A simple link between βμ matrices of the Duffin-Kemmer-Petiau theory and ρμ matrices of Tzou representations is constructed. The link consists of a constant unitary transformation of the βμ matrices ...and a projection onto a lower-dimensional subspace.
We study the 7 × 7 Hagen-Hurley equations describing spin 1 particles. We split these equations, in the interacting case, into two Dirac equations with nonstandard solutions. It is argued that these ...solutions describe decay of a virtual W boson in beta decay.
Non-linear dynamics of a bouncing ball moving vertically in a gravitational field and colliding with a moving limiter is considered. The Poincaré map, describing evolution from an impact to the next ...impact, is used to analyse the original system. Sinusoidal displacement of the table, defining the standard model, is approximated in one period of the limiter׳s motion by a cubic spline, thus making analytical computations possible. Analytical and numerical results, based on Implicit Function Theorem, obtained for this simplified model, are used to elucidate dynamics of the standard model of the bouncing ball. Finally, the same techniques are applied to investigate dynamics of the standard model.
•Two models of a bouncing ball, colliding with a moving table, are studied.•Standard sinusoidal displacement of the table (model I) is approximated by a cubic spline (model II).•Analytical and numerical results, based on Implicit Function Theorem, are obtained for both models.
We study internal structure of the Duffin-Kemmer-Petiau equations for spin 0 and spin 1 mesons. We show that in the noninteracting case full covariant solutions of the s = 0 and s = 1 DKP equations ...are generalized solutions of the Dirac equation.
Analytical Methods for High Energy Physics Zarrinkamar, Saber; Okniński, Andrzej; Jia, Chun-Sheng
Advances in High Energy Physics,
01/2019, Letnik:
2019
Journal Article
Recenzirano
Odprti dostop
...new analytical approaches, such as Lie groups, path integrals, integrable and superintegrable systems, renormalization methods, factorization methods, Green functions, special functions, homotopy ...method, integral transforms, and approximate methods based on perturbative or variational treatments to solve differential equations of high energy physics, are becoming more useful. ...there is a significant input of new formulations of quantum mechanics and field theory used in particle physics such as new kinds of interactions, wave equations in curved spaces, fractional order wave equations, noncommutative quantum mechanics, or quantum deformed algebras which may lead to better understanding of high energy physics. In this special issue we propose a selection of papers, devoted to mathematical problems of high energy and particle physics, where analytical approaches and ideas are used as the main tool to study particle physics. N. Mohajery et al. solved the six-dimensional hyperradial Schrödinger equation describing baryons consisting of two heavy quarks and one light quark to compute the mass spectra.
Nonlinear dynamics of a bouncing ball moving in gravitational field and colliding with a moving limiter is considered. Displacement of the limiter is a quadratic function of time. Several dynamical ...modes, such as fixed points, 2-cycles, grazing and chaotic bands are studied analytically and numerically. It is shown that chaotic bands appear due to homoclinic structures created from unstable 2-cycles in a corner-type bifurcation.
In the present paper we study subsolutions of the Dirac and Duffin–Kemmer–Petiau equations in the interacting case. It is shown that the Dirac equation in longitudinal external fields can be split ...into two covariant subequations (Dirac equations with built-in projection operators). Moreover, it is demonstrated that the Duffin–Kemmer–Petiau equations in crossed fields can be split into two 3 x 3 subequations. We show that all the subequations can be obtained via minimal coupling from the same 3 x 3 subequations which are thus a supersymmetric link between fermionic and bosonicdegrees of freedom.
Dynamics of nonlinear coupled driven oscillators is investigated. Recently, we have demonstrated that the amplitude profiles – dependence of the amplitude A on frequency Ω of the driving force, ...computed by asymptotic methods in implicit form as FA,Ω=0, permit prediction of metamorphoses of dynamics which occur at singular points of the implicit curve FA,Ω=0. In the present study we strive at a global view of singular points of the amplitude profiles computing bifurcation sets, i.e. sets containing all points in the parameter space for which the amplitude profile has a singular point.
•The effective equation for coupled nonlinear Duffing equations is investigated.•Implicit equations for amplitude profiles of nonlinear resonances are studied.•The bifurcation manifold in the parameter space is computed.•Examples of metamorphoses of dynamics at singular points are presented.•The method can be applied to other dynamical systems.
Dynamics of the Duffing–van der Pol driven oscillator is investigated. Periodic steady-state solutions of the corresponding equation are computed within the Krylov-Bogoliubov-Mitropolsky approach to ...yield dependence of amplitude A on forcing frequency Ω as an implicit function, FA,Ω=0, referred to as resonance curve or amplitude profile.
In singular points of the amplitude curve the conditions ∂F∂A=0, ∂F∂Ω=0 are fulfilled, i.e. in such points neither of the functions A=fΩ, Ω=gA, continuous with continuous first derivative, exists. Near such points metamorphoses of the dynamics can occur. In the present work the bifurcation set, i.e. the set in the parameter space, such that every point in this set corresponds to a singular point of the amplitude profile, is computed.
Several examples of singular points and the corresponding metamorphoses of dynamics are presented.
•The externally forced van der Pol oscillator is investigated.•Implicit equations for amplitude profiles of nonlinear resonances are derived.•The bifurcation manifold in the parameter space is computed.•Examples of dynamics near double singular points are computed.
We study a fermion-boson transformation. Our approach is based on the 3 × 3 equations which are subequations of both the Dirac and Duffin-Kemmer-Petiau equations and thus provide a link between these ...equations. We show that solutions of the free Dirac equation can be converted to solutions of spin- 0 Duffin-Kemmer-Petiau equation and vice versa. Mechanism of this transition assumes existence of a constant spinor.
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