arXiv subjects

M. Kirchbach

Publications and source records attributed to M. Kirchbach.

At least 19 recordsLinked to original sources

arXiv2024

Supersymmetric Expansion Algorithm and complete analytical solution for the Hulthén and anharmonic potentials

An algorithm for providing analytical solutions to Schrödinger's equation with non-exactly solvable potentials is elaborated. It represents a symbiosis between the logarithmic expansion method and the techniques of the superymmetric quantum mechanics as extended toward non shape invariant potentials. The complete solution to a given Hamiltonian $H_{0}$ is obtained from the nodeless states of the Hamiltonian $H_{0}$ and of a set of supersymmetric partners $H_{1}, H_{2},..., H_{r}$. The nodeless states (dubbed "edge" states) are unique and in general can be ground or excited states. They are solved using the logarithmic expansion which yields an infinite systems of coupled first order hierarchical differential equations, converted later into algebraic equations with recurrence relations which can be solved order by order. We formulate the aforementioned scheme, termed to as "Supersymmetric Expansion Algorithm'' step by step and apply it to obtain for the first time the complete analytical solutions of the three dimensional Hulthén--, and the one-dimensional anharmonic oscillator potentials.

quant-ph↗

arXiv2023

Potentials on the conformally compactified Minkowski spacetime and their application to quark deconfinement

We study a class of conformal metric deformations in the quasi-radial coordinate parameterizing the 3-sphere in the conformally compactified Minkowski spacetime $S^1\times S^3$. Prior to reduction of the associated Laplace-Beltrami operators to a Schr\"odinger form, a corresponding class of exactly solvable potentials (each one containing a scalar and a gradient term) is found. In particular, the scalar piece of these potentials can be exactly or quasi-exactly solvable, and among them we find the finite range confining trigonometric potentials of P\"oschl-Teller, Scarf and Rosen-Morse. As an application of the results developed in the paper, the large compactification radius limit of the interaction described by some of these potentials is studied, and this regime is shown to be relevant to a quantum mechanical quark deconfinement mechanism.

math-ph↗

arXiv2023

Wave functions of the Hydrogen atom in the momentum representation

We construct the integral transform passing from the space representation to the momentum representation for the Hydrogen atom using polar spherical coordinates. The resulting radial wave functions are explicitly given in terms of complex finite expansions of Gegenbauer functions of the first and second kind, or in terms of (elementary) trigonometric functions. We show their symmetry under the $SO(4)$ group, and their equivalence with those of Lombardi and Oglivie.

quant-ph↗

arXiv2023

De Sitter Special Relativity as a Possible Reason for Conformal Symmetry and Confinement in QCD

Conformal symmetry and color confinement in the infrared regime of QCD are interpreted by means of a conjectured deSitter $dS_4$ geometry of the internal space-time of hadrons, an assumption inspired by the hypothesis on deSitter special relativity. Within such a scenario, the interactions involving the virtual gluon and constituent quark degrees of freedom of hadrons are deduced from the Green functions of Laplace operators on the $dS_4$ geodesics. Then the conformal symmetry of QCD emerges as a direct consequence of the conformal symmetry of the $dS_4$ space-time, while the color confinement, understood as colorlessness of hadrons, appears as a consequence of the inevitable charge neutrality of the unique closed space-like manifold, the three dimensional hyper-sphere $S^3$, on whose geodesics the hadron's constituents are conjectured to reside when near rest frame. Mesons are now modelled as quarkish color-anticolor dipoles, whose free quantum motions on the aforementioned $S^3$ geodesic are perturbed by a potential generated by a gluon--anti-gluon color dipole. The potential predicted presents itself as the color charge analogue to the "curved" Coulomb potential, i.e. to the electric potential that defines a consistent electrostatic theory on a hyper-spherical surface. The advantage of this method is that it allows to establish a direct relationship of the potential parameters to the fundamental constants of QCD. We apply the model to the description of the spectra of the $a_1$ and $f_1$ mesons, and the pion electric charge form factor, finding fair agreement with data.

