arXiv subjects
Publications and source records attributed to A. D. Alhaidari.
Using the spectral properties of orthogonal polynomials, we introduce an algebraic version of quantum field theory for elementary particles. Closed-loop integrals in the Feynman diagrams for computing transition amplitudes are divergence-free. Consequently, the theory is finite and no renormalization scheme is required.
We introduce two-parameter classes of exactly-solvable novel systems whose Hamiltonian operators could be represented by tridiagonal symmetric matrices in some orthogonal bases. The associated wavefunction is written as point-wise convergent series in the basis elements. The expansion coefficients of the series are orthogonal polynomials in the energy that satisfy the resulting three-term recursion relation starting with two-parameter initial values. These polynomials contain all physical information about the system and they depend on the values of the two parameters. We obtain the associated two-parameter potential function induced by the change in the initial values that causes the system's wavefunction to change. We give several illustrative examples of these systems with continuous and/or discrete energy spectra. Moreover, a curious phenomenon is observed where bound states and/or resonances are induced in a system with pure continuous spectrum (e.g., a free particle) if the two parameters in the initial values exceed certain critical limits.
We introduce a perturbative formulation for a nonlinear extension of the J-matrix method of scattering in two dimensions. That is, we obtain the scattering matrix for the time-independent nonlinear Schrödinger equation in two dimensions with circular symmetry. The formulation relies on the linearization of products of orthogonal polynomials and on the utilization of the tools of the J-matrix method. Gauss quadrature integral approximation is instrumental in the numerical implementation of the approach. We present the theory for a general ψ^{2n + 1} nonlinearity, where n is a natural number, and obtain results for the cubic and quintic nonlinearities, ψ^3 and ψ^5. At certain value(s) of the energy, we observe the occurrence of bifurcation with two stable solutions. This curious and interesting phenomenon is a clear signature and manifestation of the underlying nonlinearity.
We obtain simple formulas for the matrix elements of the resolvent operator (the Green's function) in any finite set of square integrable basis. These formulas are suitable for numerical computations whether the basis elements are orthogonal or not. A byproduct of our findings is an expression for the normalized eigenvectors of a matrix in terms of its eigenvalues. We give a physical application as an illustration of how useful these results can be.
Conventional quantum field theory is a method for studying structureless elementary particles. Non-elementary particles, on the other hand, are those with internal structure or particles that are made up of elementary constituents like the hadrons, which contain quarks and gluons. We introduce a structure-inclusive algebraic formulation of quantum field theory that could handle such particles and in which orthogonal polynomials play a central role. For simplicity, we consider non-elementary scalar particles in 3+1 Minkowski space-time and, in three appendices, we treat spinors with structure, massless vector fields, and the massive vector bosons. We show how to do scattering calculation in a nonlinear scalar-spinor coupling model where we find that loop integrals in the Feynman diagrams are remarkably finite. The aim of this short exposé is to motivate further studies and research using this approach.
Within the recently proposed structure-inclusive algebraic formulation of quantum field theory, we show that a massless particle can acquire mass by special nonlinear coupling to a universal massless scalar field; establishing an alternative to the Higgs mechanism in the standard model of particle physics. We end with a conjecture concerning dark energy and dark matter.
Gauss quadrature integral approximation is extended to include integrals with a measure consisting of continuous as well as discrete components. That is, we give an approximation for the integral of a function plus its sum over a discrete weighted set.
An algebraic system is introduced, which is very useful for doing scattering calculations in quantum field theory. It is the set of all real numbers greater than or equal to -m^2 with parity designation and a special rule for addition and subtraction, where m is the rest mass of the scattered particle.
We introduce nonlinear extension of the J-matrix method of scattering. The formulation relies predominantly on the linearization of products of orthogonal polynomials. We present a toy model as an illustrative example and obtain the nonlinear scattering matrix.
We obtain exact, simple and very compact expressions for the linearization coefficients of the products of orthogonal polynomials; both the conventional Clebsch-Gordan-type and the modified version. The expressions are general depending only on the coefficients of the three-term recursion relation of the linearizing polynomials. These are more appropriate and useful for doing numerical calculations when compared to other exact formulas found in the mathematics literature, some of which apply only to special class of polynomials while others may involve the evaluation of intractable integrals. As an application in physics, we present a remarkable phenomenon where nonlinear coupling in a physical system with pure continuous spectrum generates a mixed spectrum of continuous and discrete energies.
