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