Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Discretizing parametrized systems: the magic of Ditt-invariance

Carlo Rovelli

Abstract

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.

Discretizing parametrized systems: the magic of Ditt-invariance

Abstract

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.

Paper Structure

This paper contains 7 sections, 43 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Discretization
  3. Parametrization
  4. Classical dynamics
  5. Quantum dynamics
  6. Ditt--invariance
  7. Conclusion

Figures (2)

  • Figure 1: $x_n$ as a function of $n$ from the numerical integration of equations \ref{['unodp']} and \ref{['duedp']}.
  • Figure 4: Numerical integration of the discretized path integral of the parametrized oscillator. The graph gives the Euclidean evaluation of $W(q_f,t_f;q_i,t_i)$ as a function of $q_f$, with $\omega t\sim1$, $t_i=0$ and $q_i=0$, compared with the exact result \ref{['exact']}. The number of integration points is $N= 2$.