Rotationally invariant dynamical lattice regulators for Euclidean quantum field theories

TL;DR

The paper proposes a dynamical-lattice regulator (DLR) that promotes the lattice embedding x: Λ → ℝ^d to a dynamical field, yielding exact SE(d) covariance at finite lattice spacing and enabling local twisting to suppress axis–diagonal anisotropies. It proves Osterwalder–Schrader reflection positivity for the coupled geometry-matter system under locality and a short-range geometry hypothesis (SR), and shows that integrating out the geometry produces a local Symanzik effective action with only SO(d)-invariant irrelevant operators; in d=4 the one-loop φ^4 β-function is recovered, confirming universality with standard regulators. The authors provide a concrete 2D Monte Carlo demonstration with a dynamical mesh, reporting well-behaved geometry sectors, reduced rotational artefacts, and universality with the baseline lattice. Overall, the DLR offers a conservative, regulator-friendly route to restoring Euclidean symmetry at finite lattice spacing while preserving locality, gauge invariance, and OS positivity, with potential to mitigate discretization artifacts in nonperturbative QFT simulations.

Abstract

We introduce a dynamical-lattice regulator (DLR) for Euclidean quantum field theories on a fixed hypercubic graph $Λ\simeq \mathbb{Z}^d$, in which the embedding $x:Λ\to \mathbb{R}^d$ is promoted to a dynamical field and integrated over subject to shape-regularity constraints. The total action is local on $Λ$, gauge invariant, and depends on $x$ only through Euclidean invariants built from edge vectors (local metrics, volumes, etc.), hence the partition function is exactly covariant under the global Euclidean group SE(d) at any lattice spacing. The intended symmetry-restoring mechanism is not rigid global zero modes but short-range *local twisting* of the embedding that mixes local orientations; accordingly, our universality discussion is conditioned on a short-range geometry hypothesis (SR): after quotienting the global SE(d) modes, connected correlators of local geometric observables have correlation length O(1) in lattice units. We prove Osterwalder-Schrader reflection positivity for the coupled system with embedding $x$ and generic gauge/matter fields $(U,Φ)$ in finite volume by treating $x$ as an additional multiplet of scalar fields on $Λ$. Assuming (SR), integrating out $x$ at fixed cutoff yields a local Symanzik effective action in which geometry fluctuations generate only SO(d)-invariant irrelevant operators and finite renormalizations; in particular, in $d=4$ we recover the standard one-loop $β$-function in a scalar $φ^4$ test theory. Finally, we describe a practical local Monte Carlo update and report $d=2$ proof-of-concept simulations showing a well-behaved geometry sector and a substantial reduction of axis-vs-diagonal cutoff artifacts relative to a fixed lattice at matched bare parameters.

Figures (4)

• Figure 1: Building blocks in the perturbative universality argument. Solid: $\phi$. Dashed: $\xi$.
• Figure 2: Further one--loop graphs with internal geometry lines (line conventions as in Fig. 1).
• Figure 3: Geometry diagnostics for the dynamical ensemble (all panels at identical scale).
• Figure 4: Matter, rotational-symmetry, and stability diagnostics comparing baseline and dynamical ensembles.

Theorems & Definitions (17)

• Definition 2.1
• Remark 2.2
• Lemma 2.3
• proof
• Remark 2.4
• Definition 2.5
• Lemma 4.1
• proof
• Lemma 4.2
• proof