Specific properties and spatial symmetry group of nonassociative spinor field

On the basis of nonassociative spinor field theory, the specific properties of a nonassociative spinor field are investigated. A new quantum number is introduced: the associator, which is a measure of the nonassociativeness of the field. To calculate the associator and spin in nonassociative algebra, open and closed products are introduced. It is shown that the spin consists of two components: the first half (calculated by the open-product rule) is similar to ordinary spin, while the second half (calculated by the closed-product rule) is attributed to the associator, i.e., is related to shear in the auxiliary isotopic space. The associator basis is expanded to a complete octonion basis, and the Poincaré group of four-dimensional space is expanded to a Poincaré group of eight-dimensional space. It is shown that, from these generators, in the particle rest system, the nonzero independent eigenvalues are: one, the sign of the particle energy, one of the spin components, one of the associator spatial components, and c₇. Tbilisi Medical Institute. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 101–109, October, 1998.     

P and CP violation mechanisms in nonassociative field theory

The octonion spinor field is studied. The nonassociativeness of the algebra makes it possible to introduce a new quantum number — the associator as a partner of spin — as a measure of nonassociativeness and as a generator of the Lorentz group. This makes it possible to classify a particle by spin and by the associator. In this formalism the eigenvalues of the C, P, and T operators can be determined and the superselection rules which reveal the mechanism of P and CP violation can be obtained. In particular, when the associator changes by one we have P violation and when it changes by two we have CP violation. This is confirmed by analysis of the diagrams describing the P- and CP-odd particle decays.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 92–97, October, 1990.     

CP-violation in nonassociative field theory

The octonion spinor field is studied. The nonassociativeness of the algebra makes it possible to introduce a new quantum number — the associator (partner of the spin) — as a measure of the nonassociativeness and a generator of the Lorentz group. The spin-associator interaction is introduced. This interaction could lie at the basis of the weak interaction of particles. This is proved by the connection found between the associator and the Kobayashi-Maskawa phase factor.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 39–44, October, 1990.     

Fundamentals of nonassociative classical field theory

A nonassociative classical field theory is constructed. Octonion algebra is studied. The octonion is represented as the sum of a quaternion and an associator. The octonion algebra is expanded and Lorentz group generators are specified in terms of octonion bases in one of the subalgebras. Lorentz vectors and spinors are constructed in the nonassociative algebra. The representation of the Lorentz group in terms of spin and the associator is obtained.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 22–27, November, 1986.     

Symmetries of a nonassociative octonion field

Nonassociativity is studied in the presence of a symmetry group. The parameters of the Lorentz group as the group of motions of the 4-dimensional space-time are shown to remain associative, Nonassociativity appears on the level of representations of the Lorentz group. Infinitesimally, irreducible representations of the Lorentz group can be used in describing nonassociative spinors. Two types of solutions of the octonion spinor equation are studied. It is shown that these solutions can be transformed into one another only if the associator is broken according to ¦A¦=1. In addition, a new mechanism for generating electromagnetic moments of particles is discussed.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 102–106, November, 1991.     

Light interference in octonion formalism

Considering light interference in an octonion formalism, it is shown that the interference equations in nonassociative theory obtained in [1] have an octonion solution. The associator — a new characteristic of the interference field — is a qualitative measure of the memory of the medium. Tbilisi Medical University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 54–60, August, 1996.     

Static Plane-Symmetric Nonlinear Spinor and Scalar Fields in GR

We consider a system of minimally coupled nonlinear spinor and scalar fields within the scope of a plane-symmetric gravitational field. The gravitational field plays crucial role in the formation of soliton-like solutions, i.e., solutions with limited total energy, spin, and charge. The change of the sign of the scalar field energy density of the system in question realizes physically if and only if the scalar charge does not exceed some critical value. In case of spinor field no such restriction on its parameter occurs. The choice of spinor field nonlinearity leads to the elimination of scalar field contribution to the metric functions, but leaves its contribution to the total energy unaltered. The spinor field is more sensitive to the gravitational field than the scalar field.     

A conjecture on the spinor functional determinant

We conjecture that the normalized Euclidean spinor functional determinant in an arbitrary external Yang-Mills potential be bounded above by 1. The opposite inequality has been shown to be true for the scalar determinant reflecting a diamagnetic effect of the Yang-Mills potential in that case. For the spinor situation this conjecture therefore requires that the influence of the spin be sufficiently strong to induce a net paramagnetic effect. We present arguments in favour of our conjecture including calculations to order h² and to order e² as well as the case of constant (Euclidean) electromagnetic field.     

Dynamics of spinor condensates and stochastic approach

The dynamics of the collective spin for Bose-Einstein condensates with nonlinear interactions, is studied within the framework of the two-component spinor. We discuss the spin resonance when the system is submitted to a periodically-modulated magnetic field at the zero temperature. In this case, the nonlinearity parameter controls the critical change between a localized and a homogeneous spin state. When the temperature is finite – or a random magnetic field is considered – the movement of the collective spin is governed by the Landau-Lifshitz-Gilbert equation, from which the complete Fokker-Planck equation is derived. This equation is the essential tool to describe the time-evolution of the probability distribution function for the collective spin. The functional integral approach is used to solve analytically examples of BEC spin behavior in a static magnetic field at finite temperature. We show how such a method can lead effectively to the complete solution of the Fokker-Planck equation for this kind of problems.     

Complex quantum gases: spinor Bose–Einstein condensates of trapped atomic vapors

We discuss the energy eigenstates, ground and spin mixing dynamics of a spin-1 spinor Bose–Einstein condensate for a dilute atomic vapor confined in an optical trap. Our results go beyond the mean field picture and are developed within a fully quantized framework.     

