ON THE DERIVATION OF THE WHEELER-DEWITT EQUATION IN THE HETEROTIC SUPERSTRING THEORY

Abstract

It has been shown by Pollock that the Wheeler-DeWitt equation for the wave function of the Universe Ψ cannot be derived for the D-dimensional, heterotic superstring theory, when higher-derivative terms [Formula: see text] are included in the effective Lagrangian [Formula: see text], because they occur as the Euler-number density [Formula: see text]. This means that [Formula: see text] cannot be written in the standard Hamiltonian form, and hence that macroscopic quantum mechanics does not exist at this level of approximation. It was further conjectured that the solution to this difficulty is to take into account the effect of the terms [Formula: see text], an expression for which has been obtained by Gross and Witten, and by Freeman et al. Here, this conjecture is proved, but it is pointed out that the theory must first be reduced to a lower dimensionality [Formula: see text]. When this is done, the reduced term R2 is no longer proportional to [Formula: see text], because of additional contributions arising from the dimensional reduction of [Formula: see text]. The Wheeler-DeWitt equation can now be derived in the form of a Schrödinger equation, in particular when [Formula: see text] (and [Formula: see text] is a total divergence which can be discarded), and quantum mechanics can be set up in the usual way. In the light of these results, it is argued that the non-locality of quantum mechanics is related to the cosmological horizon problem.