Unstable behaviors of classical solutions in spinor-type conformal invariant fermionic models
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, PARTICLES, FIELDS, GRAVITATION, AND ASTROPHYSICS
Unstable Behaviors of Classical Solutions in Spinor-Type Conformal Invariant Fermionic Models1 F. Aydogmus*,**,*** Department of Physics, Istanbul University, Istanbul, Turkey *e-mail: [email protected] **e-mail: [email protected] ***e-mail: [email protected] Received March 30, 2017
Abstract—It is well known that instantons are classical topological solutions existing in the context of quantum field theories that lie behind the standard model of particles. To provide a better understanding for the dynamical nature of spinor-type instanton solutions, conformal invariant pure spinor fermionic models that admit particle-like solutions for the derived classical field equations are studied in this work under cosine wave forcing. For this purpose, the effects of external periodic forcing on two systems that have different dimensions and quantum spinor numbers and have been obtained under the use of Heisenberg ansatz are investigated by constructing their Poincaré sections in phase space. As a result, bifurcations and chaos are observed depending on the excitation amplitude of the external forcing in both pure spinor fermionic models. DOI: 10.1134/S1063776117100016
1. INTRODUCTION The idea of describing bosons and fermions, having spin numbers up to 2, by means of a fermionic field had been put forward by Louis de Broglie [1]. Moreover, Dirac’s nonlinear spinor field wave equation brought out a successful interpretation of electron and anti-electron (positron) system [2]. Later, Heisenberg and Pauli developed this equation in their unified field theory [3]. After these studies, great efforts were begun among theoretical physicists in searching new nonlinear massless conformal invariant field equations without dimensional parameters that especially would have classical solutions. In this context Gursey proposed a model as a possible basis for a unitary description of elementary particles which possesses broader dynamical symmetries compared to the Dirac’s and Heisenberg et al. works [4]. This model has been the first conformally invariant 4D pure fermionic model and aroused a great interest. Later a class of its exact solutions with nonlinear (ψψ )4/3 self-coupled spinor term was found and they were shown to be instantonic in character [5, 6]. These classical solutions are similar to the solutions of pure Yang-Mills [7] equations in 4D. Another spinor field system came with the work of Thirring [8] after Gursey’s one. Thirring’s conformally invariant model describes Dirac fermions in (1 + 1) space-time dimensions with no mass; but with non-linear self-interaction term (ψψ )2. It contains 1 The article was translated by the authors.
many typical features of the quantization of relativistic quantum field theories. Classical instanton-type solutions for this 2D fermionic model were found too. It has been shown that these solutions were of the same form as the solutions of 4D model [9]. In some cases, nonlinear classical field theories provide the possibilit
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