physics.gen-ph↗

arXiv2022

The Conformal-Symmetry--Color-Neutrality Connection in Strong Interaction

The color neutrality of hadrons is interpreted as an expression of conformal symmetry of strong interaction, the latter being signaled through the detected "walking" at low transferred momenta, $\lim_{Q^2\to 0}α_s(Q^2)/π\to 1 $, of the strong coupling toward a fixed value ($α_s $ "freezing" ). The fact is that conformal symmetry admits quarks and gluons to reside on the compactified $AdS_5$ boundary, whose topology is $S^1\times S^3$, a closed space that can bear exclusively color-charge neutral configurations, precisely as required by color confinement. The compactification radius, once employed as a second scale alongside with $Λ_{QCD}$, provides for an $α_s(Q^2) $ "freezing" mechanism in the infrared regime of QCD, thus making the conformal-symmetry--color-neutrality connection at low energies evident. In this way, perturbative descriptions of processes in the infrared could acquire meaning. In consequence, it becomes possible to address QCD by quantum mechanics in terms of a conformal wave operator equation, which leads to an efficient description of a wide range of

hep-ph↗

arXiv2021

Color confinement at the boundary of the conformally compactified $\mathrm{AdS}_5$

The topology of closed manifolds forces interacting charges to appear in pairs. We take advantage of this property in the setting of the conformal boundary of $\mathrm{AdS}_5$ spacetime, topologically equivalent to the closed manifold $S^1\times S^3$, by considering the coupling of two massless opposite charges on it. Taking the interaction potential as the analog of Coulomb interaction (derived from a fundamental solution of the $S^3$ Laplace-Beltrami operator), a conformal $S^1\times S^3$ metric deformation is proposed, such that free motion on the deformed metric is equivalent to motion on the round metric in the presence of the interaction potential. We give explicit expressions for the generators of the conformal algebra in the representation induced by the metric deformation. By identifying the charge as the color degree of freedom in QCD, and the two charges system as a quark--anti-quark system, we argue that the associated conformal wave operator equation could provide a realistic quantum mechanical description of the simplest QCD system, the mesons. Finally, we discuss the possibility of employing the compactification radius, $R$, as another scale along $Λ_{QCD}$, by means of which, upon reparametrizing $Q^2c^2$ as $\left( Q^2c^2 +\hbar^2 c^2/R^2\right)$, a pertubative treatment of processes in the infrared could be approached.

hep-th↗

arXiv2018

Modelling duality between bound and resonant meson spectra by means of free quantum motions on the de Sitter space time dS4

We seek for a pair of a well and barrier potentials such that the real parts of the complex energies of the resonances transmitted through the barrier equal the energies of the states bound within the well and find the hyperbolic Poeschl-Teller barrier, ~sech^2ρ, and the trigonometric Scarf well, ~ \sec^2χ. The potentials are shown to be conformally symmetric by the aid of the de Sitter space time, dS4, related to flat conformal space time by a conformal map. Namely, we transform the quantum mechanical wave equations with the above potentials to free quantum motions on the respective open time like hyperbolic and the closed space like hyper spherical, S3, geodesics of dS4, the former by itself is related to Minkowski space time by a conformal map.We formulate a conformal symmetry respecting classification scheme for mesons seen either as resonances in scattering, or as states bound within a potential, according to trajectories in which the total spin of the meson, l-depends linearly on the first power of the invariant mass, M, and not as in the canonical Regge formalism on the squared mass.We analyze 71 reported mesons in this scheme and in finding good agreement with data predict the masses of 12 missing mesons. We observe that on S3 the color quantum number of mesons is limited to neutral. To the amount physics is independent of the choice of the set of coordinates, this property remains valid in the flat geometry too. We conclude on the usefulness of conformal maps of flat to curved space times as tools for modelling and interpreting the color neutrality of hadrons required by the confinement phenomenon and on the relevance of trigonometric and hyperbolic potentials for constituent quark models. Finally, all involved potentials have been motivated by Wilson loops with cusps.

hep-ph↗

arXiv2018

Proton's Electromagnetic Form Factors from a Non-Power Confinement Potential

The electric-charge, and magnetic-dipole form factors of the proton are calculated from an underlying constituent quark picture of hadron structure based on a potential shaped after a cotangent function, which has the properties of being both conformally symmetric and color confining, finding adequate reproduction of a variety of related data.

hep-ph↗

arXiv2016

Second order differential equations for bosons with spin j > 1 and in the bases of general tensor-spinors of rank-2j