Zero-field and Larmor spinor precessions in a neutron polarimeter experiment

We present a neutron polarimetric experiment where two kinds of spinor precessions are observed: one is induced by different total energy of neutrons (zero-field precession) and the other is induced by a stationary guide field (Larmor precession). A characteristic of the former is the dependence of the energy-difference, which is in practice tuned by the frequency of the interacting oscillating magnetic field ωR. In contrast the latter completely depends on the strength of the guide field, namely Larmor frequency ωL. Our neutron-polarimetric experiment exhibits individual tuning as well as specific properties of each spinor precession, which assures the use of both spin precessions for multi-entangled spinor manipulation.     

Spin current in a spinor Bose-Einstein condensate induced by a gradient magnetic field

We develop a research of spin currents in a ²³Na spinor Bose-Einstein condensate (BEC) by applying a magnetic field gradient. The spin current is successfully induced by the spin-dependent force arising from the magnetic field gradient. The dynamics of the spin components under the magnetic force is investigated. The study is promising to be extended to produce a longer spin-coherence and to enhance the sensitivity of the spin-mixing interferometry in a spinor BEC.     

Pseudospin symmetry for modified Rosen-Morse potential including a Pekeris-type approximation to the pseudo-centrifugal term

By a novel algebraic method we study the approximate solution to the Dirac equation with scalar and vector second P?schl-Teller potential carrying spin symmetry. The transcendental energy equation and spinor wave functions with arbitrary spin-orbit coupling quantum number k are presented. It is found that there exist only positive-energy bound states in the case of spin symmetry. Also, the energy eigenvalue approaches a constant when the potential parameter a \alpha goes to zero. The equally scalar and vector case is studied briefly.     

Space-time evolution induced by spinor fields with canonical and non-canonical kinetic terms

We study spinor field theories as an origin to induce space-time evolution. Self-interacting spinor fields with canonical and non-canonical kinetic terms are considered in a Friedman–Robertson–Walker universe. The deceleration parameter is calculated by solving the equation of motion and the Friedman equation, simultaneously. It is shown that the spinor fields can accelerate and decelerate the universe expansion. To construct realistic models we discuss the contributions from the dynamical symmetry breaking.     

Spinor fields in Bianchi type-I universe

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 studied, 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 character of evolution is analyzed. It is shown that some special choice of spinor field nonlinearity generates a regular 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.     

The spin symmetry for deformed generalized Pöschl-Teller potential

In the case of spin symmetry we solve the Dirac equation with scalar and vector deformed generalized Pöschl-Teller (DGPT) potential and obtain exact energy equation and spinor wave functions for s-wave bound states. We find that there are only positive energy states for bound states in the case of spin symmetry based on the strong regularity restriction condition λ<−η for the wave functions. The energy eigenvalue approaches a constant when the potential parameter α goes to zero. Two special cases such as generalized PT potential and standard PT potential are also briefly discussed.     

Geometric significance of the spinor Lie derivative. I

In a previous article, the writer explored the geometric foundation of the generally covariant spinor calculus. This geometric reasoning can be extended quite naturally to include the Lie covariant differentiation of spinors. The formulas for the Lie covariant derivatives of spinors, adjoint spinors, and operators in spin space are deduced, and it is observed that the Lie covariant derivative of an operator in spin space must vanish when taken with respect to a Killing vector. The commutator of two Lie covariant derivatives is calculated; it is noted that the result is consistent with the geometric interpretation of the Jacobi identity for vectors. Lie current conservation is seen to spring from the result that the operator of spinor affine covariant differentiation commutes with the operator of spinor Lie covariant differentiation with respect to a Killing vector. It is shown that differentiations of the spinor field defined geometrically are Lorentz-covariant.     

Exact plane-symmetric solutions of the nonlinear equations of a spinor field in the theory of gravitation

We obtain exact plane-symmetric solutions of the spinor field equations with a nonlinear term that is an arbitrary functions of the invariant and with the self-gravitational field taken into account. Conditions are formulated for which the initial system of Einstein's equation and the spinor field equations with a power-law nonlinearity have regular solutions with localized (negative) spinor field energy density: so-called soliton-like solutions. Exact solutions of the spinor field equations are also obtained in flat space—time in this case and it is shown that the initial system of equations does not have soliton-like solutions. Hence the self-gravitational field plays a crucial (regularizing) in soliton-like solutions of the nonlinear spinor field equations.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 63–68, August, 1995.     

Nonlinear dynamics of a spinor Bose-Einstein condensate in a double-well potential

We examine the nonlinear dynamical behavior of a spinor Bose-Einstein condensate in a double-well potential. Considering a condensate with large number of atoms, such that it can be described using the mean field theory, we separate the spinor dynamics from the spatial dynamics under the single-mode approximation. We limit ourselves to certain initial conditions under which the spatial mode is frozen so that we can focus on the spinor dynamics only. Identifying collective spin variables of our system, we derive the corresponding nonlinear equations of motion for them. Employing standard stability analysis, we find and characterize fixed points of the system. For a wide range of physical parameters such as tunneling strength and non-linear interactions, as well as for various initial preparations of the system, we identify qualitatively different dynamical regimes possible in the system. In particular, complete and incomplete oscillations of spin variables between quantum wells are found. We also show that by bringing some fixed points close to each other in the phase space of the system, it is possible to induce amplitude modulation to those otherwise regular tunneling oscillations.