Quaternionic Formalism Research Articles

Differential Geometry Using Quaternions

This paper establishes the basis of the quaternionic differential geometry (HDG) initiated in a previous article. The usual concepts of curves and surfaces are generalized to quaternionic constraints, as well as the curvature and torsion concepts, differential forms, directional derivatives and the structural equations. The analogy between the quaternionic and the real geometries were obtained using a matrix representation of quaternions. The results evidences the quaternionic formalism as a suitable language to differential geometry that can be useful in various directions of future investigation.

Periodic discrete Darboux transforms

We express Darboux transformations of discrete polarised curves as parallel sections of discrete connections in the quaternionic formalism. This immediately leads to the linearisation of the monodromy of the transformation. We also consider the integrable reduction to the case of discrete bicycle correspondence. Applying our method to the case of discrete circles, we obtain closed-form discrete parametrisations of all (closed) Darboux transforms and (closed) bicycle correspondences.

Quaternion equations for hydrodynamic two-fluid model of vortex plasma

We discuss the application of quaternionic space-time algebra for the generalization of self-consistent equations describing the hydrodynamic two-fluid model of vortex plasma. It is shown that quaternionic formalism allows one to write the system of hydrodynamic equations in a compact form as one quaternion equation, which can be easy generalized to the case of damping plasma in an external electromagnetic field. As an illustration, we apply the proposed equations for the description of sound waves in electron–ion and electron–positron plasmas.

Two-Component Spinorial Formalism Using Quaternions for Six-Dimensional Spacetimes

In this article we construct and discuss several aspects of the two-component spinorial formalism for six-dimensional spacetimes, in which chiral spinors are represented by objects with two quaternionic components and the spin group is identified with $SL(2;\mathbb{H})$, which is a double covering for the Lorentz group in six dimensions. We present the fundamental representations of this group and show how vectors, bivectors, and 3-vectors are represented in such spinorial formalism. We also complexify the spacetime, so that other signatures can be tackled. We argue that, in general, objects built from the tensor products of the fundamental representations of $SL(2;\mathbb{H})$ do not carry a representation of the group, due to the non-commutativity of the quaternions. The Lie algebra of the spin group is obtained and its connection with the Lie algebra of $SO(5,1)$ is presented, providing a physical interpretation for the elements of $SL(2;\mathbb{H})$. Finally, we present a bridge between this quaternionic spinorial formalism for six-dimensional spacetimes and the four-component spinorial formalism over the complex field that comes from the fact that the spin group in six-dimensional Euclidean spaces is given by $SU(4)$.

Quaternionic approach on the Dirac–Maxwell, Bernoulli and Navier–Stokes equations for dyonic fluid plasma

By applying the Hamilton’s quaternion algebra, we propose the generalized electromagnetic-fluid dynamics of dyons governed by the combination of the Dirac–Maxwell, Bernoulli and Navier–Stokes equations. The generalized quaternionic hydro-electromagnetic field of dyonic cold plasma consists of electrons and magnetic monopoles in which there exist dual-mass and dual-charge species in the presence of dyons. We construct the conservation of energy and conservation of momentum equations by equating the quaternionic scalar and vector parts for generalized hydro-electromagnetic field of dyonic cold plasma. We propose the quaternionic form of conservation of energy is related to the Bernoulli-like equation while the conservation of momentum is related to Navier–Stokes-like equation for dynamics of dyonic plasma fluid. Further, the continuity equation, i.e. the conservation of electric and magnetic charges with the dynamics of hydro-electric and hydro-magnetic flow of conducting cold plasma fluid is also analyzed. The quaternionic formalism for dyonic plasma wave emphasizes that there are two types of waves propagation, namely the Langmuir-like wave propagation due to electrons, and the ’t Hooft–Polyakov-like wave propagation due to magnetic monopoles.

Mathcal{N} = 2 Liouville SCFT in four dimensions

We construct a Liouville superconformal field theory with eight real supercharges in four dimensions. The Liouville superfield is an mathcal{N} = 2 chiral superfield with sixteen bosonic and sixteen fermionic component fields. Its lowest component is a logcorrelated complex scalar field whose real part carries a background charge. The theory is non-unitary with a continuous spectrum of scaling dimensions. We study its quantum dynamics on the supersymmetric 4-sphere and show that the classical background charge is not corrected quantum mechanically. We calculate the super-Weyl anomaly coefficients and find that c vanishes, while a is negative and depends on the background charge. We derive an integral expression for the correlation functions of superfield vertex operators in mathcal{N} = 2 superspace and analyze them in the semiclassical approximation by using a quaternionic formalism for the mathcal{N} = 2 superconformal algebra.

Quaternionic Formalism for Non-Abelian Gauge Theory of Dyons and Gravito-Dyons

Starting with the Quaternion formulation of SU (2 )× U (1)gauge theory of dyons and gravitodyons, it is shown that the formulation characterizes the abelian and non-Abelian gauge structure of dyons and gravito-dyons in terms of purely real and imaginary units of quaternion.

Quaternionic Formalism for Non-Abelian Gauge Theory of Dyons and Gravito-Dyons

Starting with the Quaternion formulation of SU (2 )× U (1)gauge theory of dyons and gravitodyons, it is shown that the formulation characterizes the abelian and non-Abelian gauge structure of dyons and gravito-dyons in terms of purely real and imaginary units of quaternion.

