Operational Spacetime pp 317-331 | Cite as

# Masses and Coupling Constants

## Abstract

The nonrelativistic hydrogen atom and the atomic spectrum characterize cyclic Hilbert representations for the analysis of hyperbolic position *Y* ^{3}, i.e., of the homogeneous space **SO** _{0}(1, 3)/**SO**(3) for rotation relativity. The particle spectrum is proposed to arise in an analysis of electroweak spacetime \( \mathcal{D}^4 \) as the homogeneous space for unitary relativity \( {\bf D}(1) \times y^3 \cong {\bf GL}(\mathbb{C}^2)/ {\bf U}(2) \) i.e., for position *Y* ^{3} with additional causal (dilation) operations **D**(1).

The particle spectrum \( \left( {m^2 , J, z} \right) \in \mathbb{R}_+ \times \frac{{\rm N}} {2} \times \mathbb{Z} \) for flat spacetime is characterized by the continuous mass as invariant for translations \( \mathbb{R}^4 \) in the **D**(1) × **U**(1)-extended Poincaré group \( {\bf GL} (\mathbb{C}^2) \vec \times \mathbb{R}^4 \), by (half-)integer spin or polarization for rotations **SU**(2) or **SO**(2) as translation fixgroup in the Lorentz cover group \( {\bf SL} (\mathbb{C}^2)\), and by an integer charge number for electro-magnetic windings **U**(1). The basic interactions are implemented by massless fields with characteristic coupling constants.

The eigentime \( {\bf D}(1) \cong \mathbb{R}\) invariants for the representations of electroweak spacetime \( \mathcal{D}^4 \cong {\bf D}(1) \times y^3 \cong \mathbb{R}^4_+ \) are proposed to determine the mass of relativistic particles. Since the causal spacetime group \( {\bf GL} (\mathbb{C}^2)\) has real rank 2, i.e., two characterizing continuous invariants \( \left\{ {m^2 ,\,M^2 } \right\} \), the translation invariants are related to both the embedded causal group **D**(1) and Lorentz group **SO** _{0}(1, 3)-representations of 3-position *Y* ^{3}.

## Keywords

Lorentz Group Product Representation Goldstone Mode Real Rank Dilation Property## Preview

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