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A Euclidean signature and compact gauge group

  1. Mar 8, 2016 #1
    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?

  2. jcsd
  3. Mar 8, 2016 #2


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    I believe it has to do with keeping the representation of the gauge transformation unitary and finite dimensional.
  4. Mar 8, 2016 #3
    Do you have any idea on how to show it or any source I could look at? Thanks for you reply!
  5. Mar 8, 2016 #4


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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?
  6. Mar 9, 2016 #5


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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.
  7. Mar 10, 2016 #6
    Oh I see! Thanks a lot
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