# Continuum limit of lattice field theory

Just a simple question for lattice QCD experts, is continuum limit of lattice field theory a relativistic quantum field theory?

Because i heard that lattice QCD is done in imaginary time, producing a ground-state (lowest-energy state) in infinite temporal extend limit. And that two-point correlation functions computed in imaginary time can be analytically continued to real-time. But for massive particles (such as quarks) to be relativistic, their kinetic energies should be equal to or greater than $$mc^2$$.

## 1 Answer

This isn't really specific to lattice theory - doing stuff in "imaginary time" means you're doing Euclidean QFT. This isn't so much "relativistic" or "non-relativistic" as it is just a formal computation - no one thinks this Euclidean theory directly describes any physical setting.

However, analytic continuation of the results to "real time" then gives you quantities for a relativistic Minkowskian theory. It is not at all obvious that analytically continuing the results of the a priori unphysical Euclidean theory should have anything to do with what the physical Minkowskian theory does, but this is the miracle of Wick rotation - it turns out that this actually works (although in many cases you need to think a bit carefully about what Euclidean theory corresponds to what Minkowskian theory, see e.g. this question and its answers), and the formal statement of this is called the Osterwalder-Schrader theorem.

• Well, my question was more about continuum limit of lattice field theory. Is only a data extrapolated in continuum limit gives a relativistic QFT after analytic continuation? Does it mean that a lattice with any finite lattice spacing is non-relativistic? Sep 13, 2022 at 16:50
• @Peter I don't understand the distinction you're making here - the continuum limit of a Euclidean lattice field theory is presumably a Euclidean QFT, no? Wick rotation doesn't care how you produced your Euclidean QFT. Sep 13, 2022 at 16:52
• @Peter there’s two issues that are separate, the continuum limit and wick rotation. At finite lattice spacing, your Lorentz/Euclidean group of symmetries is broken by the lattice and you get discretisation artifacts. And usually people use relativistic to mean the real-time theory, but this is separate Sep 13, 2022 at 16:53
• Thanks, now it's clear. Sep 13, 2022 at 17:04