A Euclidean signature and compact gauge group

1. Mar 8, 2016

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!

2. Mar 8, 2016

jambaugh

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

3. Mar 8, 2016

Einj

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

4. Mar 8, 2016

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?

5. Mar 9, 2016

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.

6. Mar 10, 2016

Einj

Oh I see! Thanks a lot