Abstract
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.