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Abstract

Over the past five years, there has been significant progress on the problem of quantization of diffeomorphism covariant field theories with {\it local} degrees of freedom. The absence of a background space-time metric in these theories gives rise to a host of conceptual and technical difficulties because most of the familiar methods from axiomatic, constructive and perturbative quantum field theory are no longer applicable. Perhaps the most striking examples of these problems arise in the construction of a quantum field theory of geometry. We show that these problems can be tackled using new non-perturbative methods. In particular, one can rigorously define certain geometric operators and show that their spectrum is discrete. Thus, there is a precise sense in which the geometry is quantized at the Planck scale and the continuum picture is only a coarse-grained approximation.