S. Datta Majumdar

The finite transformations of the group SU(3)

Abstract Representations of finite transformations of the group SU (3) are obtained by breaking up the elements into simpler factors. Some of the factors belong to the fully reduced SU (2) subgroup and are represented by ordinary rotation matrices. Amongst the other factors there are two numerical matrices, ( 1 ± iσ 2 ). Representations of these are obtained by subjecting the variables of a basic state to the appropriate transformation and expanding the function so obtained in a series of the basic states. The expansion coefficients involve generalized hypergeometric series of the 4 F 3 (1) type which are shown to be multiples of 6- j symbols.

The master analytic function and the Lorentz group. III. Coupling of continuous representations of O(2,1)

The Clebsch–Gordan problem for continuous representations belonging to the principal series of O(2,1) is treated by a method developed previously for the coupling of a discrete and a continuous representation. The values of the complex variable x occurring in the fundamental differential equation of the problem are restricted to lie on the unit circle, and the Clebsh–Gordan coefficients are identified with the Fourier coefficients of solutions of this equation. If j belongs to the discrete class there is only one acceptable solution of the second order equation. But, if j1,j2,j all belong to the continuous class any two independent solutions of the equation give a possible Clebsch–Gordan series. The problem of orthogonalizing the solutions in the latter case is solved and the normalization factor is determined using the Sturm–Liouville theory of differential equations. The Clebsch–Gordan coefficients generated by an orthogonal pair of solutions become automatically orthogonal. To determine the j values ap...

The master analytic function and the Lorentz group. I. Reduction of the representations of O(3,1) in O(2,1) basis

The reduction of the principal and supplementary series of representations of SL(2,C) in the SU(1,1) basis is carried out by using a basis function which formally resembles the coupled state of two angular momenta. The spectrum of the SU(1,1) representations contained in SL(2,C) and the transformation coefficients are obtained by expanding the SU(2) in terms of the SU(1,1) bases with the help of the Sommerfeld–Watson transformation. The orthogonality conditions for the principal and supplementary series are discussed. For the principal series this follows easily from the standard Sturm–Liouville theory of the second order differential equations. For the supplementary series the orthogonality condition is obtained from the fourth order differential equation satisfied by the Fourier transform of the basis function.

The Clebsch‐Gordan coefficients of SU(3) and the orthogonalization problem

The Clebsch‐Gordan coefficients of the group SU(3) are determined by integrating the product of three matrix elements of finite transformations belonging to three irreducible representations of the group. Compact expressions involving a single or a double sum over products of 3‐j and 6‐j symbols of SU(2) are obtained for several different classes of coefficients by suitably restricting the initial states but keeping the final states of the matrix elements arbitrary. To orthogonalize the CG coefficients, a linear combination of several integrals with the same final but different initial states is taken. The coefficients of the linear combination are determined by the Schmidt procedure and are found to be expressible in terms of integrals of the same type.

Cherenkov Radiation in Anisotropic Media

The problem of Cherenkov radiation in anisotropic media is studied in a Lorentz frame in which the charged particle is at rest and the medium is moving with a uniform velocity. Both electric and magnetic anisotropy are assumed to be present, but the axes of the permittivity and permeability ellipsoids are taken to be parallel to one another. The electromagnetic field generated by the charge is described by two scalar potentials. Each of these satisfies a partial differential equation of the fourth order when the velocity vector lies in a principal plane of the ellipsoids. The two equations closely resemble one another, and passage from one to the other is possible by means of certain simple symmetry operations. The equation for the scalar potential of the electric field is solved by the standard technique of Fourier transformation. In evaluating the Fourier integrals, however, it is found necessary to assume that two of the ratios Єi/μi of the principal permittivities and permeabilities are equal. With this additional restriction the integrals are evaluated easily by the residue theorem and expressions for the field and the radiated energy are obtained in closed forms.

The master analytic function and the Lorentz group. II. The Clebsch–Gordan problem for O(2,1)

The Clebsch–Gordan coefficients of the noncompact group O(2,1) representing Lorentz transformations in three‐dimensional space–time are calculated in the compact O(2) basis. Considerable simplification is achieved by introducing a variable x and replacing all algebraic equations by differential equations. The coupled state appears in the theory as a solution of an ordinary differential equation reducible to the hypergeometric equation by a simple substitution. The coefficients in the Taylor–Laurent expansion of this solution in powers of x are shown to be identical with the Clebsch–Gordan coefficients. The inverse expansion, obtained by the use of certain identities for the hypergeometric function and the Sommerfeld–Watson transformation, yields the normalization factor and the values of j appearing in the reduction.

Coupling of three angular momenta

Some earlier results on the coupling of two angular momenta are extended in the present paper to three angular momenta. As in the previous case, the eigenfunctions with sharp total angular momentum are obtained in closed forms which do not contain the Clebsch-Gordan coefficients explicitly. The occurrence of the hypergeometric function in the present formulation renders the treatment highly flexible and makes it possible to derive a number of alternative expressions for the 6 —j symbol. Moreover, recursion relations between contiguous 6 —j symbols are derived more easily from Gauss’s relations between contiguous hypergeometric functions.РезюмеВ данной работе некоторые ранние результаты по отношению связи двух угловых моментов распространяются к трем угловым моментам. Как и в предыдущем случае собственные функции с острым полным угловым моментом получены в закрытых формах, в которые явно не входят коэффициенты Клебша— Гордона. Появление гипергеометрической функции в данной формулировке сделает дискуссию высоко гибкой и дает возможность для вывода некоторого числа альтернативных выражений для символов 6—j. Далее, рекурсионные соотношения между зависящими друг от друга символами 6—j выводятся очень легко из соотношения Гаусса между связанными друг с другом гипергеометрическими функциями.

THE DIRAC EQUATION AT HIGH ENERGIES.

Abstract It is shown that the wave function for a high-energy electron scattered by a central field (such as the electrostatic field of spherical nuclei) can be obtained by solving a “Schrodinger equation” with a complex potential energy. This is found to be true for a general central field if the rest mass of the electron can be neglected in comparison with its kinetic energy. The restriction can be removed in the case of a pure Coulomb field and an exact equation, valid for all energies, can be obtained. This permits bound states in a Coulomb field to be treated on the same footing as scattering states. The present formulation is expected to simplify phase-shift calculations and to facilitate solution by other means.

Wave equations in momentum space

The behaviour of wave functions in momentum space under rotation is studied in detail and the results are used to reduce the number of independent variables in the integral wave equation for a three-particle system.РезюмеДетально изучается поведение волновых функций во врашаюшемся пространстве импульсов. Результаты использованы для уменьшения числа независимых переменных в интегральном волновом уравнении для трёхчастичной системы.