Uniform discretizations: a quantization procedure for totally constrained systems including gravity

We present a new method for the quantization of totally constrained systems including general relativity. The method consists in constructing discretized theories that have a well defined and controlled continuum limit. The discrete theories are constraint-free and can be readily quantized. This provides a framework where one can introduce a relational notion of time and that nevertheless approximates in a well defined fashion the theory of interest. The method is equivalent to the group averaging procedure for many systems where the latter makes sense and provides a generalization otherwise. In the continuum limit it can be shown to contain, under certain assumptions, the ``master constraint'' of the ``Phoenix project''. It also provides a correspondence principle with the classical theory that does not require to consider the semiclassical limit.