Quantum Kaluza-Klein theory with $M_2(\mathbb{C})$

Abstract

Following steps analogous to classical Kaluza-Klein theory, we solve for the quantum Riemannian geometry on $C^{\infty }(M)\otimes M_2(\mathbb{C})$ in terms of classical Riemannian geometry on a smooth manifold $M$, a finite quantum geometry on the algebra $M_2(\mathbb{C})$ of $2\times 2$ matrices, and a quantum metric cross term. Fixing a standard form of quantum metric on $M_2(\mathbb{C})$, we show that this cross term data amounts in the simplest case to a 1-form $A_{\mu }$ on $M$, which we regard as like a gauge-fixed background field. We show in this case that a real scalar field on the product algebra with its noncommutative Laplacian decomposes on $M$ into two real neutral fields and one complex charged field minimally coupled to $A_{\mu }$. We show further that the quantum Ricci scalar on the product decomposes into a classical Ricci scalar on $M$, the Ricci scalar on $M_2(\mathbb{C})$, the Maxwell action $||F|{|}^2$ of $A$ and a higher order $||A.F|{|}^2$ term. Another solution of the QRG on the product has $A=0$ and a dynamical real scalar field $\varphi$ on $M$ which imparts mass-splitting to some of the components of a scalar field on the product as in previous work.

References in corpus (19)

• The quantum structure of spacetime at the Planck scale and quantum fields
• Gravity coupled with matter and foundation of non-commutative geometry
• Kaluza-Klein Gravity
• Bicrossproduct structure of $\kappa$-Poincare group and non-commutative geometry
• Gravity and the standard model with neutrino mixing
• Quantization of Point Particles in 2+1 Dimensional Gravity and Space-Time Discreteness
• Connections on central bimodules
• Linear Connections in Non-Commutative Geometry
• Gravity induced from quantum spacetime
• Equivariant Lorentzian Spectral Triples
• Lorentz signature and twisted spectral triples
• Quantum gravity on a square graph
• Quantum Riemannian geometry and particle creation on the integer line
• Quantum geodesic flows and curvature
• Quantum geodesics on $\lambda$-Minkowski spacetime
• Quantum gravity on polygons and $\mathbb{R}\times {\mathbb{Z}}_n$ FLRW model
• Fuzzy and discrete black hole models
• Fermion masses, mass-mixing and the almost commutative geometry of the Standard Model
• Quantum Riemannian geometry of the discrete interval and q-deformation

Cited by in corpus (1)

• Yang-Mills field from fuzzy sphere quantum Kaluza-Klein model