C-R-T Fractionalization in the First Quantized Hamiltonian Theory

Lorentz invariant quantum field theories (QFTs) in four spacetime dimensions (4D) have a $\mathbb{Z}_4$ symmetry provided there exists a basis of operators in the QFT where all operators have even operator dimension, $d$, including those with $d > 4$. The $\mathbb{Z}_4$ symmetry is the extension of operator dimension parity by fermion number parity. If the $\mathbb{Z}_4$ is anomaly-free, such QFTs can be related to 3D topological superconductors. Additionally, imposing the $\...

We prove in this paper that the elliptic $R$--matrix of the eight vertex free fermion model is the intertwiner $R$--matrix of a quantum deformed Clifford--Hopf algebra. This algebra is constructed by affinization of a quantum Hopf deformation of the Clifford algebra.

Based on a fundamental symmetry between space, time, mass and charge, a series of group structures of physical interest is generated, ranging from C2 to E8. The most significant result of this analysis is a version of the Dirac equation combining quaternions and multivariate vectors, which is already second quantized and intrinsically supersymmetric, and which automatically leads to a symmetry breaking, with the creation of specific particle structures and a mass-generating m...

Inspired by recent developments generalizing Jordan-Wigner dualities to higher dimensions, we develop a framework of such dualities using an algebraic formalism for translation-invariant Hamiltonians proposed by Haah. We prove that given a translation-invariant fermionic system with general $q$-body interactions, where $q$ is even, a local mapping preserving global fermion parity to a dual Pauli spin model exists and is unique up to a choice of basis. Furthermore, the dual sp...

Four-dimensional spacetime, together with a natural generalisation to extra dimensions, is obtained through an analysis of the structures and symmetries deriving from possible arithmetic expressions for one-dimensional time. On taking the infinitesimal limit this simple one-dimensional structure can be consistently equated with a homogeneous form of arbitrary dimension possessing both spacetime and more general symmetries. An extended 4-dimensional manifold, with the associat...

Majorana fermions are currently of huge interest in the context of nanoscience and condensed matter physics. Different to usual fermions, Majorana fermions have the property that the particle is its own anti-particle thus, they must be described by real fields. Mathematically, this property makes nontrivial the quantization of the problem due, for instance, to the absence of a Wick-like theorem. In view of the present interest on the subject, it is important to develop differ...

Using the standard representation of the Dirac equation we show that, up to signs, there exist only TWO SETS of consistent solutions for the matrices of charge conjugation (C), parity (P), and time reversal (T). In both cases, P^2=-1, and then two succesive applications of the parity transformation to spin 1/2 fields NECESSARILY amounts to a 2\pi rotation. Each of these sets generates a non abelian group of sixteen elements, G_1 and G_2, which are non isomorphic subgroups of ...

In this paper, we present a revision of the discrete symmetries (C, P, T, CP, and CPT), within an approach that treats 2-component Weyl spinors as the fundamental building blocks. Then, we discuss some salient aspects of the discrete symmetries within quantum field theory (QFT). Besides the generic discussion, we also consider aspects arising within specific renormalizable theories (such as QED and YM). As a new application of the chiral formulation, we discuss the discrete s...

I show how the isomorphism between the Lie groups of types $B_2$ and $C_2$ leads to a faithful action of the Clifford algebra $\mathcal C\ell(3,2)$ on the phase space of 2-dimensional dynamics, and hence to a mapping from Dirac spinors modulo scalars into this same phase space. Extending to the phase space of 3-dimensional dynamics allows one to embed all the gauge groups of the Standard Model as well, and hence unify the electro-weak and strong forces into a single algebraic...

A geometric approach to the standard model in terms of the Clifford algebra $% C\ell_{7}$ is advanced. The gauge symmetries and charge assignments of the fundamental fermions are seen to arise from a simple geometric model involving extra space-like dimensions. The bare coupling constants are found to obey $g_{s}/g=1$ and $g^{\prime}/g=\sqrt{3/5}$, consistent with SU(5) grand unification but without invoking the notion of master groups. In constructing the Lagrangian density ...