Spectrum of Spacetime

Abstract

In Wigner's classification, linear spacetime and free particles originate from one operational concept and its representations, from an affine subgroup with Lorentz transformations acting on translations. Why the free particles have the characteristic invariants, i.e., the observed masses m 2 , spins J, and, for the additional internal U(1)-operations, charge numbers z, is not explained by classifying the irreducible Hilbert representations of the Poincaré group. The actual spectrum of matter (m 2 , 2J, z) ∈ R + × N × Z together with the normalization of particles and the coupling constants of interactions has to be understood by additional structures, e.g., by representations of a nonlinear spacetime model. The multilinear algebra structure of quantum operations involves typical ensembles of representations ("towers of bound states"), which are products of one basic representation, defining the relevant operation group. Characteristic examples are the free states of translations, which are familiar from the equidistant linear spectrum of the harmonic oscillator; representations of time translations R ∼ = D(1); and the bound states of the nonrelativis-tic hydrogen atom as representations of hyperbolic 3-dimensional position Y 3 ∼ = SO 0 (1, 3)/SO(3) ∼ = R 3 with the inverse squared energy spectrum. A pointwise product of positive-type functions d ∈ L ∞ (G) + of a real Lie group is a positive-type function for the product representation, d 1 · d 2 (g) = a 1 |D 1 (g)|a 1 a 2 |D 2 (g)|a 2 = a 1 , a 2 |D 1 ⊗ D 2 (g)|a 1 , a 2. For the harmonic components, one has to use the convolutioñ d 1 * ˜ d 2. The characters (representation classes, dual group) as eigenvalues of the additive groupŘgroupˇgroupŘ d-energies for time translations R and momenta for position translations R 3-give rise to convolution algebras of the corresponding distributions (functions, measures). Nonlinear spacetime D(2) ∼ = GL(C 2)/U(2) as a homogeneous space of the extended Lorentz group GL(C 2) with tangent Minkowski translations x ∈ R 4 can be represented by residues of Fourier transformed energy-momentum q ∈ ˇ R 4 functions (chapter "Residual Space-time Representations"). The representation-characterizing invariants arise as poles in the complex energy and momentum plane. Product representations come with the product of representation coefficients , i.e., in a residual formulation with the convolution * of (energy-)mo-299