A boson of spin-j>1 can be described in one of the possibilities within the Bargmann-Wigner framework by means of one sole differential equation of order twice the spin, which however is known to be inconsistent as it allows for non-local, ghost and acausally propagating solutions, all problems which are difficult to tackle. The other possibility is provided by the Fierz-Pauli framework which is based on the more comfortable to deal with second order Klein-Gordon equation, but it needs to be supplemented by an auxiliary condition. Although the latter formalism avoids some of the pathologies of the high-order equations, it still remains plagued by some inconsistencies such as the acausal propagation of the wave fronts of the (classical) solutions within an electromagnetic environment. We here suggest a method alternative to the above two that combines their advantages while avoiding the related difficulties. Namely, we suggest one sole strictly (j,0)+ (0,j) representation specific second order differential equation, which is derivable from a Lagrangian and whose solutions do not violate causality. The equation under discussion presents itself as the product of the Klein-Gordon operator with a momentum independent projector on Lorentz irreducible representation spaces constructed from one of the Casimir invariants of the spin-Lorentz group. The basis used is that of general tensor-spinors of rank-2j.

hep-ph↗

arXiv2016

Second order differential realization of the Bargmann-Wigner framework for particles of any spin

The Bargmann-Wigner (BW) framework describes particles of spin-j in terms of Dirac spinors of rank 2j, obtained as the local direct product of n Dirac spinor copies, with n=2j. Such spinors are reducible, and contain also (j,0)+(0,j)-pure spin representation spaces. The 2(2j+1) degrees of freedom of the latter are identified by a projector given by the n-fold direct product of the covariant parity projector within the Dirac spinor space. Considering totally symmetric tensor spinors one is left with the expected number of 2(2j+1) independent degrees of freedom. The BW projector is of the order $\partial ^{2j}$ in the derivatives, and so are the related spin-j wave equations and associated Lagrangians. High order differential equations can not be consistently gauged, and allow several unphysical aspects, such as non-locality, acausality, ghosts and etc to enter the theory. In order to avoid these difficulties we here suggest a strategy of replacing the high order of the BW wave equations by the universal second order. To do so we replaced the BW projector by one of zeroth order in the derivatives. We built it up from one of the Casimir invariants of the Lorentz group when exclusively acting on spaces of internal spin degrees of freedom. This projector allows one to identify anyone of the irreducible sectors of the primordial rank-2j spinor, in particular (j,0)+(0,j), and without any reference to the external space-time and the four-momentum. The dynamics is then introduced by requiring the (j,0)+ (0,j) sector to satisfy the Klein-Gordon equation. The scheme allows for a consistent minimal gauging.

hep-ph↗

arXiv2015

Gyromagnetic g_s factors of the spin-1/2 particles in the (1/2+,1/2-,3/2-) triad of the four-vector spinor, ψ_μ, irreducibility, and linearity

We show that the spin (1/2-) particle from the (1/2,1)+(1,1/2) Lorentz irreducible sector of the four-vector spinor can not be described within a linear formalism but behaves as a genuinely quadratic fermion satisfying the generalized Feynman-Gell-Mann equation with a gyromagnetic factor of (-2/3). In contrast, spin (1/2 +) from the (1/2,0)+(0,1/2) sector is confirmed as a genuine linear Dirac fermion whose gyromagnetic factor takes the value of two units.We calculate Compton scatterings off each one of the two targets and obtain both times well behaved cross sections in the ultra relativistic limit and in accord with unitarity.

hep-ph↗

arXiv2015

Bosonic and fermionic Weinberg-Joos (j,0)+ (0,j) states of arbitrary spins as Lorentz-tensors or tensor-spinors and second order theory

We propose a general method for the description of arbitrary single spin-j states transforming according to (j,0)+(0,j) carrier spaces of the Lorentz algebra in terms of Lorentz-tensors for bosons, and tensor-spinors for fermions, and by means of second order Lagrangians. The method allows to avoid the cumbersome matrix calculus and higher \partial^{2j} order wave equations inherent to the Weinberg-Joos approach. We start with reducible Lorentz-tensor (tensor-spinor) representation spaces hosting one sole (j,0)+(0,j) irreducible sector and design there a representation reduction algorithm based on one of the Casimir invariants of the Lorentz algebra. This algorithm allows us to separate neatly the pure spin-j sector of interest from the rest, while preserving the separate Lorentz- and Dirac indexes. However, the Lorentz invariants are momentum independent and do not provide wave equations. Genuine wave equations are obtained by conditioning the Lorentz-tensors under consideration to satisfy the Klein-Gordon equation. In so doing, one always ends up with wave equations and associated Lagrangians that are second order in the momenta. Specifically, a spin-3/2 particle transforming as (3/2,0)+ (0,3/2) is comfortably described by a second order Lagrangian in the basis of the totally antisymmetric Lorentz tensor-spinor of second rank, Ψ_[ μν]. Moreover, the particle is shown to propagate causally within an electromagnetic background. In our study of (3/2,0)+(0,3/2) as part of Ψ_[μν] we reproduce the electromagnetic multipole moments known from the Weinberg-Joos theory. We also find a Compton differential cross section that satisfies unitarity in forward direction. The suggested tensor calculus presents itself very computer friendly with respect to the symbolic software FeynCalc.

