The quantum geometry of random surfaces and spinning membranes

An approach to the quantisation of strings is readily adapted to other types of extended object. In this paper we consider the quantisation of p-dimensional surfaces moving through spacetime, where p > 1. The transition amplitude between given surface configurations is defined as a sum over all (p+1)-dimensional surfaces bounded by these configurations. This definition is made more precise by a careful analysis of boundary conditions. We calculate the 1-loop divergences using the semiclassical approximation and the heat-kernel expansion. Our models incorporate an Einstein Hilbert term in the action and therefore include the case of (p+1)-dimensional quantum gravity on a manifold with boundary; it should be possible to deal with more general theories of extended objects using similar techniques. We then consider the quantisation of the spinning membrane. Certain components of the fermion fields must be fixed on the boundary, and it turns out that there is a supersymmetric equivalence relation between different boundary configurations. A 1-loop calculation is performed, and it is found that the spinning membrane is not renormalisable in any number of dimensions. (author)