Rotationally invariant dynamical lattice regulators for Euclidean quantum field theories

Abstract

We introduce a dynamical lattice regulator for Euclidean quantum field theories on a fixed hypercubic graph $\hat{I}›\simeq {\mathbb{Z}}^d$, in which the embedding $x:\hat{I}›\to {\mathbb{R}}^d$ is promoted to a dynamical field and integrated over subject to shape regularity constraints. The total action is local on $\hat{I}›$, 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 $(U,\hat{I}¦)$ in finite volume by treating $x$ as an additional multiplet of scalar fields on $\hat{I}›$. 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 ${\hat{I}}^2$-function in a scalar $\ddot{I}{•}^4$ 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.

Published version, with expanded acknowledgements; 51 pages, 4 figures