ON THE HEISENBERG INDETERMINACY PRINCIPLE AND THE FORMATION OF STRUCTURE IN THE UNIVERSE

The Heisenberg indeterminacy principle Δp_aΔq^a ~ ħ, relating canonically conjugate variables p_a and q^a, is quantified for the classical action obtained by the reduction of the ten-dimensional heterotic superstring theory to four dimensions, in the mini-superspace (Friedmann space-time) . There are two coordinates, α and , representing position and velocity, respectively, the canonical momenta being and . In both cases, the result can be expressed as an indeterminacy in the time, (Δt/t)². The fluctuations connecting position and velocity decrease with time and are always undetectably small, Δt/t ≲ 10⁻⁴⁴. But the fluctuations involving velocity and acceleration increase with time, and are evaluated at the time t_e of equipartition of radiation and matter in the universe. Translated first into a metric fluctuation , this is equivalent to a Gaussian, scale-invariant spectrum of density fluctuations of magnitude , where the dimensionless constant B depends only on the compactification scheme. For a Calabi–Yau internal space, the estimate B ≈ 3 implies that ζ ≈ 2 × 10⁻⁴, which is sufficient for the creation of galaxies and in approximate agreement with observations of the anisotropy of the cosmic microwave background radiation by COBE and at Tenerife.