Quaternionic wave function

We argue that quaternions form a natural language for the description of quantum-mechanical wave functions with spin. We use the quaternionic spinor formalism which is in one-to-one correspondence with the usual spinor language. No unphysical degrees of freedom are admitted, in contrast to the majority of literature on quaternions. In this paper, we first build a Dirac Lagrangian in the quaternionic form, derive the Dirac equation and take the nonrelativistic limit to find the Schrödinger’s equation. We show that the quaternionic formalism is a natural choice to start with, while in the transition to the noninteracting nonrelativistic limit, the quaternionic description effectively reduces to the regular complex wave function language. We provide an easy-to-use grammar for switching between the ordinary spinor language and the description in terms of quaternions. As an illustration of the broader range of the formalism, we also derive the Maxwell’s equation from the quaternionic Lagrangian of Quantum Electrodynamics. In order to derive the equations of motion, we develop the variational calculus appropriate for this formalism.

Unified quaternionic description of charge and spin transport and intrinsic nonlinearity of spin currents

We present a unified theory of charge and spin transport using quaternionic formalism. It is shown that both charge and spin currents can be combined together to form a quaternionic current. The scalar and vector part of quaternionic currents correspond to charge and spin currents, respectively. We formulate a unitarity condition on the scattering matrix for quaternionic current conservation. It is shown that in the presence of spin flip interactions, a weaker quaternionic unitarity condition implying charge flux conservation but spin flux nonconservation is valid. Using this unified theory, we find that spin currents are intrinsically nonlinear. Its implication for recent experimental observation of spin generation far away from the boundaries are discussed.

Scattering Study of Fermions due to Double Dirac Delta Potential in Quaternionic Relativistic Quantum Mechanics

We have studied the scattering problem of relativistic fermions from a quaternionic double Dirac delta potential. We have used Dirac equation in the presence of the scalar and vector potentials in the quaternionic formalism of relativistic quantum mechanics to study the problem. The wave functions of different regions have been derived. Then, using the reflection coefficient, transmission coefficient, and the continuity equation, the scattering problem has been investigated in detail. It has been shown that we have faced some fluctuations in the reflection and transmission coefficients.

Quaternionic Quantum Dynamics on Complex Hilbert Spaces

We consider a quaternionic quantum formalism for the description of quantum states and quantum dynamics. We prove that generalized quantum measurements on physical systems in quaternionic quantum theory can be simulated by usual quantum measurements with positive operator valued measures on complex Hilbert spaces. Furthermore, we prove that quaternionic quantum channels can be simulated by completely positive trace preserving maps on complex matrices. These novel results map all quaternionic quantum processes to algorithms in usual quantum information theory.

Quaternionic Variational Formalism for General Relativity in Riemann and Riemann-Cartan Space-Times

It is shown that there exists a 2-dimensional matrix representation of complex quaternions over real quaternions, which allows to define Pauli matrix in 4 dimensions over the quaternionic field and leads to the quaternionic spinor group previously proposed. It is also attempted to apply complex quaternions to general relativity at the level of the variational formalism. Linear gravitational Lagrangian in Riemann-Cartan space-time U4 is derived using quaternion caluculus; namely scalar curvature in U4 is put into a quaternionic form. Consequently, Einstein-Hilbert Lagrangian in Riemann space R4 is also defined over quaternions, as first shown by Sachs. The matter fields coupled to gravity are assumed to be the scalar and the Dirac fields. The quaternionic variational formalism corresponds to the first-order formalism but with a limited pattern of allowed fields such that the quaternionic fields carry only coordinate tensor indices but no local Lorentz indices which are contracted with that possessed by the basis of complex quaternions. In particular, both the quaternionic vierbein field and Lorentz gauge field (corresponding to the spin connection) are regarded as coordinate vectors which are independently varied, obtaining Einstein and Cartan equations, respectively. It is incidentally shown that the consistent condition of Einstein equation in U4 is proved via the variational formalism and the anti-symmetric part of Einstein equation together with Cartan equation in U4 leads to an identity which expresses the anti-symmetric part of the enegy-momentum tensor by means of the covariant divergence of the spin angular momentum tensor, both of Dirac field. We also present pedagogical proofs of Bianchi and Bach-Lanczos identities in U4 using the quaternionic formalism.

Conjugacy invariants of [formula omitted

In this paper, we use the quaternionic formalism of Möbius transformations on R 4 to derive conjugacy invariants on Sl(2, H) (and hence on PSl(2, H) ). We then use these invariants to distinguish between the various conjugacy classes of PSl(2, H) .

Quaternionic formulation for electromagnetic-field equations

The quaternionic formalism for subluminal field equations (Maxwell’s equations) and its interrelationship with complex superluminal Lorentz transformations have been given and it has been shown that the quaternionic forms of relativistic equations describe tachyons.

About the Solutions of Dirac's Equations in the Presence of New Magnetic Fields

Dirac's equations are completely solved for four entirely new configurations of the external magnetic field. The use of the quaternionic formalism simplifies the problem of separation of variables. For three of those new magnetic fields the radial equations just obtained are not classical. Section II studies them and gives their solutions in term of usual transcendental functions. When energy is quantized, energy levels are given.

The exact motion of a charged particle in the magnetic field [formula omitted

The relativistic equations of motion are completely solved for a special configuration of the external magnetic field, a case which seems to have never been considered before. It is also shown how the quaternionic formalism can replace the traditional Lagrange formalism in that kind of problem. The characteristics of the movement are briefly discussed in all possible cases.