# Lattice QFT: Non-homogenous lattice spacing

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 $\tau = it$ 

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  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

A similar question on why we can't simply fix $a$ is already asked here:

• As far as I know inhomogeneous lattices are used in many cases and they lead to some kind of anomaly due to the presence of the assymetrical space. Homogeneous lattices are the norm in lattice QCD. This is an interesting question nevertheless. – Abhishek Pal Jun 17 '16 at 9:32

0) Note that in general, there is no need for the lattice spacings in different direction to be the same, as long as they all go to zero in the continuum limit. We can, for example, take the lattice spacing in the $x$ direction to be $a$, and the spacing in the $y$ direction to be $2a$. This is frequently done to achieve better resolution without the numerical cost of a finer lattice in all directions.
a) This has nothing to do with Wick rotation. Wick rotation is $t\to i\tau$, irrespective of how you choose the metric or the lattice spacing. The difficulty with a non-trivial metric is that the action may become complex.