Quantum Numbers of Eigenstates of Generalized de Broglie-Bargmann- Wigner Equations for Fermions with Partonic Substructure

2003 ◽  
Vol 58 (1) ◽  
pp. 1-12 ◽  
Author(s):  
H. Stumpf
Keyword(s):  
Angular Momentum ◽  
Integral Equations ◽  
Bound States ◽  
Solution Space ◽  
Many Body ◽  
Quantum Numbers ◽  
Relevant Group ◽  
Three Body ◽  
Relativistic Bound States ◽  
Fusion Theory

Generalized de Broglie-Bargmann-Wigner (BBW) equations are relativistically invariant quantum mechanical many body equations with nontrivial interaction, selfregularization and probability interpretation. Owing to these properties these equations are a suitable means for describing relativistic bound states of fermions. In accordance with de Broglie’s fusion theory and modern assumptions about the partonic substructure of elementary fermions, i.e., leptons and quarks, the three-body generalized BBW-equations are investigated. The transformation properties and quantum numbers of the three-parton equations under the relevant group actions are elaborated in detail. Section 3 deals with the action of the isospin group SU(2), a U(1) global gauge group for the fermion number, the hypercharge and charge generators. The resulting quantum numbers of the composite partonic systems can be adapted to those of the phenomenological particles to be described. The space-time transformations and in particular rotations generated by angular momentum operators are considered in Section 4. Based on the compatibility of the BBW-equations and the group theoretical constraints, in Sect. 5 integral equations are formulated in a representation with diagonal energy and total angular momentum variables. The paper provides new insight into the solution space and quantum labels of resulting integral equations for three parton states and prepares the ground for representing leptons and quarks as composite systems.

scholarly journals Symmetry reduction of tensor networks in many-body theory

2020 ◽  
Vol 56 (10) ◽  
Author(s):  
A. Tichai ◽  
R. Wirth ◽  
J. Ripoche ◽  
T. Duguet
Keyword(s):  
Angular Momentum ◽  
State Of The Art ◽  
Symmetry Reduction ◽  
Reduced Form ◽  
Many Body ◽  
Systematic Account ◽  
Many Body Theory ◽  
Quantum Numbers ◽  
The Many ◽  
Body Theory

AbstractThe ongoing progress in (nuclear) many-body theory is accompanied by an ever-rising increase in complexity of the underlying formalisms used to solve the stationary Schrödinger equation. The associated working equations at play in state-of-the-art ab initio nuclear many-body methods can be analytically reduced with respect to angular-momentum, i.e. SU(2), quantum numbers whenever they are effectively employed in a symmetry-restricted context. The corresponding procedure constitutes a tedious and error-prone but yet an integral part of the implementation of those many-body frameworks. Indeed, this symmetry reduction is a key step to advance modern simulations to higher accuracy since the use of symmetry-adapted tensors can decrease the computational complexity by orders of magnitude. While attempts have been made in the past to automate the (anti-) commutation rules linked to Fermionic and Bosonic algebras at play in the derivation of the working equations, there is no systematic account to achieve the same goal for their symmetry reduction. In this work, the first version of an automated tool performing graph-theory-based angular-momentum reduction is presented. Taking the symmetry-unrestricted expressions of a generic tensor network as an input, the code provides their angular-momentum-reduced form in an error-safe way in a matter of seconds. Several state-of-the-art many-body methods serve as examples to demonstrate the generality of the approach and to highlight the potential impact on the many-body community.

scholarly journals Solving relativistic three-body integral equations in the presence of bound states

2021 ◽  
Vol 104 (1) ◽  
Author(s):  
Andrew W. Jackura ◽  
Raúl A. Briceño ◽  
Sebastian M. Dawid ◽  
Md Habib E Islam ◽  
Connor McCarty
Keyword(s):  
Integral Equations ◽  
Bound States ◽  
Three Body

scholarly journals Hadronic currents and form factors in three-body semileptonic $$\tau $$ decays

