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From the discrete to the continuous - towards a cylindrically consistent dynamics

Bianca Dittrich

TL;DR

The paper proposes a coarse graining framework to achieve cylindrically consistent dynamics, aiming to reproduce continuum physics from discrete models via dynamics-informed embedding maps and fixed-point actions. Demonstrated first with a massless scalar field on 2D triangulations and then with a scalar field plus a λφ^4 potential, the approach uses tensor network renormalization ideas to shift non-localities to boundary data and to control truncations of the renormalization flow. It analyzes multiple coarse graining schemes (piecewise linear vs. piecewise constant data) and examines the implications for both classical Hamilton's principal function and quantum path integrals, emphasizing the crucial role of embedding maps. The work lays groundwork for perfect discretizations in gravity-related theories, outlines how to extend to quantum gravity settings (e.g., spin foams), and highlights open issues related to embedding selection and back-reaction bounds.

Abstract

Discrete models usually represent approximations to continuum physics. Cylindrical consistency provides a framework in which discretizations mirror exactly the continuum limit. Being a standard tool for the kinematics of loop quantum gravity we propose a coarse graining procedure that aims at constructing a cylindrically consistent dynamics in the form of transition amplitudes and Hamilton's principal functions. The coarse graining procedure, which is motivated by tensor network renormalization methods, provides a systematic approximation scheme towards this end. A crucial role in this coarse graining scheme is played by embedding maps that allow the interpretation of discrete boundary data as continuum configurations. These embedding maps should be selected according to the dynamics of the system, as a choice of embedding maps will determine a truncation of the renormalization flow.

From the discrete to the continuous - towards a cylindrically consistent dynamics

TL;DR

The paper proposes a coarse graining framework to achieve cylindrically consistent dynamics, aiming to reproduce continuum physics from discrete models via dynamics-informed embedding maps and fixed-point actions. Demonstrated first with a massless scalar field on 2D triangulations and then with a scalar field plus a λφ^4 potential, the approach uses tensor network renormalization ideas to shift non-localities to boundary data and to control truncations of the renormalization flow. It analyzes multiple coarse graining schemes (piecewise linear vs. piecewise constant data) and examines the implications for both classical Hamilton's principal function and quantum path integrals, emphasizing the crucial role of embedding maps. The work lays groundwork for perfect discretizations in gravity-related theories, outlines how to extend to quantum gravity settings (e.g., spin foams), and highlights open issues related to embedding selection and back-reaction bounds.

Abstract

Discrete models usually represent approximations to continuum physics. Cylindrical consistency provides a framework in which discretizations mirror exactly the continuum limit. Being a standard tool for the kinematics of loop quantum gravity we propose a coarse graining procedure that aims at constructing a cylindrically consistent dynamics in the form of transition amplitudes and Hamilton's principal functions. The coarse graining procedure, which is motivated by tensor network renormalization methods, provides a systematic approximation scheme towards this end. A crucial role in this coarse graining scheme is played by embedding maps that allow the interpretation of discrete boundary data as continuum configurations. These embedding maps should be selected according to the dynamics of the system, as a choice of embedding maps will determine a truncation of the renormalization flow.

Paper Structure

This paper contains 8 sections, 39 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The massless scalar field
  3. Scalar field with potential
  4. Larger coarse graining steps
  5. Other coarse graining schemes
  6. Cylindrical consistency
  7. Remarks on the quantum theory case
  8. Summary, open issues and outlook

Figures (3)

  • Figure 1: Subdivisions of a triangle.
  • Figure 2: The boundary fields on a square. To find the fixed point action with eight boundary fields $\{\phi_i,\gamma_{ij}\}$ one would have to set i.e. $\gamma_{112}=0$ as it encodes the deviation from a piecewise linear field.
  • Figure 3: The tensor network scheme for the coarse graining of a scalar field. The original square is rotated by $45^\circ$, so that now the boundary fields can be associated to the edges of an unrotated square. The gluing of these new squares corresponds to combining tensors of rank four. The corresponding tensor network is indicated by blue lines. Bulk fields $\phi_5,\ldots,\phi_8$ are now associated to the inner edges of this tensor network. Coarser boundary data correspond to piece wise constant fields. In the figure on the right we depicted the set--up for coarse graining with eight boundary fields -- fields on pairs of outer legs are set to be equal.