THE WHEELER-DEWITT EQUATION FOR THE HETEROTIC SUPERSTRING THEORY INCLUDING TERMS QUARTIC IN THE RIEMANN TENSOR

Abstract

The Wheeler-DeWitt equation for the wave function of the Universe Ψ can be derived for the heterotic superstring, after reduction of the effective action, including terms [Formula: see text] quartic in the Riemann tensor, from ten dimensions to [Formula: see text] dimensions, where [Formula: see text]. If the compactified space is Ricci flat, then no terms ℛ3 appear, since the coefficient of [Formula: see text] in the ten-dimensional action vanishes. The reduced Lagrangian, ignoring all non-gravitational fields, is then L=(16πG)−1R+a2ℛ2+a4α′2ℛ4, where G is the Newton gravitational constant, α′ is the Regge slope parameter, and a2 and a4 are dimensionless coefficients. Including only the first two terms, in the Friedmann space-time ds2=dt2−e2α(t)dx2, leads to the Schrödinger equation i∂Ψ/∂t=[−AMe−Mα∂2/∂ξ2+ VM,K(α, ξ)]Ψ, where AM is a positive constant, ξ≡dα/dt and K is the curvature of the M-space dx2. After the Wick rotation [Formula: see text], this equation becomes [Formula: see text], where [Formula: see text]. The requirement that both V and [Formula: see text] are positive semi-definite leads to the conditions M=3, K=0, which state that space is three-dimensional and flat. Here, a more complete Schrödinger equation is derived, via a perturbative treatment of the terms a4α′2ℛ4, which lifts the degeneracy of the potential V3,0 under Wick rotations, the Lorentzian signature being energetically favoured over the Euclidean signature. This corroborates results concerning supersymmetry and the quantum mechanical consistency of the string theory on the world sheet, for which the Lorentzian signature is also necessary, as it is argued to be for the Feynman path-integral formulation of Ψ.