Both in Wikipedia and on page 98 of Streater, Wightman, PCT, Spin and Statistics and all that, the second axiom postulates that a field must transform according to a representation of the Poincaré group.
I am a mathematician and I wonder if there are implicit assumptions there. Is any representation of the Poincaré group acceptable? Should such a representation be real-valued (that is, if $\rho$ is such a representation, for any element $g$ of the Poincaré group, is $\rho(g)$ a real-valued matrix)? Should it be orthogonal, unitary (that is, should the aforementioned matrices be orthogonal, unitary)?
EDIT: Let me rephrase my question.
As far as I understand, axiomatically,
- a QFT should come with a (strictly speaking, projective, but I'm not sure it's relevant here) unitary representation $U$, that is, a continuous morphism from the Poincaré group to the group of unitaries of the Hilbert space;
- a $n$-dimensional vector-valued field is described by an $n$-tuple of maps $\phi := (\phi_1,\cdots,\phi_n)$ from the Minkowski space to the set of operators on the Hilbert space (I also knew that, strictly speaking, we should consider distributions instead of maps but I don't think it is relevant here);
- now, under the action of a symmetry (that is, under conjugation by a unitary -from the unitary representation $U$) each coordinate of $\phi$ becomes a linear combination of all the coordinates, and the coefficients are stored in a matrix that Streater and Wightman call $S$ (equation 3-4 in Streater-Wightman, page 99).
This $S$ is a morphism from the Poincaré group to the group of square invertible complex matrices of size $n$, so, as a mathematician, I also call $S$ a (finite-dimensional) representation of the Poincaré group.
My question is: is there any implicit assumption on $S$?
I think my question is motivated by my fear of coordinates (I don't like the idea that a field $\phi$ should be implemented as a tuple; it looks that we are making an arbitrary choice of coordinates).