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Time evolution as refining, coarse graining and entangling

Bianca Dittrich, Sebastian Steinhaus

TL;DR

The paper addresses how to extract a continuum quantum gravity description from discrete, time-evolving systems by interpreting time evolution as refining, coarse graining, and entangling operations. It develops an inductive-limit framework for continuum Hilbert spaces, with embedding maps tied to dynamics and cylindrical consistency ensuring a well-defined continuum limit. Topological theories provide exact realizations of these ideas, while non-topological theories require coarse graining and potentially nonlocal embeddings to recover diffeomorphism-like invariances. The work also connects tensor network renormalization, Hartle-Hawking no-boundary vacua, and spin-foam formalisms, offering a concrete pathway to define physical vacua and continuum dynamics in quantum gravity, alongside open questions about embeddings, divergences, and the role of locality.

Abstract

We argue that refining, coarse graining and entangling operators can be obtained from time evolution operators. This applies in particular to geometric theories, such as spin foams. We point out that this provides a construction principle for the physical vacuum in quantum gravity theories and more generally allows to construct a (cylindrically) consistent continuum limit of the theory.

Time evolution as refining, coarse graining and entangling

TL;DR

The paper addresses how to extract a continuum quantum gravity description from discrete, time-evolving systems by interpreting time evolution as refining, coarse graining, and entangling operations. It develops an inductive-limit framework for continuum Hilbert spaces, with embedding maps tied to dynamics and cylindrical consistency ensuring a well-defined continuum limit. Topological theories provide exact realizations of these ideas, while non-topological theories require coarse graining and potentially nonlocal embeddings to recover diffeomorphism-like invariances. The work also connects tensor network renormalization, Hartle-Hawking no-boundary vacua, and spin-foam formalisms, offering a concrete pathway to define physical vacua and continuum dynamics in quantum gravity, alongside open questions about embeddings, divergences, and the role of locality.

Abstract

We argue that refining, coarse graining and entangling operators can be obtained from time evolution operators. This applies in particular to geometric theories, such as spin foams. We point out that this provides a construction principle for the physical vacuum in quantum gravity theories and more generally allows to construct a (cylindrically) consistent continuum limit of the theory.

Paper Structure

This paper contains 27 sections, 49 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Overview
  3. Time evolving phase spaces
  4. Example: evolution of a scalar field on an extending triangulation
  5. Refinement as adding degrees of freedom in the vacuum state
  6. Massless scalar field in a 2D Lorentzian space time
  7. Refinement moves in simplicial gravity: adding gauge and vacuum degrees of freedom
  8. Refining in quantum theory
  9. Inductive limit construction of a continuum Hilbert space
  10. Refining time evolution and no--boundary vacuum
  11. Path independence of evolution and consistent embedding maps
  12. Pre--constraints and coarse graining
  13. How to define a continuum theory of quantum gravity
  14. Tensor network coarse graining: time evolution in radial direction
  15. Radial evolution
  16. ...and 12 more sections

Figures (8)

  • Figure 1: A $1-3$ move in the 2D hypersurface can be obtained by gluing a tetrahedron with one of its triangles to the hypersurface.
  • Figure 2: A $2-2$ move in the 2D hypersurface can be obtained by gluing a tetrahedron with two of its triangles to the hypersurface.
  • Figure 3: The scalar field on a circle and its time evolution. The circle is drawn as an interval with periodic boundary conditions indicated by the dashed lines.
  • Figure 4: The time evolution proceeds by moving the null edge $(\nu_1,\nu_2)$ to $(\nu'_1,\nu'_2)$.
  • Figure 5: Time evolution can also proceed by gluing small diamonds to the hypersurface, which will however produce past directed null edges. Note that in this case $\phi(\nu")$ will be constrained and determined by the fields at the other vertices on the 'equal time' hypersurface.
  • ...and 3 more figures