[Paper Review] Clifford bundle formulation of BF gravity generalized to the standard model

A. Garrett Lisi|ArXiv.org|Nov 21, 2005

Noncommutative and Quantum Gravity Theories6 references3 citations

TL;DR

This paper proposes a unification of gravity and the Standard Model using a Clifford bundle formulation, where the gravitational vierbein and spin connection, along with gauge and Higgs fields, are unified into a single Clifford-algebra-valued connection. The resulting BRST-extended curvature yields a restricted BF action that reproduces the dynamics of both gravity and the Standard Model in a geometric, gauge-invariant framework.

ABSTRACT

The structure and dynamics of the standard model and gravity are described by a Clifford valued connection and its curvature.

Motivation & Objective

• To unify gravity and the Standard Model using a geometric framework based on Clifford algebras and fiber bundles.
• To reformulate General Relativity and the Standard Model fields as components of a single Clifford-valued connection.
• To incorporate fermions and gauge symmetries through BRST methods within the Clifford bundle formalism.
• To demonstrate that the dynamics of both gravity and the Standard Model emerge from a single restricted BF action on a Clifford bundle.
• To provide a minimal, geometrically consistent framework for quantum gravity and particle physics unification, avoiding excessive mathematical abstraction.

Proposed method

• Formulate gravity using a Clifford bundle with fiber algebra $\mathbb{C}l_{1,3}$, unifying the vierbein $e$ and spin connection $\omega$ into a single connection $\Omega = e + \omega$.
• Extend the connection to include Standard Model gauge fields $Z$ and Higgs fields $\phi$, forming $A = \phi e + \omega + Z$.
• Introduce fermions via BRST methods, embedding them as anti-commuting Clifford elements $\psi$ in an extended connection $\tilde{A} = \phi e + \omega + Z + \psi$.
• Construct the curvature $\tilde{F} = d\tilde{A} + \tilde{A} \cdot \tilde{A}$ using the Clifford algebra product and graded Lie bracket.
• Apply a restricted BF action $\mathcal{L} \propto \tilde{F} \cdot \tilde{F}$ to derive the full dynamics of gravity and the Standard Model.
• Use matrix representations of $\mathbb{C}l_{8}$ to describe the gauge and fermionic content, with eigenvalues and eigenvectors labeling particle charges and states.

Experimental results

Research questions

• RQ1 Can gravity and the Standard Model be described using a single geometric object—a Clifford-algebra-valued connection—on a fiber bundle?
• RQ2 How can fermions and their gauge interactions emerge naturally from a BRST-extended connection in a Clifford bundle framework?
• RQ3 Can the dynamics of both gravity and the Standard Model be derived from a single restricted BF action on a Clifford bundle?
• RQ4 What is the role of the Higgs field and mass generation in this geometric unification scheme?
• RQ5 Is it possible to interpret the full connection as a Cartan connection or Kaluza-Klein-like structure in higher-dimensional geometry?

Key findings

• The gravitational vierbein and spin connection are unified into a single Clifford-valued connection $\Omega = e + \omega$ using the MacDowell-Mansouri approach.
• The inclusion of Standard Model gauge fields and Higgs fields into a Clifford bundle connection $A = \phi e + \omega + Z$ allows for a geometric unification of all bosonic fields.
• Fermions emerge naturally as anti-commuting components $\psi$ in a BRST-extended connection $\tilde{A} = A + \psi$, with gauge symmetry acting from left and right.
• The curvature $\tilde{F}$ of the BRST-extended connection generates a restricted BF Lagrangian that reproduces the dynamics of both gravity and the Standard Model.
• The matrix representation of $\mathbb{C}l_8$ provides a framework for labeling fermion generations and charges via eigenvalues of gauge bivectors.
• The model suggests a potential geometric origin for particle masses through the structure of $\mathbb{C}l_8$, though this remains speculative and untested.