Hopf algebraic structures will replace groups and group representations as the leading paradigm in forthcoming times. K -theory, co-homology, entanglement, statistics, representation categories, quantized or twisted structures as well as more geometric topics of invariant theory, e.g., the Graßmann-Cayley bracket algebra are all covered by the Hopf algebraic framework. The new branch of experim...

We construct the Extended Relativity Theory in Born-Clifford-Phase spaces with an upper and lower length scales (infrared/ultraviolet cutoff ). The invariance symmetry leads naturally to the real Clifford algebra Cl(2, 6, R) and complexified Clifford ClC(4) algebra related to Twistors. We proceed with an extensive review of Smith’s 8D model based on the Clifford algebra Cl(1, 7) that reproduces...

Recently several generalizations to higher dimension of the classical Fourier transform (FT) using Clifford geometric algebra have been introduced, including the two-dimensional (2D) Clifford Fourier transform (CFT). Based on the 2D CFT, we establish the two-dimensional Clifford windowed Fourier transform (CWFT). Using the spectral representation of the CFT, we derive several important properti...

The ZX calculus is a diagrammatic language for quantum mechanics and quantum information processing. We prove that the ZX-calculus is not complete for the Clifford+T quantum mechanics. The completeness for this fragment has been stated as one of the main current open problems in categorical quantum mechanics [8]. The ZX calculus was known to be incomplete for quantum mechanics [7], on the other...

Abstract In this paper a new proof is given for the supermodularity of information content. Using the decomposability of the information content an algorithm is given for discovering the Markov network graph structure endowed by the pairwise Markov property of a given probability distribution. A discrete probability distribution is given for which the equivalence of Hammersley-Clifford theorem ...

The main objective of this paper is to clarify the ontology of DiracHestenes spinor fields (DHSF ) and its relationship with sum of even multivector fields, on a general Riemann-Cartan spacetime M=(M, g,∇, τg, ↑) admitting a spin structure and to give a mathematically rigorous derivation of the so called Dirac-Hestenes equation (DHE ) when M is a Lorentzian spacetime. To this aim we introduce t...

It is a well known fact from the group theory that irreducible tensor representations of classical groups are suitably characterized by irreducible representations of the symmetric groups. However, due to their different nature, vector and spinor representations are only connected and not united in such description. Clifford algebras are an ideal tool with which to describe symmetries of multip...

This paper is concerned with the modular representation theory of the affine Hecke-Clifford superalgebra, the cyclotomic Hecke-Clifford superalgebras, and projective representations of the symmetric group. Our approach exploits crystal graphs of affine Kac-Moody algebras.

Fault-tolerant quantum computation is a basic problem in quantum computation, and teleportation is one of the main techniques in this theory. Using teleportation on stabilizer codes, the most well-known quantum codes, Pauli gates and Clifford operators can be applied fault-tolerantly. Indeed, this technique can be generalized for an extended set of gates, the so called Ck hierarchy gates, intro...