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We discuss in detail the uniform discretization approach to the quantizationof totally constrained theories. This approach allows to construct thecontinuum theory of interest as a well defined, controlled, limit of wellbehaved discrete theories. We work out several finite dimensional examples thatexhibit behaviors expected to be of importance in the quantization of gravity.We also work out the case of BF theory. At the time of quantization, one cantake two points of view. The technique can be used to define, upon taking thecontinuum limit, the space of physical states of the continuum constrainedtheory of interest. In particular we show in models that it agrees with thegroup averaging procedure when the latter exists. The technique can also beused to compute, at the discrete level, conditional probabilities and theintroduction of a relational time. Upon taking the continuum limit one can showthat one reproduces results obtained by the use of evolving constants, andtherefore recover all physical predictions of the continuum theory. This secondpoint of view can also be used as a paradigm to deal with cases where thecontinuum limit does not exist. There one would have discrete theories that atleast at certain scales reproduce the semiclassical properties of the theory ofinterest. In this way the approach can be viewed as a generalization of theDirac quantization procedure that can handle situations where the latter fails.Comment: 17 pages, Revtex, no figures, published versio

Uniform discretizations: A new approach for the quantization of totally constrained systems