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

It has been shown by Pollock that the Wheeler-DeWitt equation for the wave function of the Universe psi cannot be derived for the D-dimensional, heterotic superstring theory, when higher-derivative terms alpha'R2 are included in the effective Lagrangian L, because they occur as the Euler-number density R(E)2. This means that L 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 alpha'3R4, 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 D < D. When this is done, the reduced term R2 is no longer proportional to R(E)2, because of additional contributions arising from the dimensional reduction of R4. The Wheeler-DeWitt equation can now be derived in the form of a Schrodinger equation, in particular when D = 4 (and R(E)2 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.