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Physicists are familiar working with Yang-Mills theory with compact and semi-simple gauge groups $G$ (Lie groups).

However, it is not entirely clear the formulation of Yang-Mills theory with non-compact or non-semi-simple gauge groups.

One issue is that physical system governed by quantum theory and QFT, we hope to have

  • unitarity (say the partition function and the probability will be always conserved)

  • locality

However, it looks that Unitary for "Yang-Mills theory with non-compact or non-semi-simple gauge groups" may be an issue.

  1. Opposition for Unitarity:

See discussion here: "a necessary condition for unitarity is that the Yang Mills (YM) gauge group $G$ with corresponding Lie algebra $g$ should be real and have a positive (semi)definite associative/invariant bilinear form " --- Why is the Yang-Mills gauge group assumed compact and semi-simple?

  1. Neutral opinion for Unitary:

On gauge theories for non-semisimple groups A.A. Tseytlin The non-positivity of the metric implies that these theories are apparently non-unitary. However, the special structure of interaction terms (degenerate compared to non-compact YM theories) suggests that there may exist a unitary `truncation'. "Yang-Mills theories for non-semisimple real Lie algebras which admit invariant non-degenerate metrics. These 4-dimensional theories have many similarities with corresponding WZW models in 2 dimensions and Chern-Simons theories in 3 dimensions."

  1. Supportive opinion for Unitary (Yang-Mills theory with non-compact gauge groups G):

or at least some attempts:

YANG-MILLS FIELD QUANTIZATION WITH NON-COMPACT GAUGE GROUP Article in Modern Physics Letters A 07(29) · November 2011

Unitary gauge theories of noncompact groups - Kevin Cahill and Sertaç Özenli - Phys. Rev. D 27, 1396 two pape papers] "It is noted that the use of an internal metric field allows one to gauge noncompact internal-symmetry groups without sacrificing unitarity. The possibility that such theories could be rendered renormalizable is discussed."

Question: So are their sharp mathematical statement to be made for "Yang-Mills theory with non-compact or non-semi-simple gauge groups" --- will the unitarity and locality be an issue?

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  • $\begingroup$ Are (feynman-integral) and (path-integral) two different things? (I just wonder whether we need two separate tags for them.) The tag (feynman-integral) already exists on the site. $\endgroup$ – Martin Sleziak Nov 23 '18 at 7:18
  • $\begingroup$ Witten (1991, Introduction) says the reason is not unitarity but positive energy. $\endgroup$ – Francois Ziegler Nov 23 '18 at 13:41
  • $\begingroup$ @Martin Sleziak, thanks, path integral is more general, I think we can merge (feynman-integral) into (path-integral). $\endgroup$ – wonderich Nov 23 '18 at 15:50
  • $\begingroup$ "The basic idea of the path integral formulation can be traced back to Norbert Wiener, who introduced the Wiener integral for solving problems in diffusion and Brownian motion. This idea was extended to the use of the Lagrangian in quantum mechanics by P. A. M. Dirac in his 1933 article. A specific method was developed much later in 1948 by Richard Feynman." $\endgroup$ – wonderich Nov 24 '18 at 0:14
  • $\begingroup$ @wonderich I have asked for advice about these tags also in chat but I got no response so far. If you think that there should be a synonym, probably the best thing would be to post that suggestion in the thread on meta which was created for tag synonyms: Help improve tagging! (My main objection to the synonym is that the phrase path integral is commonly used also with different meaning.) $\endgroup$ – Martin Sleziak Nov 24 '18 at 5:42

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