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)
However, it looks that Unitary for "Yang-Mills theory with non-compact or non-semi-simple gauge groups" may be an issue.
- 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?
- 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."
- Supportive opinion for Unitary (Yang-Mills theory with non-compact gauge groups G):
or at least some attempts:
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?