Unified Split Octonion Formulation of Dyons

Demonstrating the split octonion formalism for unified fields of dyons (electromagnetic fields) and gravito-dyons (gravito-Heavisidian fields of linear gravity), relevant field equations are derived in compact, simpler and manifestly covariant forms. It has been shown that this unified model reproduces the dynamics of structure of fields associated with individual charges (masses) in the absence of others.     

Sedenionic Formulation for Generalized Fields of Dyons

In this paper, the conic sedenionic formulation is presented for the unification of generalized field equations of dyons (electromagnetic theory) and gravito-dyons (linear gravity). Sedenionic definitions of generalized potential and current equations for dyons are discussed in consistent manner. The sixteen dimensional field equations related to linear gravity and electromagnetism have been written as one sedenionic equation in elegant and compact form.     

Generalized Gravi—Electromagnetism

A self consistant and manifestly covariant theory for the dynamics of four charges (masses) (namely electric, magnetic, gravitational, Heavisidian) has been developed in simple, compact and consistent manner. Starting with an invariant Lagrangian density and its quaternionic representation, we have obtained the consistent field equation for the dynamics of four charges. It has been shown that the present reformulation reproduces the dynamics of individual charges (masses) in the absence of other charge (masses) as well as the generalized theory of dyons (gravito-dyons) in the absence gravito-dyons (dyons).     

Generalized Hyperbolic Octonion Formulation for the Fields of Massive Dyons and Gravito-Dyons

The close analogy between electromagnetic theory and linear gravity is discussed by the hyperbolic (split) octonion formalism. Using the similarities between the relevant field equations of massive dyons in electromagnetic theory and gravito-dyons in linear gravity, a new mathematical model is proposed to formulate these fields in a compact and simple form. The generalized wave equation including both massive dyon and monopole terms is derived. Similarly, the most generalized form of hyperbolic octonionic Klein–Gordon equation is obtained for the hypothetical particle carrying simultaneously both electromagnetic and gravitational charges (masses).     

Generalization of Schwinger-Zwanziger Dyon to?Quaternion

Postulating the existence of magnetic monopole in electromagnetism and Heavisidian monopoles in gravitational interactions, a unified theory of gravi-electromagnetism has been developed on generalizing the Schwinger-Zwanziger formulation of dyon to quaternion in simple and consistent manner. Starting with the four Lorentz like forces on different charges, we have generalized the Schwinger-Zwanziger quantization parameters in order to obtain the angular momentum for unified fields of dyons and gravito-dyons (i.e. Gravi-electromagnetism). Taking the unified charge as quaternion, we have reformulated manifestly covariant and consistent theory for the dynamics of four charges namely electric, magnetic, gravitational and Heavisidian associated with gravi electromagnetism.     

Generalized Split-Octonion Electrodynamics

Starting with the usual definitions of octonions and split octonions in terms of Zorn vector matrix realization, we have made an attempt to write the consistent form of generalized Maxwell’s equations in presence of electric and magnetic charges (dyons). We have thus written the generalized potential, generalized field, and generalized current of dyons in terms of split octonions and accordingly the split octonion forms of generalized Dirac Maxwell’s equations are obtained in compact and consistent manner. This theory reproduces the dynamic of electric (magnetic) in the absence of magnetic (electric) charges.     

Quaternion-Octonion Analyticity for Abelian and Non-Abelian Gauge Theories of Dyons

Einstein-Schrödinger (ES) non-symmetric theory has been extended to accommodate the Abelian and non-Abelian gauge theories of dyons in terms of the quaternion-octonion metric realization. Corresponding covariant derivatives for complex, quaternion and octonion spaces in internal gauge groups are shown to describe the consistent field equations and generalized Dirac equation of dyons. It is also shown that quaternion and octonion representations extend the so-called unified theory of gravitation and electromagnetism to the Yang-Mill’s fields leading to two SU(2) gauge theories of internal spaces due to the presence of electric and magnetic charges on dyons.     

Gauge Formulation for Two Potential Theory of Dyons

Dual electrodynamics and corresponding Maxwell’s equations (in the presence of monopole only) are revisited from the symmetry of duality and gauge invariance. Accordingly, the manifestly covariant, dual symmetric and gauge invariant two potential theory of generalized electromagnetic fields of dyons has been developed consistently from U(1)×U(1) gauge symmetry. Corresponding field equations and equation of motion are derived from Lagrangian formulation adopted for U(1)×U(1) gauge symmetry for the justification of two four potentials of dyons.     

