[Paper Review] Kaluza-Klein Aspects of Noncommutative Geometry

J. Madore|arXiv (Cornell University)|Oct 13, 2015

Noncommutative and Quantum Gravity Theories4 references9 citations

TL;DR

This paper proposes a noncommutative geometry framework that replaces extra compactified dimensions in Kaluza-Klein theory with an algebraic internal structure via matrix algebras $M_n$, unifying gravity and gauge fields. By replacing the algebra of functions on spacetime with a noncommutative algebra, it generates emergent spin and isospin degrees of freedom, yielding a Yang-Mills-Higgs model with finite spectral dimension and renormalizable structure, where the Higgs potential arises from curvature of the internal connection.

ABSTRACT

Using some elementary methods from noncommutative geometry a structure is given to a point of space-time which is different from and simpler than that which would come from extra dimensions. The structure is described by a supplementary factor in the algebra which in noncommutative geometry replaces the algebra of functions. Using different examples of algebras it is shown that the extra structure can be used to describe spin or isospin.

Motivation & Objective

• To generalize Kaluza-Klein theory by replacing extra compactified dimensions with a noncommutative algebraic structure instead of a manifold.
• To show that internal degrees of freedom such as spin and isospin can emerge from the algebraic structure of matrix algebras $M_n$ in noncommutative geometry.
• To construct a unified gauge theory of gravity and Yang-Mills fields using a noncommutative spacetime algebra, preserving 4D momentum space and renormalizability.
• To demonstrate that the Higgs potential arises naturally from the curvature of a connection on the noncommutative factor, without introducing external scalar fields.

Proposed method

• Replace the algebra of functions on spacetime $\mathbb{R}^4$ with a noncommutative algebra $\mathcal{A} = \mathcal{C} \otimes M_n$, where $M_n$ is the algebra of $n \times n$ complex matrices.
• Use the differential calculus of noncommutative geometry to define connections and curvature forms on the algebra $\mathcal{A}$, generalizing the Levi-Civita and gauge connections.
• Introduce a noncommutative connection with torsion via $\tilde{d}\tilde{\theta}^i + \tilde{\omega}^i{}_j \wedge \tilde{\theta}^j = \tilde{\Theta}^i$, where $\tilde{\Theta}^a = -\kappa(D\phi - \phi^2)$.
• Construct the action as $S = \frac{1}{4} \operatorname{Tr} \int \Omega_{ij} \Omega^{ij}$, with curvature $\Omega_{ij}$ from the connection $\tilde{\omega}$, leading to a Yang-Mills-Higgs theory.
• Define the Higgs potential as $V(\phi) = -\frac{1}{4} \operatorname{Tr}(\Omega_{ab} \Omega^{ab})$, which is a quartic polynomial in $\phi_a$ arising from the internal curvature.
• Use the trace over $M_n$ to replace integration over the internal manifold, ensuring finite-dimensional momentum space and renormalizability.

Experimental results

Research questions

• RQ1 Can the internal degrees of freedom such as spin or isospin be encoded algebraically in a noncommutative geometry framework without postulating extra dimensions?
• RQ2 How does replacing the internal manifold $F$ in Kaluza-Klein theory with a noncommutative algebra $M_n$ affect the structure of gauge and gravitational fields?
• RQ3 Can the Higgs mechanism and mass generation arise naturally from the curvature of a noncommutative connection, rather than from an ad hoc scalar sector?
• RQ4 What are the implications for renormalizability and the spectrum of the theory when the internal space is replaced by a finite-dimensional matrix algebra?
• RQ5 How does the noncommutative structure affect the definition of spinors and the Dirac operator in the model?

Key findings

• The Higgs potential $V(\phi)$ arises naturally from the curvature of the noncommutative connection, specifically as $V(\phi) = -\frac{1}{4} \operatorname{Tr}(\Omega_{ab} \Omega^{ab})$, a quartic polynomial in $\phi_a$ with no free parameters beyond the mass scale $m$.
• In the symmetric phase ($\phi_a = 0$), the Higgs scalar modes have mass $m_H^2 = n m^2$, with all masses degenerate and real, indicating a stable minimum.
• In the broken phase ($\phi_a = m \lambda_a$), the gauge bosons acquire mass $m_A^2 = 2n m^2$, showing spontaneous symmetry breaking with a finite mass gap.
• The model yields a finite-dimensional momentum space due to the finite-dimensional space of derivations of $M_n$, ensuring renormalizability and avoiding the infinite tower of Kaluza-Klein modes.
• For $n=2$, the model describes a neutral, mass-degenerate vector boson sector without a Weinberg angle, differing from the Standard Model's electroweak sector.
• The fermionic sector couples universally to the gauge bosons in the broken phase, with the interaction Lagrangian $\mathcal{L}_I = \operatorname{Tr}(\bar{\psi} \gamma^\alpha A_\alpha \psi)$, indicating coupling to all fermions including the massless one.