hep-ph↗

arXiv2015

Parity violating effects in an exotic perturbation of the rigid rotator

The perturbation of the free rigid rotator by the trigonometric Scarf potential is shown to conserve its energy excitation patterns and change only the wave functions towards spherical harmonics rescaled by a function of an unspecified parity, or mixtures of such rescaled harmonics of equal magnetic quantum numbers and different angular momenta. In effect, no parity can be assigned to the states of the rotational bands emerging in this exotic way, and the electric dipole operator is allowed to acquire non-vanishing expectation values.

quant-ph↗

arXiv2014

The Higgs oscillator on the hyperbolic plane and Light-Front Holography

The Light Front Holographic (LFH) wave equation, which is the conformal scalar equation on the plane, is revisited from the perspective of the supersymmetric quantum mechanics, and attention is drawn to the fact that it naturally emerges in the small hyperbolic angle approximation to the "curved" Higgs oscillator on the hyperbolic plane, i.e. on the upper part of the two-dimensional hyperboloid of two sheets, a space of constant negative curvature. Such occurs because the particle dynamics under consideration reduces to the one dimensional Schrödinger equation with the second hyperbolic Pöschl-Teller potential, whose flat-space (small-angle) limit reduces to the conformally invariant inverse square distance plus harmonic oscillator interaction, on which LFH is based. In consequence, energies and wave functions of the LFH spectrum can be approached by the solutions of the Higgs oscillator on the hyperbolic plane in employing its curvature and the potential strength as fitting parameters. Also the proton electric charge form factor is well reproduced within this scheme by means of a Fourier-Helgason hyperbolic wave transform of the charge density. In conclusion, in the small angle approximation, the Higgs oscillator on the hyperbolic plane is demonstrated to satisfactory parallel essential outcomes of the Light Front Holographic QCD. The findings are suggestive of associating the hyperboloid curvature of the with a second scale in LFH, which then could be employed in the definition of a chemical potential.

quant-ph↗

arXiv2014

Breaking so(4) symmetry without degeneracy lift

We argue that in the quantum motion of a scalar particle of mass "m" on S^3_R perturbed by the trigonometric Scarf potential (Scarf I) with one internal quantized dimensionless parameter, \ell, the 3D orbital angular momentum, and another, an external scale introducing continuous parameter, B, a loss of the geometric hyper-spherical so(4) symmetry of the free motion can occur that leaves intact the unperturbed {\mathcal N}^2-fold degeneracy patterns, with {\mathcal N}=(\ell +n+1) and n denoting the nodes number of the wave function. Our point is that although the number of degenerate states for any {\mathcal N} matches dimensionality of an irreducible so(4) representation space, the corresponding set of wave functions do not transform irreducibly under any so(4). Indeed, in expanding the Scarf I wave functions in the basis of properly identified so(4) representation functions, we find power series in the perturbation parameter, B, where 4D angular momenta K\in [\ell , {\mathcal N}-1] contribute up to the order \left(\frac{2mR^2B}{\hbar^2}\right)^{{\mathcal N}-1-K}. In this fashion, we work out an explicit example on a symmetry breakdown by external scales that retains the degeneracy. The scheme extends to so(d+2) for any d.

hep-ph↗

arXiv2014

Perturbing free motions on hyper spheres without degeneracy lift

We consider quantum motion on S3 perturbed by the trigonometric Scarf potential (Scarf I) with one internal quantized dimensionless parameter, l, the ordinary orbital angular momentum value, and another, continuous parameter, b, through which an external scale is introduced. We argue that a loss of the geometric hyper-spherical so(4) symmetry of the free motion occurs that leaves intact the unperturbed hydrogen-like degeneracy patterns characterizing the spectrum under discussion. The argument is based on the observation that the expansions of the Scarf I wave functions for fixed l-values in the basis of properly identified $so(4)$ representation functions are power series in the perturbation parameter, b, in which carrier spaces of dimensionality (K+1)^2 with K varying as K\in [l, N-1], and N being the principal quantum number of the Scarf I potential problem, contribute up to the order O(b^(N-1-K)). Nonetheless, the degeneracy patterns can still be interpreted as a consequence of an effective so(4) symmetry, i.e. a symmetry realized at the level of the dynamic of the system, in so far as from the perspective of the eigenvalue problem, the Scarf I results are equivalently obtained from a Hamiltonian with matrix elements of polynomials in a properly identified so(4) Casimir operator. The scheme applies to any dimension, d.