2021 ◽  
Vol 81 (12) ◽  
Author(s):  
Fabian Krinner ◽  
Stephan Paul
Keyword(s):  
Angular Momentum ◽  
Orbital Angular Momentum ◽  
Dipole Moments ◽  
Form Factors ◽  
Decay Chain ◽  
Full Description ◽  
Final States ◽  
Arbitrary Mass ◽  
Quantum Numbers ◽  
Three Body

AbstractThree-body semileptonic $$\tau $$ τ -decays offer a path to understand the properties of light hadronic systems and CP symmetry violations through searches for electric dipole moments. In studies of electro-weak physics, the hadronic part of the final states has traditionally been described using the language of form factors. Spectroscopic information, resolved in terms of orbital angular momentum quantum-numbers, is best being derived from an explicit decomposition of the hadronic current in the orbital angular momentum basis. Motivated by the upcoming large data samples from $$\mathrm {B}$$ B factories, we present the full description of the hadronic currents decomposed into quantum numbers of the hadronic final state using the isobar picture. We present formulas for orbital angular momenta up to three and apply the rules derived from hadron spectroscopy to formulate the decay chain of hadronic three-body systems of arbitrary mass. We also translate this formalism to the language of form factors and thereby correct insufficiencies found in previous analyses of three-body hadronic final states.

scholarly journals Relativistic Bound States of Spinless Particle by the Cornell Potential Model in External Fields

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Sameer M. Ikhdair
Keyword(s):  
Bound States ◽  
Dimensional Space ◽  
Spinless Particle ◽  
Relativistic Energy ◽  
External Fields ◽  
Ansatz Method ◽  
Cornell Potential ◽  
Quantum Numbers ◽  
Relativistic Bound States ◽  
Aharonov Bohm

The two-dimensional (2D) relativistic bound states of a spinless particle placed in scalarS(r)and vectorV(r)Cornell potentials (withS(r)>V(r)) are obtained under the influence of external magnetic and Aharonov-Bohm (AB) flux fields using the wave function ansatz method. The relativistic energy eigenvalues and wave functions are found for any arbitrary state with principalnand magneticmquantum numbers. Further, we obtain the eigensolutions in any dimensional spaceDwithout external fields. We also find the relativistic and nonrelativistic bound states for Coulomb, harmonic oscillator, and Kratzer potentials.

scholarly journals Stationary scalar and vector clouds around Kerr–Newman black holes

2020 ◽  
Vol 29 (11) ◽  
pp. 2041013
Author(s):  
Nuno M. Santos ◽  
Carlos A. R. Herdeiro
Keyword(s):  
Black Holes ◽  
Angular Momentum ◽  
Bound States ◽  
Total Angular Momentum ◽  
Radial Direction ◽  
Three Dimensional ◽  
Proca Equation ◽  
Dimensional Parameter ◽  
Atomic Orbitals ◽  
Quantum Numbers

Massive bosons in the vicinity of Kerr–Newman black holes can form pure bound states when their phase angular velocity fulfills the synchronization condition, i.e. at the threshold of superradiance. The presence of these stationary clouds at the linear level is intimately linked to the existence of Kerr black holes with synchronized hair at the nonlinear level. These configurations are very similar to the atomic orbitals of the electron in a hydrogen atom. They can be labeled by four quantum numbers: [Formula: see text], the number of nodes in the radial direction; [Formula: see text], the orbital angular momentum; [Formula: see text], the total angular momentum; and [Formula: see text], the azimuthal total angular momentum. These synchronized configurations are solely allowed for particular values of the black holes mass, angular momentum and electric charge. Such quantization results in an existence surface in the three-dimensional parameter space of Kerr–Newman black holes. The phenomenology of stationary scalar clouds has been widely addressed over the last years. However, there is a gap in the literature concerning their vector cousins. Following the separability of the Proca equation in Kerr(–Newman) spacetime, this paper explores and compares scalar and vector stationary clouds around Kerr and Kerr–Newman black holes, extending previous research.

