[Paper Review] Particular boundary condition ensures that a fermion in d=1+5, compactified on a finite disk, manifests in d=1+3 as massless spinor with a charge 1/2, mass protected and chirally coupled to the gauge field

N. S. Mankoč Borštnik, H. B. Nielsen|arXiv (Cornell University)|Oct 10, 2007

Algebraic and Geometric Analysis4 citations

TL;DR

This paper proposes a specific boundary condition in a (1+5)-dimensional Kaluza-Klein model compactified on a finite disk, ensuring that a single massless spinor with charge 1/2 emerges in 4D spacetime, protected from mass generation and chirally coupled to the gauge field. The mechanism selects one chiral state and one charge from a higher-dimensional spectrum, resolving the issue of unwanted massless states and double counting in compactification.

ABSTRACT

The genuine Kaluza-Klein-like theories--with no fields in addition to gravity--have difficulties with the existence of massless spinors after the compactification of some space dimensions \cite{witten}. We proposed in previous paper a boundary condition for spinors in d=(1+5) compactified on a flat disk that ensures masslessness of spinors (with all positive half integer charges) in d=(1+3) as well as their chiral coupling to the corresponding background gauge gravitational field. In this paper we study the same toy model, proposing a boundary condition allowing a massless spinor of one handedness and only one charge (1/2) and infinitely many massive spinors of the same charge, allowing disc to be curved. We define the operator of momentum to be Hermitean on the vector space of spinor states--the solutions on a disc with the boundary.

Motivation & Objective

• To resolve the problem of massless spinor doubling and lack of chiral coupling in standard Kaluza-Klein compactifications of higher-dimensional gravity-only theories.
• To ensure that only one chiral fermion state (with charge 1/2) survives after compactification on a finite disk in d=1+5.
• To establish a mass protection mechanism for the emergent 4D fermion, preventing radiative corrections from generating mass.
• To define a Hermitean momentum operator on the spinor Hilbert space constrained by the boundary condition.
• To generalize the model to curved compactified dimensions while preserving the desired physical properties of the zero-mode fermion.

Proposed method

• Introduces a covariant boundary condition using the operator $\hat{\cal R} = \frac{1}{2}(1 - i n^{(\rho)}_a n^{(\phi)}_b \gamma^a \gamma^b)$, which projects spinor states at the boundary of the compactified disk.
• Applies the boundary condition $\hat{\cal R} \psi|_{\text{wall}} = 0$ to select only one chiral state (left-handed) and one charge (1/2) from the higher-dimensional spectrum.
• Constructs the momentum operator $p_{0a} = f^\alpha_a p_{0\alpha}$ with $p_{0\alpha} = p_\alpha - \frac{1}{2} S^{cd} \omega_{cda}$, ensuring Hermiteicity on the spinor state space.
• Derives the Lagrangian density $\mathcal{L} = \psi^\dagger \gamma^0 \gamma^a \left[ E(p_a - \frac{1}{2} S^{cd} \omega_{cda}) + \frac{1}{2} \{p_\alpha, E f^\alpha_a \}_{-} \right] \psi$ to describe the dynamics of the spinor in curved background.
• Considers the case of a curved compactified disk by allowing $f^{\sigma}_s = \delta^\sigma_s f(\rho)$, generalizing the flat-space result.
• Demonstrates that the current through the boundary vanishes for both massless and massive modes, ensuring consistency of the boundary condition.

Experimental results

Research questions

• RQ1 How can a single chiral fermion with charge 1/2 be selected from the spectrum of a (1+5)-dimensional Weyl spinor after compactification on a finite disk?
• RQ2 What boundary condition ensures that the zero-mode fermion remains massless and protected from quantum corrections in a Kaluza-Klein-like theory?
• RQ3 Can the momentum operator be defined as Hermitean on the Hilbert space of spinor states constrained by a non-trivial boundary condition?
• RQ4 How does the chiral coupling of the emergent 4D fermion to the gauge field arise from the compactification procedure?
• RQ5 Is it possible to generalize the model to curved compactified dimensions while preserving the massless, chiral, and charge-protected properties of the zero mode?

Key findings

• The proposed boundary condition $\hat{\cal R} \psi|_{\text{wall}} = 0$ selects exactly one chiral fermion state (left-handed) and one charge (1/2) from the (1+5)-dimensional spectrum.
• The zero-mode fermion is massless and remains protected from mass generation due to the topological and algebraic structure of the boundary condition.
• The fermion couples chirally to the gauge field in 4D, as required by the Standard Model, without introducing additional chiral states.
• The momentum operator defined on the spinor Hilbert space is Hermitean, ensuring unitary time evolution and consistent quantization.
• The current through the boundary vanishes for all solutions (massless and massive), confirming the consistency of the boundary condition.
• The model generalizes to curved compactified dimensions, allowing the disk to be non-flat while preserving the key physical properties of the zero mode.