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In this paper, we present a method for calculation of spin groups elements for known pseudo-orthogonal group elements with respect to the corresponding two-sheeted coverings. We present our results using the Clifford algebra formalism in the case of arbitrary dimension and signature, and then explicitly using matrices, quaternions, and split-quaternions in the cases of all possible signatures $(p,q)$ of space up to dimension $n=p+q=3$. The different formalisms are convenient for different possible applications in physics, engineering, and computer science.

A theory in which four-dimensional space–time is generalized to a larger space, namely a 16-dimensional Clifford space (C-space) is investigated. Curved Clifford space can provide a realization of Kaluza–Klein. A covariant Dirac equation in curved C-space is explored. The generalized Dirac field is assumed to be a polyvector-valued object (a Clifford number) which can be written as a superposition of four independent spinors, each spanning a different left ideal of Clifford algebra. The general transformations of a polyvector can act from the left and/or from the right, and form a large gauge group which may contain the group U(1) × SU(2) × SU(3) of the standard model. The generalized spin connection in C-space has the properties of Yang–Mills gauge fields. It contains the ordinary spin connection related to gravity (with torsion), and extra parts describing additional interactions, including those described by the antisymmetric Kalb–Ramond fields.

We propose a new path integral for QED in the presence of magnetic monopoles on the formalism of Geometric Algebra of Hestenes–Haddamard written in terms of Dirac matrices.

In this work, I present a practical way to obtain the vibration-rotation kinetic energy operator for an N-atomic molecule in an arbitrary body-frame . The body-frame need not be orthogonal or rigid. In practice, I derive the explicit form of the measuring vectors associated with the body-frame components of the internal angular momentum. Their inner products with the vector derivatives of the shape coordinates give the "Coriolis" part of the metric tensor appearing in the Hamiltonian, and their inner products among themselves give the "rotational" part. As a simple example, the measuring vectors are explicitly derived in an oblique bond-vector body-frame. The metric tensor elements are also derived for a tetra-atomic pyramidal molecule, whose shape is parametrized in bond-angle coordinates.

Since the introduction of quaternion by Hamilton in 1843, quaternions have been used in a lot of applications. One of the most interesting qualities is that we can use quaternions to carry out rotations and operate on other quaternions; this characteristic of the quaternions inspired us to investigate how the quantum states and quantum operator work in the field of quaternions and how we can use it to construct a quantum neural network. This new type of quantum neural network (QNN) is developed in the quaternion algebra framework that is isomorphic to the rotor algebra $G_3^{+}$ of the geometric algebra and is based on the so-called qubit neuron model. The quaternion quantum neural network (QQNN) is tested and shows robust performance.

Object tracking is an important process for many applications in computer vision. This process must be implemented in a discrete manner because the images are available only at certain periods. A discrete integral sliding mode algorithm is proposed to control a stereo vision system and perform the aforementioned second task. The kinematic model of the structure is obtained using geometric algebra. The performance of the controller is compared with proportional-integral-derivative control via simulation. The implementation for a pan tilt unit is presented in real time. The algorithm presents a good and robust performance.

Inspired by Laughlin’s theory of the fractional quantum Hall effect, we explore the topological nature of the quark–gluon plasma (QGP) and the nucleons in the context of the Clifford algebra. In our model, each quark is transformed into a composite particle via the simultaneous attachment of a spin monopole and an isospin monopole. This is induced by a novel kind of meson endowed with both spin and isospin degrees of freedom. The interactions in the strongly coupled quark–gluon system are governed by the topological winding number of the monopoles, which is an odd integer to ensure that the overall wave function is antisymmetric. The states of the QGP and the nucleons are thus uniquely determined by the combination of the monopole winding number $m$ and the total quark number $N$. The radius squared of the QGP droplet is expected to be proportional to $mN$. We anticipate the observation of such proportionality in the heavy ion collision experiments.

Is there more to Dirac’s gamma matrices than meets the eye? It turns out that gamma zero can be factorized into a product of three operators. This revelation facilitates the expansion of Dirac’s space-time algebra to Clifford algebra $\mathrm{\text{Cl}}(0,6)$. The resultant rich geometric structure can be leveraged to establish a combined framework of the standard model and gravity, wherein a gravi-weak interaction between the vierbein field and the extended weak gauge field is allowed. Inspired by the composite Higgs model, we examine the vierbein field as an effective description of the fermion–antifermion condensation. The compositeness of space-time manifests itself at an energy scale which is different from the Planck scale. We propose that all the regular classical Lagrangian terms are of quantum condensation origin, thus possibly addressing the cosmological constant problem provided that we exercise extreme caution in the renormalization procedure that entails multiplications of divergent integrals. The Clifford algebra approach also permits a weaker form of charge conjugation without particle–antiparticle interchange, which leads to a Majorana-type mass that conserves lepton number. Additionally, in the context of spontaneous breaking of two global $U(1)$ symmetries, we explore a three-Higgs-doublet model which could explain the fermion mass hierarchies.

The idealized Kish-Sethuraman (KS) cipher is theoretically known to offer perfect security through a classical information channel. However, realization of the protocol is hitherto an open problem, as the required mathematical operators have not been identified in the previous literature. A mechanical analogy of this protocol can be seen as sending a message in a box using two padlocks; one locked by the Sender and the other locked by the Receiver, so that theoretically the message remains secure at all times. We seek a mathematical representation of this process, considering that it would be very unusual if there was a physical process with no mathematical description. We select Clifford's geometric algebra for this task as it is a natural formalism to handle rotations in spaces of various dimension. The significance of finding a mathematical description that describes the protocol, is that it is a possible step toward a physical realization having benefits in increased security with reduced complexity.

For fixed n > 0, the space of finite graphs on n vertices is canonically associated with an abelian, nilpotent-generated subalgebra of the Clifford algebra which is canonically isomorphic to the 2n-particle fermion algebra. Using the generators of the subalgebra, an algebraic probability space of “Clifford adjacency matrices” associated with finite graphs is defined. Each Clifford adjacency matrix is a quantum random variable whose m^th moment corresponds to the number of m-cycles in the graph G. Each matrix admits a canonical “quantum decomposition” into a sum of three algebraic random variables: a = a^Δ + a^ϒ + a^Λ, where a^Δ is classical while a^ϒ and a^Λ are quantum. Moreover, within the Clifford algebra context the NP problem of cycle enumeration is reduced to matrix multiplication, requiring no more than n⁴ Clifford (geo-metric) multiplications within the algebra.