[Paper Review] Spacetime as the Manifold of the Internal Symmetry Orbits in the External Symmetries

Heinrich Saller|ArXiv.org|Mar 7, 2001

Astro and Planetary Science2 references3 citations

TL;DR

This paper proposes that spacetime arises as the manifold of orbits from internal symmetry groups—specifically hypercharge-isospin (U(2))—acting on the external Lorentz group (SL(ℂ²)). By interpreting spacetime coordinates as parameters of induced representations from internal to external symmetries, the framework unifies particle states and interactions through a geometric realization of the Higgs mechanism and Lorentz boosts, with standard model fields as transmutators mapping internal and external symmetry representations.

ABSTRACT

Interactions and particles in the standard model are characterized by the action of internal and external symmetry groups. The four symmetry regimes involved are related to each other in the context of induced group representations. In addition to Wigner's induced representations of external Poincaré group operations, parametrized by energy-momenta, and the induced internal hyperisospin representations, parametrized by the standard model Higgs field, the external operations, including the Lorentz group, can be considered to be induced also by representations of the internal hypercharge-isospin group. In such an interpretation nonlinear spacetime is parametrized by the orbits of the internal action group in the external action group.

Motivation & Objective

• To establish a geometric and group-theoretic foundation for spacetime as the orbit space of internal symmetry groups acting on external symmetries.
• To unify the description of particles and interactions in the Standard Model through induced representations of internal (U(2)) and external (SL(ℂ²)) symmetry groups.
• To demonstrate that spacetime coordinates, momenta, and Higgs fields emerge as parameters of induced representations linking internal and external symmetry regimes.
• To show that standard model fields (Weyl, Dirac, gauge) act as transmutators between internal and external symmetry representations.
• To provide a new operational triunit framework linking Higgs, Weyl, and Pauli transmutators via coset spaces and induced group actions.

Proposed method

• Uses induced unitary representations of the Poincaré group and internal hyperisospin groups (U(2)) to parametrize spacetime as the orbit space of internal group actions on external symmetry groups.
• Applies Wigner's method of induced representations to both external (Poincaré) and internal (U(2)) symmetries, with energy-momenta (q) and Higgs fields (Φ) as group parameterization variables.
• Introduces transmutators as field mappings from coset tangent spaces (e.g., SL(ℂ²)/SU(2)) to tensor products of internal and external representation spaces (D[U] ⊗ T[G]).
• Constructs three fundamental transmutators: the Weyl field (L), right-handed Weyl field (R), and gauge field (A), each transforming under specific internal-external symmetry pairs.
• Parametrizes the transition from particle states (with U(1) and SO(2) symmetries) to interactions (with U(2) and SL(ℂ²)) via Higgs degrees of freedom (Φ/M) and momentum cosets (q/m).
• Defines an operational triunit structure linking Higgs (U(1) × U(2)/U(1)₊), Weyl (SU(2) × SL(ℂ²)/SU(2)), and Pauli (SO(2) × SU(2)/SO(2)) transmutators through coset representations.

Experimental results

Research questions

• RQ1 How can spacetime be geometrically derived from the orbits of internal symmetry groups acting on external symmetries?
• RQ2 What is the role of the Higgs field in parametrizing the transition from internal particle symmetries to external interaction symmetries?
• RQ3 How do standard model fields like Weyl spinors and gauge bosons emerge as mappings between internal and external symmetry representations?
• RQ4 Can the Lorentz group and spacetime coordinates be induced from internal hyperisospin (U(2)) representations via induced representation theory?
• RQ5 What is the significance of the operational triunit structure in unifying particle and interaction symmetries?

Key findings

• Spacetime is realized as the orbit manifold of the internal U(2) group action on the external SL(ℂ²) Lorentz group, with spacetime coordinates x≻ parametrizing the induced representation from internal to external symmetries.
• The Higgs field Φ/M parametrizes the coset space U(2)/U(1)₊, enabling the transition from internal particle symmetries (U(1)) to interaction symmetries (U(2)) via induced representations.
• The Weyl field L(x) is a transmutator mapping the coset SL(ℂ²)/SU(2) (boost manifold) to the tensor product space ℂ²⊗ℂ², transforming under U(2) × UL(ℂ²) with a faithful action.
• The right-handed Weyl field R(x) and the gauge field A(x) are transmutators with U(1) × UL(ℂ²) and SO(3) × SO₀(1,3) symmetry actions, respectively, each with complete internal-external representation spaces.
• The Pauli transmutator u(𝐪/|𝐪|) parametrizes the direction of momentum via the coset SU(2)/SO(2), linking spin quantization to spatial orientation in momentum space.
• The full framework realizes a triunit structure: Higgs (U(1) × U(2)/U(1)₊), Weyl (SU(2) × SL(ℂ²)/SU(2)), and Pauli (SO(2) × SU(2)/SO(2)) transmutators, each encoding a distinct symmetry transition in the Standard Model.