# Wick rotation and spinors

I am quite familiar with use of Wick rotations in QFT, but one thing annoys me: let's say we perform it for treating more conveniently (ie. making converge) a functional integral containing spinors; when we perform this Wick rotation, in a way we change the metric to (-,+,+,+) to (+,+,+,+), so the invariant group is no more SO(3,1) but SO(4) and (SO(4) being compact and the spinor representation non unitary) spinors don't carry finite dimensional representation of this group. So I feel like we shouldn't be talking anymore about this objects, but only about vectors of SO(4).

Is my fear justified? or where am I wrong in my reasoning?

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You may find this paper interesting: arxiv.org/abs/hep-th/9611043 – Qmechanic Feb 21 '12 at 16:48
Could you specify more precisely where is the problem? Probably, illustrate it with some functional integral. – Misha Feb 21 '12 at 17:06

When I studied at first course and investigated the special theory of relativity the lecturer said about old interpretation of relativity. In this approach instead pseudo-euclidean metric and four-vectors $(t,\bf x)$ people use euclidean metric and four-vectors $(it,\bf{x})$. But it does not mean that we use SO(4) group! We use also SO(3,1) group but we do some change of variables.
This “change of variables” is imaginary, and this means we can no longer be using $\mathrm{SO}(3,1)$, which is a real group. We can either use the complex version, the rather unfamilliar ${}^{\mathbf C}\mathrm{SO}(3,1) = \mathrm{SO}(4,\mathbf C)$, or choose an appropriate real version of it, which is exactly $\mathrm{SO}(4)$. (You can’t always get away with being cavalier about complexifications.) – Alex Shpilkin Apr 17 at 13:21