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In the context of describing Real-time dynamics of Lattice gauge theories, have they specifically mentioned real-time in order to differentiate it from imaginary-time. Or does it have any other interpretation?

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Yes, the name "real time" is simply to differentiate it from "imaginary time," the two being related by Wick rotation. The same language is used, with the same meaning, both in continuum QFT and lattice QFT. Sometimes the adjectives "Lorentzian" and "Euclidean" are used as synonyms for "real time" and "imaginary time," respectively, because of what Wick rotation does to the metric signature.

I looked in four different books for a direct statement about this (three of the books were specifically about lattice QFT), but I didn't find a statement that spelled it out this directly. However, in my experience, this is what the words mean. I don't recall seeing any other interpretation.

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