Quantization of systems with temporally varying discretization II: Local evolution moves

TL;DR

This work develops a quantum formalism for local, temporally varying discretization moves by extending the global-move quantization to four local move types (I–IV), each corresponding to Pachner-like refinements or coarse-grainings of the discretization. A central finding is that non-trivial coarse-graining moves induce non-unitary projections on physical Hilbert spaces, reflecting irreversible information loss of fine-grained Dirac observables, while refining moves can embed states into finer discretizations unitarily. The framework uses extended configuration spaces with auxiliary variables, momentum updating maps, and constrained propagators to construct a path-integral-like evolution that preserves post-constraints and handles gauge divergences via projection or gauge fixing. The analysis connects to simplicial quantum gravity by highlighting how Pachner moves modify the set of quantum Dirac observables and by outlining how divergences from non-compact gauge orbits may be regularized, offering insights for 4D Regge calculus and spin-foam models. Overall, the results clarify how discretization-changing dynamics can be consistently encoded in quantum theories and point toward practical schemes for cylindrically consistent vacua and effective continuum limits.

Abstract

Several quantum gravity approaches and field theory on an evolving lattice involve a discretization changing dynamics generated by evolution moves. Local evolution moves in variational discrete systems (1) are a generalization of the Pachner evolution moves of simplicial gravity models, (2) update only a small subset of the dynamical data, (3) change the number of kinematical and physical degrees of freedom, and (4) generate a dynamical coarse graining or refining of the underlying discretization. To systematically explore such local moves and their implications in the quantum theory, this article suitably expands the quantum formalism for global evolution moves, constructed in a companion paper, by employing that global moves can be decomposed into sequences of local moves. This formalism is spelled out for systems with Euclidean configuration spaces. Various types of local moves, the different kinds of constraints generated by them, the constraint preservation and possible divergences in resulting state sums are discussed. It is shown that non-trivial local coarse graining moves entail a non-unitary projection of (physical) Hilbert spaces and `fine grained' Dirac observables defined on them. Identities for undoing a local evolution move with its (time reversed) inverse are derived. Finally, the implications of these results for a Pachner move generated dynamics in simplicial quantum gravity models are commented on.

Figures (12)

• Figure 1: Schematic illustration of two global evolution moves $0\rightarrow1$ and $1\rightarrow2$. In discrete gravity models, an evolution move corresponds to a region of triangulated space-time. Composing the moves is equivalent to gluing the corresponding regions together at time $1$. This process requires a momentum matching ${}^+p^1={}^-p^1$ and an implementation of both pre-- and post--constraints at step $1$.
• Figure 2: Illustration of the four types of variables labeled by $e,b,o,n$ appearing in local evolution moves. The picture shows a local evolution move for a scalar field living on the vertices of a 2D discretized spacetime. The local evolution move corresponds to gluing a new piece of discrete 2D spacetime with corresponding scalar field action $S_{k+1}$ to the 1D hypersurface $\Sigma_k$ which constitutes the time step $k$. The 'old' field variable $\phi^o_k$ disappears from $\Sigma$ and becomes bulk in this move, while a new vertex with a 'new' field variable $\phi^n_{k+1}$ is introduced. Clearly, the neighbouring time steps overlap, $\Sigma_k\cap\Sigma_{k+1}\neq\emptyset$. The two field variables $\phi^{e_1}_k,\phi^{e_2}_k$ occur before and after the move and contribute to $S_{k+1}$. The remaining variables $\phi^{b_i}_k$ occur at both time steps, however, do not contribute to $S_{k+1}$.
• Figure 3: The 1--2 Pachner move for a scalar field on a 2D triangulated space-time is of type I. It introduces one new field variable $\phi^v_{k+1}$ at step $k+1$.
• Figure 4: The 2--1 Pachner move for a scalar field on a 2D space-time triangulation is of type II. It corresponds to gluing a triangle onto two edges in $\Sigma_k$ which annihilates a vertex $v^*$ with the variable $\phi^{v^*}_{k}$.
• Figure 5: An example of a type III move for a scalar field on a 2D quadrangulation. It corresponds to gluing a square onto a 1D zig-zag line $\Sigma_k$ and removes the field variable $\phi^{v^*}_k$ and introduces $\phi^v_{k+1}$.

Theorems & Definitions (23)

• Remark
• Theorem 4.1
• proof
• Example 4.1
• Theorem 4.2
• proof
• Example 4.2
• Theorem 4.3
• proof
• Example 4.3