[Paper Review] Dirac operator on the Riemann sphere

А. А. Абрикосов|ArXiv.org|Dec 11, 2002

Algebraic and Geometric Analysis3 references19 citations

TL;DR

This paper provides an explicit, group-theoretic solution to the Dirac operator on the Riemann sphere $S^2$, deriving its spectrum and eigenfunctions as two-component spinors. It shows that eigenvalues are nonzero integers $\lambda$, with corresponding eigenfunctions transforming under $SU(2)$ representations with half-integer angular momentum $l = |\lambda| - \frac{1}{2}$, forming a complete orthonormal basis alternative to standard spherical spinors.

ABSTRACT

We solve for spectrum, obtain explicitly and study group properties of eigenfunctions of Dirac operator on the Riemann sphere $S^2$. The eigenvalues $λ$ are nonzero integers. The eigenfunctions are two-component spinors that belong to representations of SU(2)-group with half-integer angular momenta $l = |λ| - \half$. They form on the sphere a complete orthonormal functional set alternative to conventional spherical spinors. The difference and relationship between the spherical spinors in question and the standard ones are explained.

Motivation & Objective

• To derive the complete spectrum and eigenfunctions of the Dirac operator on the Riemann sphere $S^2$ using a self-contained, classical approach.
• To establish the group-theoretic properties of the eigenfunctions, showing they transform under $SU(2)$ representations with half-integer angular momentum $l = |\lambda| - \frac{1}{2}$.
• To provide an explicit construction of the eigenfunctions in spherical coordinates, using Jacobi polynomials and orthogonal polynomials.
• To clarify the relationship between the newly derived spinors and conventional spherical spinors, resolving discrepancies in transformation behavior and normalization.
• To enable practical application in quantum field theory and condensed matter physics, particularly in systems with chiral symmetry or spherical electron dynamics, such as fullerenes.

Proposed method

• Parameterize the Riemann sphere using standard spherical coordinates $\theta, \phi$, and define the zweibein and spin connection to construct the covariant derivative for spinors.
• Express the Dirac operator in terms of Pauli matrices and the spin connection, leading to a system of coupled first-order differential equations in $\theta$ and $\phi$.
• Separate variables by assuming a $\phi$-dependence of the form $e^{im\phi}$, reducing the problem to a second-order differential equation in $\theta$ solvable via Jacobi polynomials.
• Use the properties of Jacobi polynomials to derive explicit expressions for the eigenfunctions as two-component spinors, with normalization and orthogonality verified.
• Apply $SU(2)$ transformation rules to confirm the spinor eigenfunctions transform under irreducible representations with half-integer $l = |\lambda| - \frac{1}{2}$.
• Compare the new eigenfunctions with standard spherical spinors by transforming them into Cartesian coordinates and deriving the exact transformation relation.

Experimental results

Research questions

• RQ1 What is the complete spectrum of the Dirac operator on the Riemann sphere $S^2$?
• RQ2 How do the eigenfunctions of the Dirac operator on $S^2$ transform under the $SU(2)$ group, and what is their representation content?
• RQ3 What is the precise mathematical relationship between the new eigenfunctions and the conventional spherical spinors used in quantum mechanics?
• RQ4 How do the eigenfunctions behave under time-reversal symmetry, and what is their orthogonality and normalization structure?
• RQ5 In what physical contexts—such as spectral boundary conditions or electron states in fullerenes—can these eigenfunctions serve as a fundamental basis?

Key findings

• The eigenvalues of the Dirac operator on $S^2$ are nonzero integers $\lambda$, with no zero eigenvalues present.
• The eigenfunctions are two-component spinors that form a complete orthonormal basis on the sphere, transforming under $SU(2)$ representations with half-integer angular momentum $l = |\lambda| - \frac{1}{2}$.
• The eigenfunctions are explicitly constructed using Jacobi polynomials, with the radial dependence encoded in $P^{(m, m)}_{l-m}(\cos\theta)$, and exhibit proper normalization and orthogonality.
• The eigenfunctions are related to standard spherical spinors by a specific $SU(2)$-covariant transformation, differing in phase and normalization due to distinct spinor transformation rules.
• The time-reversal operation maps eigenfunctions with eigenvalue $\lambda$ to those with eigenvalue $-\lambda$, preserving the structure of the eigenbasis.
• The eigenfunctions provide an alternative orthonormal basis for half-integer angular momentum states, suitable for problems involving chiral symmetry or electron dynamics on spherical surfaces, such as in $C_{60}$ molecules.