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

TL;DR

The paper develops uniform discretizations as a nonperturbative framework for quantizing totally constrained systems, recasting the problem of dynamics in contexts where the continuum Dirac approach is problematic. By promoting certain Lagrange multipliers to dynamical variables and defining a global Hamiltonian $H=f(\phi_1,\dots,\phi_N)$, it yields a well-defined discrete evolution with a controlled continuum limit that reproduces the target theory when possible. The quantum formulation uses a unitary evolution $\hat{U}=e^{-i\hat{H}/\hbar}$ and a projector $\hat{P}$, linking to group averaging and evolving constants to recover continuum predictions, or providing meaningful discrete/relational descriptions when the continuum limit fails. A range of finite-dimensional models, BF theory, and 3D gravity illustrate the method, showing agreement with established results where available and offering a viable, physically interpretable alternative in cases lacking a well-defined continuum limit.

Abstract

We discuss in detail the uniform discretization approach to the quantization of totally constrained theories. This approach allows to construct the continuum theory of interest as a well defined, controlled, limit of well behaved discrete theories. We work out several finite dimensional examples that exhibit 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 can take two points of view. The technique can be used to define, upon taking the continuum limit, the space of physical states of the continuum constrained theory of interest. In particular we show in models that it agrees with the group averaging procedure when the latter exists. The technique can also be used to compute, at the discrete level, conditional probabilities and the introduction of a relational time. Upon taking the continuum limit one can show that one reproduces results obtained by the use of evolving constants, and therefore recover all physical predictions of the continuum theory. This second point of view can also be used as a paradigm to deal with cases where the continuum limit does not exist. There one would have discrete theories that at least at certain scales reproduce the semiclassical properties of the theory of interest. In this way the approach can be viewed as a generalization of the Dirac quantization procedure that can handle situations where the latter fails.