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General Relativity and Quantum Cosmology, gr-qc,High Energy Physics - Lattice, hep-lat
Abstract:
To obtain a well defined path integral one often employs discretizations. In
the case of gravity and reparametrization invariant systems, the latter of
which we consider here as a toy example, discretizations generically break
diffeomorphism and reparametrization symmetry, respectively. This has severe
implications, as these symmetries determine the dynamics of the corresponding
system. Indeed we will show that a discretized path integral with
reparametrization invariance is necessarily also discretization independent and
therefore uniquely determined by the corresponding continuum quantum mechanical
propagator. We use this insight to develop an iterative method for constructing
such a discretized path integral, akin to a Wilsonian RG flow. This allows us
to address the problem of discretization ambiguities and of an anomaly--free
path integral measure for such systems. The latter is needed to obtain a path
integral, that can act as a projector onto the physical states, satisfying the
quantum constraints. We will comment on implications for discrete quantum
gravity models, such as spin foams.
