[Paper Review] The External-Internal Group Quotient Structure for the Standard Model in Analogy to General Relativity

Heinrich Saller|ArXiv.org|May 11, 1998

Particle physics theoretical and experimental studies5 references4 citations

TL;DR

This paper proposes a geometric unification framework for the Standard Model by drawing an analogy to general relativity's $ frac{ ext{GL}(b R^4)}{ ext{O}(1,3)}$ structure, introducing $ frac{ ext{GL}(b C^2)}{ ext{U}(2)}$ as a coset space for internal symmetries. It reinterprets leptons, quarks, Higgs, and gauge fields as first-order terms in a flat spacetime expansion of a fundamental field, analogous to the tetrad, offering a novel geometric foundation for the Standard Model's gauge structure.

ABSTRACT

In analogy to the class structure $\GL(\R^4)/Ø(1,3)$ for general relativity with a local Lorentz group as stabilizer and a basic tetrad field for the parametrization, a corresponding class structure $\GL(\C^2)/\U(2)$ is investigated for the standard model with a local hyperisospin group $\U(2)$. The lepton, quark, Higgs and gauge fields, used in the standard model, cannot be basic in a coset interpretation, they may to be taken as first order terms in a flat spacetime, particle oriented expansion of a basic field (as the analogue to the tetrad) and its products.

Motivation & Objective

• To develop a geometric unification of the Standard Model's internal symmetries by analogy with general relativity's spacetime geometry.
• To address the lack of a fundamental field underlying the Standard Model's gauge, Higgs, and matter fields.
• To propose a coset structure $ frac{ ext{GL}(b C^2)}{ ext{U}(2)}$ as a geometric foundation for the Standard Model's internal space.
• To reinterpret known fields (leptons, quarks, Higgs, gauge bosons) as first-order terms in an expansion of a basic field, analogous to the tetrad in general relativity.
• To establish a formalism where the Standard Model's gauge structure emerges from a deeper geometric principle, akin to the role of the tetrad in gravity.

Proposed method

• Adopt the coset space $ frac{ ext{GL}(b C^2)}{ ext{U}(2)}$ as the geometric framework for the Standard Model's internal symmetries, analogous to $ frac{ ext{GL}(b R^4)}{ ext{O}(1,3)}$ in general relativity.
• Introduce a fundamental field, analogous to the tetrad in general relativity, as the basic geometric object in the internal space.
• Expand the fundamental field in a flat spacetime, particle-oriented series, where the first-order terms correspond to the known Standard Model fields.
• Use the local $ ext{U}(2)$ group as the stabilizer of the coset, analogous to the Lorentz group in general relativity.
• Derive the transformation properties of leptons, quarks, Higgs, and gauge fields from the expansion of the fundamental field under the $ ext{GL}(b C^2)$ group.
• Apply the formalism to reconstruct the Standard Model's gauge structure from geometric principles, treating all fields as derived from a single geometric source.

Experimental results

Research questions

• RQ1 Can the Standard Model's internal gauge symmetry structure be derived from a geometric coset space analogous to general relativity's spacetime geometry?
• RQ2 What is the role of a fundamental field in the internal space, analogous to the tetrad in general relativity, in generating the known fields of the Standard Model?
• RQ3 How do the lepton, quark, Higgs, and gauge fields emerge as first-order terms in an expansion of a basic field in a flat spacetime limit?
• RQ4 What is the significance of the $ frac{ ext{GL}(b C^2)}{ ext{U}(2)}$ coset structure in unifying the Standard Model's internal symmetries?
• RQ5 Can the Standard Model's field content and gauge structure be reconstructed from a single geometric principle, rather than postulated?

Key findings

• The coset space $ frac{ ext{GL}(b C^2)}{ ext{U}(2)}$ is proposed as the geometric foundation for the Standard Model's internal symmetries, analogous to $ frac{ ext{GL}(b R^4)}{ ext{O}(1,3)}$ in general relativity.
• The known fields of the Standard Model—leptons, quarks, Higgs, and gauge bosons—are interpreted as first-order terms in a flat spacetime expansion of a fundamental field, analogous to the tetrad in gravity.
• The local $ ext{U}(2)$ group acts as the stabilizer of the coset, playing a role analogous to the Lorentz group in general relativity.
• The formalism provides a geometric origin for the Standard Model's gauge structure, suggesting that all fields arise from a single fundamental field through a series expansion.
• The framework offers a unified geometric perspective where the Standard Model's internal space is treated on the same footing as spacetime in general relativity, with the same mathematical structure.