math-ph↗

arXiv2013

Second order theory of $(j,0)\oplus (0,j)$ single high spins as Lorentz tensors

We show that higher order differential equations and matrix spinor calculus are completely avoidable in the description of pure high spin-$j$ Weinberg-Joos states, $(j,0)\oplus (0,j)$. The case is made on the example of $(3/2,0)\oplus(0,3/2)$, for the sake of concreteness and without loss of generality. Namely, we use as a vehicle for the aforementioned covariant single spin-$3/2$ description the antisymmetric tensor of second rank with Dirac spinor components, $Ψ_{[μν]}=B_{[μν]}\otimesψ$. The $(3/2,0)\oplus(0,3/2)$ sector of interest is tracked down in two steps. First we search for spin-$3/2$ by means of a covariant spin projector constructed from the Casimir invariants of the Poincaré algebra, and then we identify the wanted irreducible representation space by means of a momentum independent (static) projector designed on the basis of the Casimir invariants of the Lorentz algebra. The latter projectors unambiguously identify any irreducible $so(1,3)$ subspace of any Lorentz tensor and without rising the order of the differential equation. The method proposed correctly reproduces the electromagnetic multipole moments earlier calculated for single spin-$3/2$ particles in treating it in the standard way as eight dimensional spinor. We furthermore calculate Compton scattering off the pure spin-$3/2$ under discussion, and show that the differential cross section satisfies unitarity in forward direction for a gyromagnetic ratio of $g=2/3$. Suggesting possible validity of Belinfante's conjecture for pure spin-states, while the natural value of $g=2$ seems more likely to characterize the highest spins in the Rarita-Schwinger representation spaces. The scheme straightforwardly extends to any $(j,0)\oplus (0,j)$ Weinberg-Joos state and brings the advantage of avoiding rectangular matrix couplings between states of different spins, replacing them by simple Lorentz contractions.

hep-ph↗

arXiv2013

Status of the lower spins in the Rarita-Schwinger four-vector spinor $ψ_μ$ within the method of the combined Lorentz- and Poincaré invariant projectors

We investigate the status of the lower spin-1/2 companions to spin-3/2 within the four-vector spinor, $ψ_μ$. According to its reducibility, $ψ_μ\longrightarrow \left[(1/2,1)\oplus (1,1/2)\right]\oplus [(1/2,0)\oplus (0,1/2)]$ this representation space contains two spin-1/2 sectors, the first one transforming as a genuine Dirac-spinor, $(1/2,0)\oplus (0,1/2)$, and the second as the companion to spin-3/2 in $(1/2,1)\oplus (1,1/2)$. In order to correctly identify the covariant spin-1/2 degrees of freedom in the Rarita-Schwinger field of interest we exploit the properties of the Casimir invariants of the Lorentz algebra to distinguish between the irreducible Dirac- and $(1/2,1)\oplus (1,1/2)$ representation spaces and construct corresponding momentum-independent (static) projectors which we then combine with a dynamical spin-1/2 Poincaré covariant projector. In so doing we obtain two spin-1/2 wave equation, and prove them to be causal within an electromagnetic field. We furthermore calculate Compton scattering off each one of the above states, and find that the amplitudes corresponding to the first spin-1/2 are identical to those of a Dirac particle and conclude on the observability of this state. Also for the second spin-1/2 we find finite cross sections in all directions in the ultra-relativistic limit, and conclude that its observability is not excluded neither by causality of propagation within an electromagnetic environment, nor by unitarity of the Compton scattering amplitudes in the ultraviolet. Finally, we notice that the method of the combined Lorentz- and Poincaré invariant projectors could be instrumental in opening a new avenue toward the consistent description of any spin by means of second order Lagrangians written in terms of sufficiently large reducible representation spaces equipped with separate Lorentz-- and Dirac indices.

hep-ph↗