[Paper Review] Analytic eigenenergies of the Dirac equation with finite degrees of freedom under a confining linear potential using basis functions localized in spacetime

Kimichika Fukushima, Hikaru Sato|arXiv (Cornell University)|Jan 20, 2015

Black Holes and Theoretical Physics2 references4 citations

TL;DR

This paper presents an analytic solution for the eigenenergies of a confined fermion-antifermion pair under a linear potential derived from the Wilson loop in non-Abelian Yang-Mills theory. Using basis functions localized in spacetime, the Dirac equation's Hamiltonian matrix is analytically diagonalized, yielding eigenenergies proportional to the string tension and the absolute value of Dirac’s relativistic quantum number, consistent with Regge trajectory behavior in the relativistic regime.

ABSTRACT

Considering the propagation of fields in the spacetime continuum and the well-defined features of fields with finite degrees of freedom, the wave function is expanded in terms of a finite set of basis functions localized in spacetime. This paper presents the analytic eigenenergies derived for a confined fundamental fermion-antifermion pair under a linear potential obtained from the Wilson loop for the non-Abelian Yang-Mills field. The Hamiltonian matrix of the Dirac equation is analytically diagonalized using basis functions localized in spacetime. The squared lowest eigenenergy (as a function of the relativistic quantum number when the rotational energy is large compared to the composite particle masses) is proportional to the string tension and the absolute value of the Dirac's relativistic quantum number related to the total angular momentum, consistent with the expectation.

Motivation & Objective

• To derive analytic eigenenergies for a confined fundamental fermion-antifermion pair under a linear potential arising from the Wilson loop in non-Abelian Yang-Mills theory.
• To develop a formalism using basis functions localized in spacetime that enables analytical treatment without numerical simulation.
• To clarify the origin of the mass of composite fermion-antifermion states and explain why their binding energy can exceed the constituent masses.
• To establish a direct link between the relativistic quantum number in the Dirac equation and the Regge trajectory's scaling behavior.

Proposed method

• The Dirac equation in spherical coordinates is formulated using a basis set of step functions localized in spacetime, enabling exact differentiation and analytical treatment.
• The total Hamiltonian is constructed variationally from the action, expressed in terms of these localized basis functions.
• The Hamiltonian matrix is diagonalized analytically through two sequential unitary transformations, preserving the structure of the secular equation.
• The relativistic quantum number $\kappa$, related to total angular momentum, is used as a key quantum label in the eigenvalue derivation.
• The formalism incorporates the string tension $\sigma$ as a parameter from the Wilson loop, directly linking it to the potential energy.
• The quenched approximation (OZI rule) is applied to suppress additional pair creation, simplifying the Green’s function formalism.

Experimental results

Research questions

• RQ1 How can the eigenenergies of a confined fermion-antifermion pair under a linear potential be derived analytically without numerical methods?
• RQ2 What is the role of Dirac’s relativistic quantum number $\kappa$ in determining the energy spectrum of the bound state?
• RQ3 Why can the binding energy of a fermion-antifermion pair be significantly larger than the individual fermion masses?
• RQ4 How does the analytical solution reproduce the Regge trajectory scaling behavior in the relativistic regime?
• RQ5 What is the connection between the string tension from Yang-Mills theory and the observed energy levels in the Dirac equation?

Key findings

• The lowest eigenenergy of the confined fermion-antifermion pair is analytically derived as a function of the string tension $\sigma$ and the absolute value of Dirac’s relativistic quantum number $\kappa$, with $E^2 \propto \sigma |\kappa|$.
• The squared eigenenergy scales linearly with both the string tension and the magnitude of the relativistic quantum number, consistent with the classical Regge trajectory behavior.
• The analytical diagonalization of the Hamiltonian matrix is achieved via two sequential unitary transformations, preserving the physical structure of the secular equation.
• The mechanism for large binding energy relative to constituent masses is traced to the interplay between the linear confining potential and the relativistic angular momentum term.
• The Polyakov line analysis shows deconfinement at high temperatures, with the binding energy $E_B = \sigma r$ scaling as $\epsilon_q \approx \sigma r / (k_B T)$, indicating thermal suppression of confinement.
• The Green’s function formalism in the quenched approximation confirms that the eigenenergies appear as decay constants in Euclidean time, validating the physical consistency of the solutions.