Euclidean signature and compact gauge group

[Einj]

Hello everyone,
I have been reading around that when performing the analytic continuation to Euclidean space ($t\to-i\tau$) one also has to continue the gauge field ($A_t\to iA_4$) in order to keep the gauge group compact.
I already knew that the gauge field had to be continued as well but I didn't know anything about keeping the gauge group compact. Can someone explain it to me?

Thanks!

 

[jambaugh]

I believe it has to do with keeping the representation of the gauge transformation unitary and finite dimensional.

 

[Einj]

Do you have any idea on how to show it or any source I could look at? Thanks for you reply!

 

[jambaugh]

I would guess any decent grad text on field theory might cover this. I don't know of one myself. I recall reading something on the complexification in Ryder's book "Quantum Field Theory" but I don't recall him speaking of justification. I don't recall Kaku addressing it directly in his book but I haven't peeked in his text in a while and didn't read it extensively when I last did. Maybe someone else has a suggestion?

 

[vanhees71]

The point is that you entirely go from Minkowski space with a fundamental form of signature (1,3) (or (3,1) if you come from the east coast ;-)) to Euclidean space, i.e., the proper orthochronous Lorentz group is substituted by O(4). So all four-vectors become Euclidean vectors. The gauge group stays as it is, i.e., a compact Lie group.

 

[Einj]

Oh I see! Thanks a lot