Bound States in Three Dimensions: The Atom

2021 ◽  
pp. 82-92
Author(s):  
Duncan G. Steel
Keyword(s):  
Angular Momentum ◽  
Quantum Number ◽  
Bound States ◽  
Laguerre Polynomials ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Spherical Coordinates ◽  
Quantum Numbers ◽  
Energy Plus ◽  
Charged Nucleus

In this chapter, we go to three dimensions in space and look at the solution of the time independent Schrödinger equation for the hydrogen atom. The Hamiltonian is then the kinetic energy plus the potential energy due to the Coulomb coupling between the positively charged nucleus and the electron. We construct the angular momentum operator and find that the partial differential equation for the angular momentum eigenfunctions of the spherical coordinates θ,ϕ is the same as the angular part of the ∇2 operator in spherical coordinates. The angular momentum eigenfunctions are the spherical harmonics, with two quantum numbers, l and m, and the solution to the radial part of the Hamiltonian including the Coulomb potential are Laguerre polynomials with one quantum number, called the principle quantum number, n. The hydrogen wave function is the product of a Laguerre polynomial and a spherical harmonic with three quantum numbers. Since these are two- and three-dimensional functions for angular momentum and hydrogen respectively, they are best understood in a series of plots. The chapter concludes by giving the historical letter names to specific orbitals, since they continue to be used today.

scholarly journals The Genuine Resonance of Full-Charm Tetraquarks

Universe ◽  
2021 ◽  
Vol 7 (5) ◽  
pp. 155
Author(s):  
Xiaoyun Chen
Keyword(s):  
Quark Model ◽  
Bound States ◽  
Expansion Method ◽  
Chiral Quark Model ◽  
Stabilization Method ◽  
Scaling Method ◽  
Resonance States ◽  
Gaussian Expansion Method ◽  
Color Structure ◽  
Quantum Numbers

In this work, the genuine resonance states of full-charm tetraquark systems with quantum numbers JPC=0++,1+−,2++ are searched in a nonrelativistic chiral quark model with the help of the Gaussian Expansion Method. In this calculation, two structures, meson-meson and diquark–antidiquark, as well as their mixing with all possible color-spin configurations, are considered. The results show that no bound states can be formed. However, resonances are possible because of the color structure. The genuine resonances are identified by the stabilization method (real scaling method). Several resonances for the full-charm system are proposed, and some of them are reasonable candidates for the full-charm states recently reported by LHCb.

THE ANGULAR MOMENTUM DEPENDENT CALCULATION OF THE GROUND STATE ENERGY OF LIQUID 3He

2005 ◽  
Vol 19 (30) ◽  
pp. 1793-1802 ◽  
Author(s):  
M. MODARRES
Keyword(s):  
Angular Momentum ◽  
Ground State Energy ◽  
Ground State ◽  
Correlation Functions ◽  
State Energy ◽  
Interatomic Potentials ◽  
Channel Correlation ◽  
P Waves ◽  
Effective Interactions ◽  
Three Body

We investigate the possible angular momentum, l, dependence of the ground state energy of normal liquid 3 He . The method of lowest order constrained variational (LOCV) which includes the three-body cluster energy and normalization constraint (LOCVE) is used with angular momentum dependent two-body correlation functions. A functional minimization is performed with respect to each l-channel correlation function. It is shown that this dependence increases the binding energy of liquid 3 He by 8% with respect to calculations without angular momentum dependent correlation functions. The l=0 state has completely different behavior with respect to other l-channels. It is also found that the main contribution from potential energy comes from the l=1 state (p-waves) and the effect of l≥11 is less than about 0.1%. The effective interactions and two-body correlations in different channels are being discussed. Finally we conclude that this l-dependence can be verified experimentally by looking into the magnetization properties of liquid helium 3 and interatomic potentials.

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