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Unitarity is a cornerstone of quantum theory, ensuring the conservation of probability and information. Although non-Hermitian Hamiltonians are typically associated with open or dissipative systems, pseudo-Hermitian quantum mechanics shows…
In math.RT/0302174 we developed a framework to study representations of groups of the form $G((t))$, where $G$ is an algebraic group over a local field $K$. The main feature of this theory is that natural representations of groups of this…
We construct an $SO(10)$ grand unified theory in the formulation of non-com-\break mutative geometry. The geometry of space-time is that of a product of a continuos four dimensional manifold times a discrete set of points. The properties of…
When joined the unified gauge picture of fundamental interactions, the gravitation theory leads to geometry of a space-time which is far from simplicity of pseudo-Riemannian geometry of Einstein's General Relativity. This is geometry of the…
In part I of the foundation of the hyperunified field theory, we have shown the presence of entangled hyperqubit-spinor field as fundamental building block with appearance of Minkowski hyper-spacetime as free-motion spacetime and emergence…
In this article, we develop an algebraic framework of axioms which abstracts various high-level properties of multi-qudit representations of generalized Clifford algebras. We further construct an explicit model and prove that it satisfies…
In this paper, we develop a general framework of geometric functorial field theories, meaning that all bordisms in question are endowed with geometric structures. We take particular care to establish a notion of smooth variation of such…
We show that Clifford algebras provide a natural language to describe the physics of liquid crystal defects in 3D. This framework shows that most of these defects have fermionic nature, as the director field profile on a 2D cross section…
The aim of this project is to attach a geometric structure to the ring of integers. It is generally assumed that the spectrum $\mathrm{Spec}(\mathbb{Z})$ defined by Grothendieck serves this purpose. However, it is still not clear what…
We consider a four dimensional space-time symmetry which is a non trivial extension of the Poincar\'e algebra, different from supersymmetry and not contradicting {\sl a priori} the well-known no-go theorems. We investigate some field…
We study conformal field theories (CFTs) and their classifications from a modern perspective based on the abstract algebraic formalism of symmetries or conserved charges, known as symmetry topological field theories (SymTFTs). By studying…
In this paper we show how a gravitational field generated by a given energy-momentum distribution (for all realistic cases) can be represented by distinct geometrical structures (Lorentzian, teleparallel and non null nonmetricity…
Gravitation theory meets spontaneous symmetry breaking when the structure group of the principal linear frame bundle $LX$ over a world manifold $X^4$ is reducible to the Lorentz group $SO(3,1)$. The physical underlying reason of this…
We give a complete classification of topological field theories with reflection structure and spin-statistics in one and two spacetime dimensions. Our answers can be naturally expressed in terms of an internal fermionic symmetry group $G$…
Different ways of representing the group $SU(3)$ within a Geometric Algebra approach are explored. As part of this we consider characteristic multivectors for $SU(3)$, and how these are linked with decomposition of generators into commuting…
We formulate the unitary rational orbifold conformal field theories in the algebraic quantum field theory framework. Under general conditions, we show that the orbifold of a given unitary rational conformal field theories generates a…
The mathematical foundations of relativistic quantum mechanics is largely based upon the discovery of the Pauli and Dirac matrices. An algebra which lies at an even more fundamental level is the geometric Clifford algebra with metric…
We describe a theory amalgamating quantum theory and general relativity through the identification of a continuous 4-dimensional spacetime arena constructed from the substructures of a generalised multi-dimensional form for proper time. In…
UNIFORM algebras have been extensively investigated because of their importance in the theory of uniform approximation and as examples of complex Banach algebras. An interesting question is whether analogous algebras exist when a complete…
The motivations for the construction of an 8-component representation of fermion fields based on a two dimensional representation of time reversal transformation and CPT invariance are discussed. Some of the elementary properties of the…