I am interested why we fix the lattice spacing, $a$ to be homogenous in all dimensions. After a Wick rotation, $a=i \epsilon$ where $\epsilon=t_{i+1} - t_{i}$ and with euclidean time given by $\tau = it$ [1]
a. Why does this Wick rotation necessarily set $a$ as homogenous throughout all dimensions of the lattice?
Is the answer due to the fact that the curvature of the Minkowski metric is defined by the difference between the time and euclidean spatial dimensions. I think that rotating to euclidean time then ensures that all dimensions are euclidean and so should have equal separation (spacings) by definition.
b. Does this reasoning hold when extending QFT to QCD? I think this is true since QCD is still defined in the Poincaré group and naturally one would still work with Minkowski metrics.
c. Can non-homogenous lattice spacing be used to study gravitation in QFT/QCD? After some (brief) searching I found an article [2] which uses the following lattice,
which seems like a very logical approach to studying the effects of curved space-time and QFT: What are the drawbacks to non-homogenous lattice studies vs. continuum approaches?
References
[1] A Statistical Approach to Quantum Mechanics, Creutz & Freedman :: Alternate Link
[2] Lattice QCD in curved spacetimes, Yamamoto :: ArXiV link
A similar question on why we can't simply fix $a$ is already asked here: