Lattice QFT: Non-homogenous lattice spacing

Question

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:

• Why can't we set the lattice spacing 'a' in lattice QCD?

Answer 1

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.

Note that QCD is subtle, beacuse the classical action is scale invariant. This implies that different lattice spacings are actually encoded in different coupling constants for plaquettes in different 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.

b) I don't understand what you are asking. Obviously QCD is a subset of all QFTs, not the other way around.

c) Yes, you should be able to study QCD in a non-trivial background metric, as indicated in the paper you link to. Obviously, we don't know how to extend this to dynamical backgrounds (this would solve the problem of quantum gravity).