Unit determinant for relevant symmetry groups in QFT

Question

When treating QFT we want our theory to be invariant under different symmetry groups, for example, the Standard Model is a non-abelian gauge theory with the symmetry group $U(1)×SU(2)×SU(3)$. Moreover, it is invariant under Lorentz transformations which form the Lorentz group $O(1,3)$ with the proper transformations subgroup $SO(1,3)$ In this example $S$ stands for Special, meaning that these transformations are represented by matrices with determinant $1$.

My questions are:

1. why is this requirement so important?
2. In particular, what would happen if we allow the determinant to be any real (or complex) number?

Answer 1

To preserve the notion of probabilities in QFT, symmetries must be implemented as unitary or antiunitary operations. See also Wigner's theorem. For finite-dimensional representations, the determinant of such operations must be just a phase factor.

Example: $U(1)$ symmetry.