Magnetic symmetry and dyons

A self-consistent theory of dyons in Abelian and non-Abelian limits has been formulated in terms of an extra magnetic symmetry and topological magnetic charge. It has been shown that the restricted gauge potential describes the fields of dyons in terms of two regular (time-like) potentials only when recourse is made to the duality of topological (magnetic) and isocolour (electric) charges. Choosing a suitable Lagrangian density for the system of dyons in non-Abelian gauge theory, the field equations, energy-momentum tensor, Hamiltonian and momentum densities have also been derived and the conservation of the four-linear momentum and the total angular momentum has been demonstrated.     

Octonion massive electrodynamics

In this paper, we have made an attempt to reformulate the generalized field equation and various quantum equations of massive dyons in terms of octonion eight dimensional space as the combination of two (external and internal) four dimensional spaces. The octonion forms of generalized potential and current equations of massive dyons are discussed in consistent manner. It has been shown that due to the non associativity of octonion variables it is necessary to impose certain constraints to describe generalized octonion massive electrodynamics in manifestly covariant and consistent manner.     

Discrete Symmetries and Generalized Fields of Dyons

We have studied the different symmetric properties of the generalized Maxwell’s–Dirac equation along with their quantum properties. Applying the parity (℘), time reversal ( T\mathcal{T} ), charge conjugation (C\mathcal{C}) and their combined effect like parity time reversal (PT\mathcal{PT}), charge conjugation and parity (CP\mathcal{CP}) and CPT\mathcal{CP}T transformations to various equations of generalized fields of dyons, it is shown that the corresponding dynamical quantities and equations of dyons are invariant under these discrete symmetries.     

Supersymmetrization of Quaternion Dirac Equation for Generalized Fields of Dyons

The quaternion Dirac equation in presence of generalized electromagnetic field has been discussed in terms of two gauge potentials of dyons. Accordingly, the supersymmetry has been established consistently and thereafter the one, two and component Dirac Spinors of generalized quaternion Dirac equation of dyons for various energy and spin values are obtained for different cases in order to understand the duality invariance between the electric and magnetic constituents of dyons.     

Einstein field equations in spinor formalism: a clifford-algebra approach

Einstein equations describing gravitational fields are expressed as a compact exterior system of spinor-valued forms. It is pointed out that Clifford algebra-valued differential forms provide a natural formalism for Einstein gravitational fields. Finally we try to express these equations in a real spacetime.     

Generalized Octonion Electrodynamics

We have made an attempt to reformulate the generalized field equation of dyons in terms of octonion variables. Octonion forms of generalized potential and current equations are discussed in consistent manner. It has been shown that due to the non associativity of octonion variables it is necessary to impose certain constraints to describe generalized octonion electrodynamics in manifestly covariant and consistent manner.     

Spin-Statistics Transmutation in Quantum Field Theory

Spin-statistics transmutation is the phenomenon occurring when a “dressing” transformation introduced for physical reasons (e.g. gauge invariance) modifies the “bare” spin and statistics of particles or fields. Historically, it first appeared in Quantum Mechanics and in semiclassical approximation to Quantum Field Theory. After a brief historical introduction, we sketch how to describe such phenomenon in Quantum Field Theory beyond the semiclassical approximation, using a path-integral formulation of euclidean correlation functions, exemplifying with anyons, dyons and skyrmions.     

Semi-classical spectrum of the homogeneous sine-Gordon theories

The semi-classical spectrum of the homogeneous sine-Gordon theories associated with an arbitrary compact simple Lie group G is obtained and shown to be entirely given by solitons. These theories describe quantum integrable massive perturbations of Gepner's G-parafermions whose classical equations-of-motion are non-abelian affine Toda equations. One-soliton solutions are constructed by embeddings of the SU(2) complex sine-Gordon soliton in the regular SU(2) subgroups of G. The resulting spectrum exhibits both stable and unstable particles, which is a peculiar feature shared with the spectrum of monopoles and dyons in N = 2 and N = 4 supersymmetric gauge theories.     

Gauge-invariant charged,monopole and dyon fields in gauge theories

We propose explicit recipes to construct the Euclidean Green functions of gauge-invariant charged, monopole and dyon fields in four-dimensional gauge theories whose phase diagram contains phases with deconfined electric and/or magnetic charges. In theories with only either abelian electric or magnetic charges, our construction is an Euclidean version of Dirac's original proposal, the magnetic dual of his proposal, respectively. Rigorous mathematical control is achieved for a class of abelian lattice theories. In theories where electric and magnetic charges coexist, our construction of Green functions of electrically or magnetically charged fields involves taking an average over Mandelstam strings or the dual magnetic flux tubes, in accordance with Dirac's flux quantization condition. We apply our construction to 't Hooft-Polyakov monopoles and Julia-Zee dyons. Connections between our construction and the semiclassical approach are discussed.