Abstract
We analyze the spectrum of time observable in noncommutative cosmological model introduced in [5], defined by $(\rho, s=\frac 12)\,$ representation of the de Sitter group. We find that time has peculiar property: it is not self-adjoint, but appropriate restrictions to the space of physical states give self-adjoint extensions. Extensions have discrete spectrum with logarithmic distribution of eigenvalues, $\,t_n \sim \ell\, \log\, n$+const, where $\ell$ characterizes noncommutativity and the usual assumption is $\,\ell=\ell_{Planck}$. When calculated on physical states, radius of the universe is bounded below by $\, \ell\, \sqrt{\frac 34\, \left( \frac 14 +\rho^2\right)}\, $, which resolves the big bang singularity. An immediate consequence of the model is a specific breaking of the original symmetry at the Planck scale.
Highlights
The expression “quantum space” was introduced in the early days of quantum mechanics by Heisenberg, along with “quantum derivative” introduced by Dirac who observed that a commutator is a derivation; “points” of the quantum space are “q numbers” or operators
II we introduce fuzzy de Sitter space as a unitary irreducible representation of the de Sitter group, i.e., identify its coordinates and differential structure
Let us verify that fuzzy de Sitter space corresponds to an expanding cosmology and discuss the absence of the big bang singularity
Summary
The expression “quantum space” was introduced in the early days of quantum mechanics by Heisenberg, along with “quantum derivative” introduced by Dirac who observed that a commutator is a derivation; “points” of the quantum space are “q numbers” or operators. A may not have a Schrödinger-type representation of momenta through the partial derivatives; some representations might be finite dimensional In this picture, position algebra (1.3) determines the structure of the points of noncommutative space, i.e., the algebraic properties of coordinates, while (1.2) and the related commutators between momenta define the differential-geometric structure and enable us to introduce connection and curvature. In various descriptions of noncommutative spaces, more or less invariant, while the differential-geometric part is specific in every approach: we use the noncommutative frame formalism of Madore. Are basic variables that are quantized in a background-space independent way In these approaches quantum space with its properties is a derived quantity or notion. In the last section we discuss physical properties and some cosmological implications of the given fuzzy geometry
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