In the Clay institute problem description of the Yang-Mills existence and mass gap problem it states that the quantum Yang Mills needs to be formulated in $\mathbb{R}^4$ space. I was wondering whether this meant it needed to be formulated in Euclidean space or Minkowski space? (It seems like Euclidean but the majority of QFTs are in Minkowski space, right?)
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2 Answers
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In QFT, we like to relate 4D Euclidean and Minkowski spacetimes by a Wick rotation. Rarely, if ever, does one break assumptions made in a proof by this (complex) coordinate transformation.
For many reasons it can be easier to work in Euclidean space and then Wick-rotate to get physical results. This is for example standard practice in solving loop integrals.
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$\mathbb{R}^{4}$ refers to four dimensional euclidean space.
answered May 26, 2014 at 23:17
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1$\begingroup$ is there any particular reason that the problem requests euclidean space rather than minkowski space? $\endgroup$ Commented May 26, 2014 at 23:29
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$\begingroup$ For lack of a better explanation, it seems to be for the sake of keeping the yang mills theory more general. Minkowski space is not the same as four dimensional euclidean space but it can be generated from it. Consider that in SR there are time-like light-like and space-like separations. If we treat all of these types of separations as if they exist on equal footing then one can form a 4-D euclidean map of causally and non causally related events. $\endgroup$ Commented May 27, 2014 at 0:14