Euclidean signature and compact gauge group

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  • Thread starter Einj
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Hello everyone,
I have been reading around that when performing the analytic continuation to Euclidean space ([itex]t\to-i\tau[/itex]) one also has to continue the gauge field ([itex]A_t\to iA_4[/itex]) 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!
 

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  • #2
jambaugh
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I believe it has to do with keeping the representation of the gauge transformation unitary and finite dimensional.
 
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Do you have any idea on how to show it or any source I could look at? Thanks for you reply!
 
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jambaugh
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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?
 
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vanhees71
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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.
 
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Oh I see! Thanks a lot
 

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