Discretizing parametrized systems: The magic of Dittrich invariance

Peculiar phenomena appear in the discretization of a system invariant under reparametrization. The structure of the continuum limit is markedly different from the usual one, as in lattice QCD. First, the continuum limit does not require tuning a parameter in the action to a critical value. Rather, there is a regime where the system approaches a sort of asymptotic topological invariance (Ditt-invariance). Second, in this regime the expansion in the number of discretization points provides a good approximation to the transition amplitudes. These phenomena are relevant for understanding the continuum limit of quantum gravity. I illustrate them here in the context of a simple system.

Automated summary

Peculiar phenomena occur in the discretization of a system invariant under reparametrization. The structure of the continuum limit is markedly different from the usual case, such as in lattice QCD. These phenomena are crucial for understanding the continuum limit of quantum gravity.