction via the braid product, if we embed the braid in a trivalent scheme, and supplement the local Pachner moves, with a non-local and graph changing 1-handle attachment. Moreover, we use Kauffman-Lins recoupling theory to obtain invariants of braided networks that distinguish topological configurations associated to particles in the Bilson-Thompson model.

gr-qc↗

arXiv2021

Deep Neural Networks as the Semi-classical Limit of Quantum Neural Networks

Our work intends to show that: (1) Quantum Neural Networks (QNN) can be mapped onto spinnetworks, with the consequence that the level of analysis of their operation can be carried out on the side of Topological Quantum Field Theories (TQFT); (2) Deep Neural Networks (DNN) are a subcase of QNN, in the sense that they emerge as the semiclassical limit of QNN; (3) A number of Machine Learning (ML) key-concepts can be rephrased by using the terminology of TQFT. Our framework provides as well a working hypothesis for understanding the generalization behavior of DNN, relating it to the topological features of the graphs structures involved.

cond-mat.dis-nn↗

arXiv2018

Knotted boundaries and braid only form of braided belts

The Helon model identifies Standard Model quarks and leptons with certain framed braids joined together at both ends by a connecting node (disk). These surfaces with boundary are called braided 3-belts (or simply belts). Twisting and braiding of ribbons composing braided 3-belts are interchangeable, and it was shown in the literature that any braided 3-belt can be written in a pure twist form, specified by a vector of three multiples of half integers [a,b,c], a topological invariant. This paper identifies the set of braided 3-belts that can be written in a braid only form in which all twisting is eliminated. For these braids an algorithm to calculate the braid word is determined which allows the braid only word of every braided 3-belt to be written in a canonical form. It is furthermore demonstrated that the set of braided 3-belts do not form a group, due to a lack of isogeny. The conditions under which the boundary of a braided 3-belt is a knot are determined, and a formula for the Jones polynomial for knotted boundaries is derived. Considering knotted boundaries makes it possible to relate the Helon model to a model of quarks and leptons in terms of quantum trefoil knots, understood as representation of the quantum group SUq(2). Associating representations of a quantum group to the boundary of braided belts provides a possible means of developing the gauge symmetries of interacting braided belts in future work.

math.GT↗