Rotationally invariant dynamical lattice regulators for Euclidean quantum field theories

TL;DR

This paper introduces a dynamical lattice regulator for Euclidean quantum field theories that promotes the embedding to a dynamical field, ensuring exact covariance and improved symmetry restoration through local geometric fluctuations.

Contribution

The authors develop a novel dynamical lattice regulator with a shape regularity constraint, proving reflection positivity and demonstrating reduced cutoff artifacts in simulations.

Findings

Proved Osterwalder-Schrader reflection positivity for the coupled system.

Derived a local Symanzik effective action with geometry fluctuations generating irrelevant operators.

Performed proof-of-concept simulations showing improved geometric behavior and reduced artifacts.

Abstract

We introduce a dynamical lattice regulator for Euclidean quantum field theories on a fixed hypercubic graph $\Lambda\simeq\mathbb{Z}^d$, in which the embedding $x:\Lambda\to\mathbb{R}^d$ is promoted to a dynamical field and integrated over subject to shape regularity constraints. The total action is local on $\Lambda$, 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 special 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. 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 and matter fields $(U,\Phi)$ in finite volume by treating $x$ as an additional multiplet of scalar fields on $\Lambda$. 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. For example, in d=4 we recover the standard one-loop $\beta$-function in a scalar $\phi^4$ 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 artefacts relative to a fixed lattice at matched bare parameters.