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Perfect discretization of reparametrization invariant path integrals

Benjamin Bahr, Bianca Dittrich, Sebastian Steinhaus

TL;DR

This paper shows that enforcing reparametrization invariance at the discrete level forces the path integral to be discretization independent and to reproduce the continuum propagator. It develops a Wilsonian renormalization group–style iterative procedure to construct perfect discretizations, determining both the action and the measure to maintain gauge invariance and avoid anomalies. The authors demonstrate this explicitly for the parametrized harmonic oscillator (and variants) and extend the method to the anharmonic oscillator, obtaining fixed-point propagators with a finite set of continuum parameters. The results suggest similar mechanisms could fix discretization ambiguities in discrete gravity and spin foam models by enforcing vertex-translation (diffeomorphism) symmetry, highlighting the need to solve dynamics and coarse-graining problems to achieve anomaly-free, gauge-invariant quantum theories.

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.

Perfect discretization of reparametrization invariant path integrals

TL;DR

This paper shows that enforcing reparametrization invariance at the discrete level forces the path integral to be discretization independent and to reproduce the continuum propagator. It develops a Wilsonian renormalization group–style iterative procedure to construct perfect discretizations, determining both the action and the measure to maintain gauge invariance and avoid anomalies. The authors demonstrate this explicitly for the parametrized harmonic oscillator (and variants) and extend the method to the anharmonic oscillator, obtaining fixed-point propagators with a finite set of continuum parameters. The results suggest similar mechanisms could fix discretization ambiguities in discrete gravity and spin foam models by enforcing vertex-translation (diffeomorphism) symmetry, highlighting the need to solve dynamics and coarse-graining problems to achieve anomaly-free, gauge-invariant quantum theories.

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.

Paper Structure

This paper contains 9 sections, 84 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Reparametrization invariance in the discrete and discretization independence
  3. The harmonic oscillator linearized around $\bar{q}_n=0$
  4. Linearized dynamics around a solution with $\bar{q}_n\neq 0$
  5. The anharmonic oscillator
  6. Summary and Discussion
  7. Hamilton's principal function for the anharmonic oscillator
  8. On the uniqueness of the fixed point solutions
  9. Recursion relations for the $\beta_i$

Figures (1)

  • Figure 1: The refinement process of the propagator $K^{(n)}$ involves subdividing the discretization intervals and integrating over the new variables, obtaining a new propagator $K^{(n+1)}$. This process can be iterated, leading in the limit to the perfect propagator, which is in particular invariant under refinement of the discretization.