S13 ISSN 1063-7796, Physics of Particles and Nuclei, 2006, Vol. 37, Suppl. 1, pp. S13–S44. © Pleiades Publishing, Inc., 2006. 1. INTRODUCTION Nonlinear phenomena have been one of the most popular topics during last years. Nevertheless, it must be admitted that nonlinear classical fields have not received general consideration. This is probably due to the mathematical difficulties which arise because of the nonrenormalizability of the Fermi and other nonlinear couplings [1]. Nonlinear self-coupling of the spinor fields may arise as a consequence of the geometrical structure of space-time and, more precisely, because of the existence of torsion. Ivanenko [2, 3] and Rodichev [4] showed that a relativistic theory imposes in some cases a fourth-order self-coupling. In 1950, Weyl [5] proved that if the affine and the metric properties of space-time are taken as independent, the spinor field obeys either a linear equation in a space with torsion or a nonlinear one in a Reimannian space. As the self- action is of the spin–spin type, it allows the assignment of a dynamical role to the spin and offers a clue about the origin of the nonlinearities. This question was fur- ther clarified in some important papers by Utiyama, Kibble, and Sciama [6–8]. In the simplest scheme, the self-action is of the pseudovector type, but it can be shown that one can also get a scalar coupling [9]. An excellent review of the problem may be found in [10]. Nonlinear quantum Dirac fields were used by Heisen- berg [11, 12] in his ambitious unified theory of elemen- tary particles. They have been the object of renewed interest since the publication of the widely known paper by Gross and Neveu [13]. A nonlinear spinor field, suggested by the symmetric coupling between nucleons, muons, and leptons, was investigated by Finkelstein et al. [14] in the classical approximation. Although the existence of a spin-1/2 fermion is both theoretically and experimentally undisputed, these are described by quantum spinor fields. Possible justifica- tions for the existence of classical spinors were addressed in [15]. The quantum field theory in curved space-time has been a matter of great interest in recent years because of its applications to cosmology and astrophysics. The evidence for the existence of strong gravitational fields in our Universe led to the study of the quantum effects of material fields in external classical gravitational field. In [16, 17], Parker considers spin-0 and spin-1/2 fields respectively, in a FRW space-time that is not quantized. The equations governing the spin-0 field are the covariant generalization of the special-relativistic free-field equations, whereas the spin-1/2 field satisfies the fully covariant generalization of the Dirac equation. In [16], the author showed that massless particles of arbitrary nonzero spin, such as photons or gravitons, Spinor Fields in Bianchi Type-I Universe* B. Saha Laboratory of Information Technologies, Joint Institute for Nuclear Research, 141980 Dubna, Moscow oblast, Russia; e-mail: saha@thsun1.jinr.ru; bijan@jinr.ru URL: http://thsun1.jinr.ru/~saha Abstract—A system of minimally coupled nonlinear spinor and scalar fields within the scope of a Bianchi type-I (BI) cosmological model in the presence of a perfect fluid and a cosmological constant (Λ term) is stud- ied, and solutions to the corresponding field equations are obtained. The problem of initial singularity and the asymptotical isotropization process of the Universe are thoroughly studied. The effect of the Λ term on the char- acter of evolution is analyzed. It is shown that some special choice of spinor field nonlinearity generates a reg- ular solution, but the absence of singularity results in violating the dominant energy condition in the Hawking– Penrose theorem. It is also shown that a positive Λ, which denotes an additional gravitational force in our case, gives rise to an oscillatory or a non-periodic mode of expansion of the Universe depending on the choice of problem parameter. The regular oscillatory mode of expansion violets the dominant energy condition if the spinor field nonlinearity occurs as a result of self-action, whereas, in the case of a linear spinor field or nonlinear one that occurs due to interaction with a scalar field, the dominant condition remains unbroken. A system with time-varying gravitational (G) and cosmological (Λ) constants is also studied to some extent. The introduction of magneto-fluid in the system generates nonhomogeneity in the energy-momentum tensor and can be exactly solved only under some additional condition. Though in this case, we indeed deal with all four known fields, i.e., spinor, scalar, electromagnetic, and gravitational, the over-all picture of evolution remains unchanged. PACS numbers: 04.20.Ha, 03.65.Pm, 04.20.Jb, 98.80.Cq DOI: 10.1134/S1063779606070021 * The text was submitted by the author in English.