# Euclidean signature and compact gauge group

• A

## Main Question or Discussion Point

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!

Related Quantum Physics News on Phys.org
jambaugh
Gold Member
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
Gold Member
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