ON THE POSITIVITY OF THE GRAVITATIONAL POTENTIAL IN THE QUANTUM-COSMOLOGICAL SCHRÖDINGER EQUATION

Abstract

Quantization of the D(=M+1)-dimensional gravitational theory with higher-derivative terms [Formula: see text] in the Friedmann space-time ds2=dt2−e2α(t)dx2, where the M-space dx2 has curvature K, yields the Schrödinger equation (Wheeler-DeWitt equation) i∂Ψ/∂t=[−AMe−Mα∂2/∂ξ2+VM,K(α, ξ)]Ψ, where ξ≡dα/dt, provided that [Formula: see text] differs from the Euler-number density. The coefficient AM is positive if there is no spin-0 tachyon in [Formula: see text] and the potential VM,K is positive semi-definite if M=3 and K=0. The theory is classically stable if the spin-2 tachyon is also absent. All of these conditions are satisfied by the heterotic superstring, after reduction to four dimensions, but not by the bosonic string, which contains a spin-2 tachyon, nor by the type-II superstring, which contains a spin-0 tachyon. After generalization to the anisotropic space-time ds2= dt2−e2β(t)dy2−e2γ(t)dz2, where dy2 and dz2 have dimensions one and two, respectively, the Schrödinger equation becomes i∂Ψ/∂t=[–(4B)−1e−(β+2γ) (X∂2/∂ζ2+Y∂2+ 2Z∂2/∂ζ∂η)+…+V(β, γ; ζ, η)]Ψ, where ζ≡dβ/dt, η≡dγ/dt. The potential V is unbounded both from above and from below for all β≠γ, for all three superstring theories, and in fact for all dimensionalities. This explains why the Universe is isotropic, and why dimensional reduction and compactification occur.