Starting with an orthogonal polynomial sequence $\{p_n(s)\}_{n=0}^\infty$ that has a discrete spectrum, we design an energy spectrum formula, $E_k = f (s_k)$, where $|{s_k\}$ is the finite or infinite discrete spectrum of the polynomial. Using a recent approach for doing quantum mechanics based, not on potential functions but, on orthogonal energy polynomials, we give a local numerical realization of the potential function associated with the chosen energy spectrum. In this work, we select the three-parameter continuous dual Hahn polynomial as an example. Exact analytic expressions are given for the corresponding bound states energy spectrum, scattering states phase shift, and wavefunctions. However, the potential function is obtained only numerically for a given set of physical parameters.
The interaction of an electron with a local static charge distribution (e.g., an atom or molecule) is dominated at large distances by the radial 1/r Coulomb potential. The second order effect comes from the non-central electric dipole contribution cos(theta)/r^2. Moreover, the third order effect is due to the electric quadrupole potential, [3*cos^2(theta)-1]/2*r^3. We use the tridiagonal representation approach to give a reasonably accurate account for the combined effects of all these contributions to the binding energy of the electron but with an effective quadrupole interaction. As an application, we obtain the bound states of a valence electron in an atom with both electric dipole and quadrupole moments.
We use the tridiagonal representation approach to solve the radial Schrödinger equation for the continuum scattering states of the Kratzer potential. We do the same for a radial power-law potential with inverse-square and inverse-cube singularities. These solutions are written as infinite convergent series of Bessel functions with a discrete index. As physical application of the latter solution, we treat electron scattering off a neutral molecule with electric dipole and electric quadrupole moments.
With the present trend in experimental particle physics of probing yet shorter distances and with the requirement on the theoretical side of renormalizability, conformal invariance becomes an attractive symmetry for particle interactions. That is because conformal field theories form a large class of massless no-scale renormalizable quantum field theories. It is generally believed, however, that renormalization breaks conformal invariance and with the absence of a reported success to find a conformally invariant renormalization scheme, an alternative approach is followed in this thesis. We begin with a conformally invariant theory and study the implications of such idealization. Conformal field theories are populated with new gauge degrees of freedom and the inclusion, in a natural way, of these auxiliary fields may improve the renormalization program. Another remarkable property is the presence of the vacuum mode in the physical subspace of the gauge potential, believed to be associated with spontaneously generated gauge theories called "zero-center modules." In spite of all the work that has been done over the years on conformal field theories, conformal electrodynamics has remained only partially developed. New results...(more)
We use the tridiagonal representation approach to solve the radial Schrödinger equation for the continuum scattering states of the Coulomb problem in a complete basis set of discrete Bessel functions. Consequently, we obtain a new representation of the confluent hypergeometric function as an infinite sum of Bessel functions, which is numerically very stable and more rapidly convergent than another well-known formula.
We use the "tridiagonal representation approach" to solve the time-independent Schrödinger equation for bound states in a basis set of finite size. We obtain two classes of solutions written as finite series of square integrable functions that support a tridiagonal matrix representation of the wave operator. The differential wave equation becomes an algebraic three-term recursion relation for the expansion coefficients of the series, which is solved in terms of finite polynomials in the energy and/or potential parameters. These orthogonal polynomials contain all physical information about the system. The basis elements in configuration space are written in terms of either the Romanovski-Bessel polynomial or the Romanovski-Jacobi polynomial. The maximum degree of both polynomials is limited by the polynomial parameter(s). This makes the size of the basis set finite but sufficient to give a very good approximation of the bound states wavefunctions that improves with an increase in the basis size.
We use the "tridiagonal representation approach" to solve the time-independent Schrödinger equation for the bound states of generalized versions of the trigonometric and hyperbolic Pöschl-Teller potentials. These new solvable potentials do not belong to the conventional class of exactly solvable problems. The solutions are finite series of square integrable functions written in terms of the Jacobi polynomial.
We solve the one-dimensional Schrödinger equation for the bound states of two potential models with a rich structure as shown by their "spectral phase diagram". These potentials do not belong to the well-known class of exactly solvable problems. The solutions are finite series of square integrable functions written in terms of the Jacobi polynomials.