Quaternion Dirac Equation and Supersymmetry

Rawat, Seema; Negi, O. P. S.

doi:10.1007/s10773-009-0003-4

Cited by 22 publications

(20 citation statements)

References 18 publications

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“…To explain the dual nature of quaternionic unified rotational energy-momentum solutions, we start with quaternionic spinor (Ψ) with scalar and vector fields as [6],…”

Section: Quaternionic Rotational Dirac (Qrd) Equationmentioning

confidence: 99%

A comparative study of quaternionic rotational Dirac equation and its interpretation

Chanyal¹,

Sandhya²

2020

Int. J. Geom. Methods Mod. Phys.

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In this study, we develop the generalized Dirac like four-momentum equation for rotating spin-half particles in four-dimensional quaternionic algebra. The generalized quaternionic Dirac equation consists the rotational energy and angular momentum of particle and anti-particle. Accordingly, we also discuss the four vector form of quaternionic relativistic mass, moment of inertia and rotational energy-momentum in Euclidean space-time. The quaternionic four angular momentum (i.e. the rotational analogy of four linear momentum) predicts the dual energy (rest mass energy and pure rotational energy) and dual momentum (linear like momentum and pure rotational momentum). Further, the solutions of quaternionic rotational Dirac energy-momentum are obtained by using one, two and fourcomponent of quaternionic spinor. We also demonstrate the solutions of quaternionic plane wave equation which gives the rotational frequency and wave propagation vector of Dirac particles and anti-particles in terms of quaternionic form. PACS: 03.65.−w, 03.65.Fd, 02.10.Ud relativistic velocity. To combine the special theory of relativity with quantum mechanics, there has been developed the relativistic quantum mechanics. In the same way, Klein-Gordon and Dirac independently investigated the relativistic wave equations by combining special relativity with quantum mechanics. These relativistic wave equations describe the various phenomenons that occur in high energy physics and are invariant under Lorentz transformations. Dirac [1] discussed a relativistic quantum wave equation by using Hamiltonian operator to overcome the difficulties arising in Klein-Gordon equation. As we know that the conservation of energy and angular momentum are one of the mandatory conservation laws to check the validity of Dirac particles. Keeping in mind the conservation laws of energy and angular momentum of a rotating particle, in this paper, we proposed a quaternionic Dirac equation that consists not only the energy representation but also shows the angular momentum representation of spin-1/2 particles. The quaternion number [2] is basically an extension of complex numbers. Although, there are four types of norm-division algebras, i.e. real, complex, quaternion and octonion algebra. The division algebra is defined as an algebra in which all non-zero elements have their inverse under multiplication [3]. Nowadays, the quaternionic algebra is a popular algebra to study the various theories in modern theoretical physics. The quaternionic algebra is associative and commutative under addition but not commutative under multiplication. Thus, this algebra form a group under multiplication but not an Abelian group, is also called the division ring. Many researchers have attempted the formulation of usual Dirac equation for free particles in terms of quaternionic algebra. Firstly, Rotelli [4] developed the Dirac equation in quaternionic four fields. The another version of quaternionic Dirac equation has been studied by Rawat et. al [5] with the description of quaternionic spinor...

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“…To explain the dual nature of quaternionic unified rotational energy-momentum solutions, we start with quaternionic spinor (Ψ) with scalar and vector fields as [6],…”

Section: Quaternionic Rotational Dirac (Qrd) Equationmentioning

confidence: 99%

A comparative study of quaternionic rotational Dirac equation and its interpretation

Chanyal¹,

Sandhya²

2020

Int. J. Geom. Methods Mod. Phys.

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“…where the spinor field ψ , is the described [16,17] as a quaternion ψ = ψ 0 + e k ψ k (∀k = 1, 2, 3)…”

Section: Quaternionic Lagrangian Formulationmentioning

confidence: 99%

“…Equations (15) and (16) are respectively recalled as the Dirac equations [16] for the spinors ψ a and ψ b . Similarly if we take the variations with respect to ψ a and ψ b we get…”

Section: Quaternionic Dirac Equationmentioning

confidence: 99%

Quaternion Octonion Reformulation of Quantum Chromodynamics

Pushpa

Bisht

et al. 2010

Int J Theor Phys

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We have made an attempt to develop the quaternionic formulation of Yang - Mill's field equations and octonion reformulation of quantum chromo dynamics (QCD). Starting with the Lagrangian density, we have discussed the field equations of SU(2) and SU(3) gauge fields for both cases of global and local gauge symmetries. It has been shown that the three quaternion units explain the structure of Yang- Mill's field while the seven octonion units provide the consistent structure of SU(3)_{C} gauge symmetry of quantum chromo dynamics

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“…These algebras offer important and alternative notations of physical systems for representing them in a more compact, effective, and elegant way. Particularly, there are many studies on electromagnetism, linear gravity, and quantum mechanics that were performed in terms of hypercomplex values; see [12]- [48].…”

Section: Introductionmentioning

confidence: 99%

Octonion form of duality-invariant field equations for dyons

Kansu¹,

Tanışlı²,

Demir³

2020

Turk J Phys

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The hypothetical particles dyons, which carry both electric and magnetic charges simultaneously, are widely discussed in application to electromagnetic theory and magnetohydrodynamics. Particularly, the duality-invariant field equations were suggested with suitable definitions of the dyon's electromagnetic characteristics. In this study, we propose an alternative formulation of the duality-invariant field equations for dyons based on octonion algebra. Octonions have been used to express the equations for potentials, field strengths, and sources in a more compact and consistent manner.Additionally, the octonionic form of the energy conservation law for dyons has been derived.

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Quaternion Dirac Equation and Supersymmetry

Cited by 22 publications

References 18 publications

A comparative study of quaternionic rotational Dirac equation and its interpretation

A comparative study of quaternionic rotational Dirac equation and its interpretation

Quaternion Octonion Reformulation of Quantum Chromodynamics

Octonion form of duality-invariant field equations for dyons

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