Clifford operators in SU(N)1; N not odd prime

Clifford odd and even objects in even and odd dimensional spaces describing internal spaces of fermion and boson fields

In a long series of works, it has been demonstrated, that the {\it spin-charge-family} theory offers the explanation for all in the {\it standard model} assumed properties of the second quantized fermion and boson fields, offering several predictions as well as explanations for several of the observed phenomena. The theory assumes a simple starting action in even dimensional spaces with $d \ge (13 +1)$ with massless fermions interacting with gravity only. The internal spaces of fermion and boson fields are described by the Clifford odd and even objects, respectively. This contribution discusse…

On Supergroups with Odd Clifford Parameters and Supersymmetry with Modified Leibniz Rule

We investigate supergroups with Grassmann parameters replaced by odd Clifford parameters. The connection with non-anticommutative supersymmetry is discussed. A Berezin-like calculus for odd Clifford variables is introduced. Fermionic covariant derivatives for supergroups with odd Clifford variables are derived. Applications to supersymmetric quantum mechanics are made. Deformations of the original supersymmetric theories are encountered when the fermionic covariant derivatives do not obey the graded Leibniz property. The simplest non-trivial example is given by the N=2 SQM with a real $(1,2,1)…

Clifford odd and even objects offer the description of the internal space of fermions and bosons, respectively, opening new insight into the second quantization of fields

In a long series of works the author has demonstrated that the model named the {\it spin-charge-family} theory offers the explanation for all in the {\it standard model} assumed properties of the fermion and boson fields, as well as for many of their so far observed properties if the space-time is $\ge (13 +1)$ while fermions interact with gravity only. In this paper, I briefly report on the so far achievements of the theory. The main contribution demonstrates the offer of the Clifford odd and even objects for the description of the internal spaces of fermion (Clifford odd) and boson (Clifford…

Clifford代数如何帮助理解二次量子化的夸克和轻子及相应的矢量和标量玻色子场，{\it 开启超越标准模型的新一步}

How Clifford algebra helps understand second quantized quarks and leptons and corresponding vector and scalar boson fields, {\it opening a new step beyond the standard model}

本文利用Clifford代数的奇偶基向量描述高维空间中费米子与玻色子场的内部空间，证明其在$d=(3+1)$中能展现所有夸克、轻子及规范玻色子场的性质并做出预测，为超越标准模型提供新途径。

Understanding the second quantization of fermions in Clifford and in Grassmann space -- New way of second quantization of fermions, Part II

We present in Part II the description of the internal degrees of freedom of fermions by the superposition of odd products of the Clifford algebra elements, either $γ^a$'s or $\tildeγ^a$'s, which determine with their oddness the anticommuting properties of the creation and annihilation operators of the second quantized fermion fields in even $d$-dimensional space-time, as we do in Part I of this paper by the Grassmann algebra elements $θ^a$'s and $\frac{\partial}{\partial θ_a}$'s. We discuss: {\bf i.} The properties of the two kinds of the odd Clifford algebras, forming two independent spaces, …

Hypergraph States in SU(N)1, N odd prime, Chern-Simons Theory

Graph states and hypergraph states can be constructed from products of basic operations that appear in SU(N)1. The level-rank dual of a theorem of Salton, Swingle, and Walter implies that these operations can be prepared topologically in the n-torus Hilbert space of Chern-Simons theory for N neq 5 mod 4. For SU(N)1, N = 5 mod 4, only stabilizer states can be prepared on the n-torus Hilbert space, which restricts the construction to graph states.