On the quantisation of gravity by embedding spacetime in a higher dimensional space

Abstract

Certain difficulties of quantum gravity can be avoided if we embed the spacetime $V_4$ into a higher dimensional space $V_N$; then our spacetime is merely a 4-surface in $V_N$.What remains is conceptually not so difficult: just to quantise this 4-surface. Our formal procedure generalises our version of Stueckelberg's proper time method of worldline quantisation. We write the equations of $V_4$ in the covariant canonical form starting from a model Lagrangian which contains the classical Einstein gravity as a particular case. Then we perform quantisation in the Schr\"odinger picture by using the concepts of a phase functional and wave functional. As a result we obtain the uncertainty relations which imply that an observer is `aware' either of a particular spacetime surface and has no information about other spacetime surfaces (which represent alternative histories); or conversely, he loses information about a particular $V_4$ whilst he obtains some information about other spacetimes (and histories). Equivalently, one cannot measure to an arbitrary precision both the metric on $V_4$ and matter distribution on various alternative spacetime surfaces. We show how this special case in the `coordinate' representations can be generalised to an arbitrary vector in an abstract Hilbert space.