Particles before symmetry

TL;DR

The paper develops a geometry-first reformulation of gauge theory, showing how the Standard Model’s Higgs and Yukawa mechanisms can be recast without postulating fundamental symmetry at the ground level. It contrasts the conventional symmetry-first (PFB-POV) viewpoint with a geometry-first (VB-POV) framework where a small set of fundamental vector bundles generate all fields via tensor operations, and the gauge group arises from automorphisms of these bundles. The Higgs mechanism is reinterpreted as a mass effect arising from the extrinsic curvature associated with a background Higgs direction, while Yukawa couplings become natural fiberwise contractions within the tensor algebra, clarifying their geometric meaning and the origin of generation mixing via the CKM matrix. The paper argues that restricting gauge structure to automorphism groups of fundamental bundles tightens the link between geometry and physics, excludes many slack models, and yields concrete, testable explanations (e.g., charge quantisation) in a principled, ontology-driven way.

Abstract

The Standard Model of particle physics is usually cast in symmetry-first terms. On this approach, one begins with a symmetry group and postulates matter fields as objects transforming under its representations, without requiring that the group be grounded in, or derived from, independent geometric structures. Recently, a geometry-first formulation has been proposed, in which the relevant symmetries are not fundamental. In this paper I extend this approach to two central mechanisms of the Standard Model: spontaneous symmetry breaking and the Yukawa coupling, both essential for particles to acquire mass. These reformulations offer alternative explanations cast in purely geometric terms. In particular, the quantisation of charge arises here as a purely geometric consequence of the tensorial construction of matter fields from the fundamental bundles -- a mechanism that is both more general and more transparent than the usual topological account based on the compactness of symmetry groups. More generally, I argue that a symmetry-first account in terms of principal and associated bundles admits a genuine geometry-first counterpart only under certain strict conditions.

Figures (2)

• Figure 1: The principal $G$-bundle , with structure group $G$, over the manifold $M$, with a principal connection $\varpi$ (a $\mathfrak{g}$-valued one-form on $P$), abbreviated by $P(M,G,\omega)$, and its associated vector bundles $E_i:=P\times_{\rho_i}V_i$, where $\rho_i: G\rightarrow V_i$ is a representation of the Lie group---determined by a particle's quantum numbers---onto the vector space representing the typical fibre, $V_i$ which is linearly isomorphic to $\pi_i^{-1}(x)$, for $x\in M$ and $\pi_i:E\rightarrow M$ the projection of the vector bundle onto its base space (spacetime). The covariant derivatives $\nabla_i$ are the ones induced by $\varpi$, as per Equation \ref{['eq:PFB_cov']}. See Appendix \ref{['app:PFB']} for more details.
• Figure 2: An example of the kinds of bundles that can be formed from a fundamental vector bundle. Each such tensor product---which could include arbitrary symmetrisations---inherits an affine structure from $\nabla$, and each corresponds to a representation of $\mathsf{Aut}(V)$. In the VB-POV, particle types must arise through this kind of structure.

Theorems & Definitions (7)

• Definition 1: Principal fibre Bundle
• Definition 2: An principal connection-form
• Definition 3: Associated Vector Bundle
• Definition 4: Vector Bundle
• Definition 5: A section of $E$
• Definition 6: Parallel transport in a vector bundle
• Definition 7: Curvature