On Parametrization of the Linear GL(4,C) and Unitary SU(4) Groups in Terms of Dirac Matrices

Parametrization of 4 × 4-matrices G of the complex linear group GL(4,C) in terms of four complex 4-vector parameters (k,m,n,l) is investigated. Additional restrictions separating some subgroups of GL(4,C) are given explicitly. In the given parametrization, the problem of inverting any 4 × 4 matrix G is solved. Expression for determinant of any matrix G is found: det G = F(k,m,n,l). Unitarity conditions G⁺ = G⁻¹ have been formulated in the form of non-linear cubic algebraic equations including complex conjugation. Several simplest solutions of these unitarity equations have been found: three 2-parametric subgroups G₁, G₂, G₃ - each of subgroups consists of two commuting Abelian unitary groups; 4-parametric unitary subgroup consisting of a product of a 3-parametric group isomorphic SU(2) and 1-parametric Abelian group. The Dirac basis of generators Λk, being of Gell-Mann type, substantially differs from the basis λi used in the literature on SU(4) group, formulas relating them are found - they permit to separate SU(3) subgroup in SU(4). Special way to list 15 Dirac generators of GL(4,C) can be used {Λk} = {αiÅβjÅ(αiVβj = KÅL ÅM )}, which permit to factorize SU(4) transformations according to S = eiaα eibβeikKeilLeimM, where two first factors commute with each other and are isomorphic to SU(2) group, the three last ones are 3-parametric groups, each of them consisting of three Abelian commuting unitary subgroups. Besides, the structure of fifteen Dirac matrices Λk permits to separate twenty 3-parametric subgroups in SU(4) isomorphic to SU(2); those subgroups might be used as bigger elementary blocks in constructing of a general transformation SU(4). It is shown how one can specify the present approach for the pseudounitary group SU(2,2) and SU(3,1)

Classical Particle in Presence of Magnetic Field, Hyperbolic Lobachevsky and Spherical Riemann Models

Motion of a classical particle in 3-dimensional Lobachevsky and Riemann spaces is studied in the presence of an external magnetic field which is analogous to a constant uniform magnetic field in Euclidean space. In both cases three integrals of motions are constructed and equations of motion are solved exactly in the special cylindrical coordinates on the base of the method of separation of variables. In Lobachevsky space there exist trajectories of two types, finite and infinite in radial variable, in Riemann space all motions are finite and periodical. The invariance of the uniform magnetic field in tensor description and gauge invariance of corresponding 4-potential description is demonstrated explicitly. The role of the symmetry is clarified in classification of all possible solutions, based on the geometric symmetry group, SO(3,1) and SO(4) respectively

Noncommutativity and Tachyon Condensation

We study the fuzzy or noncommutative Dp-branes in terms of infinitely many unstable D0-branes, from which we can construct any Dp-branes. We show that the tachyon condensation of the unstable D0-branes induces the noncommutativity. In the infinite tachyon condensation limit, most of the unstable D0-branes disappear and remaining D0-branes are actually the BPS D0-branes with the correct noncommutative coordinates. For the fuzzy S^2 case, we explicitly show only the D0-branes corresponding to the lowest Landau level survive in the limit. We also show that a boundary state for a Dp-brane satisfying the Dirichlet boundary condition on a curved submanifold embedded in the flat space is not localized on the submanifold. This implies that the Dp-brane on it is ambiguous at the string scale and solves the problem for a spherical D2-brane with a unit flux on the world volume which should be equivalent to one D0-brane. We also discuss the diffeomorphism in the D0-brane picture.Comment: 30 pages, references added, minor corrections and clarifications, version to appear in JHE

Spin 1/2 Particle with Two Masses in External Magnetic Field

Equation for spin 1/2 particle with two mass states is investigated in presence of magnetic field. The problem reduces to a system of 4 linked 2-nd order differential equations. After diagonalization of the mixing term, separate equations for four different functions are derived, in which the spectral parameters coincide with the roots of a 4-th order polynomial. Solutions are constructed in terms of confluent hyper-geometric functions; four series of energy spectrum are found. Numerical study of the spectra is performed. Physical energy levels for the two mass fermion differ from those for the ordinary Dirac fermion

Оn describing bound states for a spin 1 particle in the external coulomb field

Исследуется система из 10 радиальных уравнений для векторной частицы в кулоновском поле. С использованием оператора пространственной четности система разбивается на две, по 4 и 6 уравнений каждая. Система из 4 уравнений решается в гипергеометрических функциях, приводя к известному спектру энергий. Комбинированием 6 уравнений удается получить для некоторых радиальных функций дифференциальные уравнения второго порядка. В частности, одно из уравнений оказывается уравнением Гойна, это позволило на основе выделения так называемых трансцендентных вырожденных функций Гойна получить условие квантования и соответствующий спектр энергий. Система 6 уравнений после исключения недифференциальных соотношений приведена к связанным уравнениям 1-го порядка для функций 1234. f fff Выведены уравнения 4-го порядка для каждой из этих функций, описаны их сингулярности. Предложен метод описания проекций векторов решений – линий в 4-мерном пространстве 1234 {()()()()} f rfrfrfr на различные плоскости 0.The system of 10 radial equationsfor a spin 1 particle in the external Coulomb field, is studied. With the use of the space reflection operator, the system is split to subsystems, consisted of 4 and 6 equations respectively. The system of 4 equations is solved in terms of hypergeometric functions, which gives the known energy spectrum. Combining the 6-equation system, we derive several equations of the 2-nd order for some separate functions. On of them may be recognized as a confluent Heun equation. A series of bound states is constructed in terms of the so calledtranscendental confluent Heun functions, which provides us with solutions for the second class of bound states, with corresponding formula for energy levels. The subsystem of 6 is equations reduced to the system of the 1-st order equations for 4 functions , 1,2,3,4.i fi We derive explicit form of a corresponding of the 4-th order equation for eachfunction. From four independent solutions of each 4-th order equation, only two solutions may be referred to series of bound states