[Paper Review] Dirac-type tensor equations with non-Abelian gauge symmetries on pseudo-Riemannian space

Н. Г. Марчук|arXiv (Cornell University)|May 15, 2002

Cosmology and Gravitation Theories2 references4 citations

TL;DR

This paper introduces a Dirac-type tensor equation with non-Abelian gauge symmetries on pseudo-Riemannian spacetime, generalizing the spinor Dirac equation to include unitary gauge groups such as U(1), U(1)×SU(2), and SU(3). The key contribution is the explicit construction of the tensor $ B_{ u} $ in terms of the metric and its first derivatives, enabling a geometric formulation of non-Abelian gauge theories in curved spacetime while preserving gauge invariance and charge conservation.

ABSTRACT

We suggest a so-called Dirac type tensor equation with nonabelian gauge symmetry on pseudo-Riemannian space. This equation reproduce some of the properties of spinor Dirac equation. A geometrical interpretation of results in terms of Riemannian geometry is given.

Motivation & Objective

• To generalize the Dirac-type tensor equation to include non-Abelian gauge symmetries, extending previous work in Minkowski space to curved pseudo-Riemannian spacetime.
• To incorporate unitary gauge groups relevant to the Standard Model, such as U(1), U(1)×SU(2), and SU(3), into a geometric framework compatible with general relativity.
• To establish a consistent formulation of gauge invariance and charge conservation laws in curved spacetime using differential forms and Clifford algebraic structures.
• To provide a geometric interpretation of the results in terms of Riemannian geometry, linking gauge symmetry to the underlying spacetime structure.

Proposed method

• The paper employs differential forms and the Hodge star operator on a four-dimensional pseudo-Riemannian manifold with signature -2, using complex-valued forms to represent wave functions.
• It introduces a Clifford product and a bilinear operation Com on 2-forms to define algebraic structures necessary for gauge-covariant derivatives.
• The tensor $ B_{ u} $ is derived explicitly from the metric $ g_{ u au} $ and its first derivatives, enabling the construction of gauge-covariant derivatives in curved space.
• The method generalizes previous results from Minkowski space to pseudo-Riemannian geometry by using a systematic approach based on the tensor $ B_{ u} $, with explicit formulas provided for general and special cases (e.g., temporal gauge, diagonal metric).
• Lagrangians are constructed to ensure gauge invariance and derive field equations, with conservation laws for non-Abelian charges derived from Noether's theorem.
• A geometric interpretation is developed via Riemannian geometry, showing how gauge symmetries emerge from the spacetime metric and its connection to differential forms.

Experimental results

Research questions

• RQ1 How can the Dirac-type tensor equation be generalized to include non-Abelian gauge symmetries on a pseudo-Riemannian spacetime?
• RQ2 What is the explicit form of the tensor $ B_{ u} $ in terms of the metric and its first derivatives, enabling gauge-covariant derivatives in curved space?
• RQ3 How do non-Abelian charge conservation laws emerge in this geometric framework, and are they consistent with the Standard Model gauge groups?
• RQ4 Can the proposed formalism reproduce key features of the spinor Dirac equation while extending to SU(3) and other non-Abelian groups?
• RQ5 What is the geometric meaning of the gauge symmetry in terms of Riemannian and differential form structures on curved spacetime?

Key findings

• The tensor $ B_{ u} $ is explicitly expressed in terms of the metric $ g_{ u au} $ and its first derivatives, with closed-form expressions derived for general, temporal gauge, and diagonal metric cases.
• For the SU(3) gauge group, the Dirac-type tensor equation has 16 complex components, differing from the 12-component chromospinor Dirac equation, indicating a distinct system not equivalent to the standard model's fermion description.
• The equation preserves non-Abelian gauge symmetry and gives rise to consistent charge conservation laws, generalizing the Abelian case to non-Abelian groups like SU(2) and SU(3).
• In the temporal gauge and diagonal metric cases, the components of $ B_{ u} $ simplify to $ b_{ u au au} = -\frac{1}{2}\partial_{[ u}g_{ au]\tau} $, providing a compact and geometrically meaningful expression.
• The Lagrangian formulation ensures gauge invariance and leads to field equations consistent with the geometric structure of the pseudo-Riemannian manifold.
• A geometric interpretation is established via Riemannian geometry, showing that the gauge symmetry and field equations arise naturally from the metric and differential form structure of spacetime.