[Paper Review] Fermion doubling problem and noncommutative geometry II

A. P. Balachandran, T. R. Govindarajan+1 more|ArXiv.org|Jun 28, 2000

Advanced Operator Algebra Research4 citations

TL;DR

This paper resolves the fermion doubling problem in lattice field theories using noncommutative geometry via fuzzy spheres and their products, demonstrating that the Ginsparg-Wilson relation emerges naturally in this framework. It shows that Dirac operators on fuzzy spaces preserve chiral symmetry and anomaly structure, offering a finite-dimensional matrix model for chiral gauge theories without fermion doubling.

ABSTRACT

In our previous paper (hep-th/9911087), we proposed a resolution for the fermion doubling problem in discrete field theories based on the fuzzy sphere and its cartesian products. In this paper after a review of that work, we bring out its relationship to the Ginsparg-Wilson approach.

Motivation & Objective

• To resolve the fermion doubling problem in discrete field theories while preserving chiral symmetry.
• To establish a connection between the fuzzy sphere approach and the Ginsparg-Wilson algebraic framework.
• To extend the noncommutative geometric formulation to higher-dimensional manifolds like $S^2 \otimes S^2$ and $\mathbb{C}P^2$.
• To construct finite-dimensional matrix models for quantum fields on quantized adjoint orbits of compact Lie groups.
• To demonstrate that chiral anomalies and topological features are consistently realized in the fuzzy geometry framework.

Proposed method

• Constructs fuzzy spheres $S_F^2$ using $SU(2)$-covariant matrix algebras with noncommutative coordinates $x_i$ satisfying $[x_i, x_j] = i \epsilon_{ijk} x_k / \sqrt{l(l+1)}$.
• Defines Dirac operators $D_1 = \vec{\sigma} \cdot \vec{\cal L} + \mathbf{1}$ and $D_2 = \epsilon_{ijk} \sigma_i \hat{n}_j {\cal J}_k$ on fuzzy $S^2$, where $\vec{\cal L}$ is the orbital angular momentum operator.
• Introduces a chirality operator $\Gamma = \vec{\sigma} \cdot \hat{n}$ that anticommutes with both $D_1$ and $D_2$, ensuring chiral symmetry.
• Uses the total angular momentum $\vec{J} = \vec{\cal L} + \vec{\sigma}/2$ to diagonalize $D_1$, showing its spectrum is $\{ \pm(j + 1/2) \}$ for $j \in \{1/2, 3/2, \dots, 2l - 1/2\}$ and an additional $+(2l + 1/2)$ state.
• Demonstrates that $D_1$ and $D_2$ are unitarily equivalent in the continuum limit and that $|D_1| = |D_2|$ due to rotational invariance.
• Establishes a link to the Ginsparg-Wilson relation by showing that the discrete Dirac operator $D'_1$ satisfies $aD'_1 = 2(\Gamma_1^2 + \Gamma_1\Gamma_2)$, with spectrum lying on a circle in the complex plane.

Experimental results

Research questions

• RQ1 Can the fermion doubling problem be resolved in lattice field theories using noncommutative geometry without breaking chiral symmetry?
• RQ2 How does the fuzzy sphere construction relate to the Ginsparg-Wilson algebraic approach to chiral fermions?
• RQ3 What is the spectrum and chirality structure of Dirac operators on fuzzy $S^2$ and $S^2 \otimes S^2$?
• RQ4 Can fuzzy monopoles and instantons be consistently constructed in this framework using projectors in matrix algebras?
• RQ5 Does the noncommutative geometry approach preserve chiral anomalies and topological invariants in finite-dimensional models?

Key findings

• The Dirac operator $D_1 = \vec{\sigma} \cdot \vec{\cal L} + \mathbf{1}$ on the fuzzy sphere has a spectrum $\{ \pm(j + 1/2) \}$ for $j = 1/2, 3/2, \dots, 2l - 1/2$, plus one additional positive eigenvalue at $j = 2l + 1/2$.
• The chirality operator $\Gamma = \vec{\sigma} \cdot \hat{n}$ anticommutes with both $D_1$ and $D_2$, ensuring chiral symmetry is preserved in the discrete setting.
• The operators $D_1$ and $D_2$ are unitarily equivalent in the continuum limit, with $|D_1| = |D_2|$, and satisfy $\Gamma = i \frac{D_1}{|D_1|} \frac{D_2}{|D_2|}$.
• The Ginsparg-Wilson operator $D'_1$ has a spectrum on each angular momentum subspace $V_j$ given by $1 + \exp(\pm 2i\theta_j)$, lying on a circle of radius 1 centered at 1.
• On the maximal total angular momentum subspace $W$ (with $j = 2l + 1/2$), $D_2$ vanishes and $|D_1|$ reaches its maximum value, indicating a breakdown of chirality in this sector.
• Fuzzy monopoles of charge $\pm N$ are constructed via projectors $p^{(\pm N)}$ acting on $A^{2^N}$, projecting to the highest or lowest $SU(2)$ representation, with $\vec{K} = \vec{L}^L + \sum_i \vec{\tau}^{(i)}/2$ generating the total angular momentum algebra.