Rotationally invariant dynamical lattice regulators for Euclidean quantum field theories

A bstract We introduce a dynamical lattice regulator for Euclidean quantum field theories on a fixed hypercubic graph Λ ≃ ℤ d , in which the embedding x : Λ → 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 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, 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. For example, 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 O (1)-scale geometry correlations, a direct SO (2)-connection diagnostic of short-range local twisting, and evidence for reduced axis-vs-diagonal cutoff artefacts relative to a fixed lattice at matched bare parameters.