Spin and Lorentz Transformations
We could try the same approach to determine the behavior of spin under general Lorentz transformations, but we would immediately discover a major obstacle. Unlike SO(3), the group of Lorentz transformations SO(3,1) is non-compact and therefore does not have any faithful, unitary, finite-dimensional representations.
In case of spin 1/2 particles, it is possible to find a construction that includes both a finite-dimensional representation and a scalar product that is preserved by this representation. We associate a 4-component Dirac spinor with each particle. These spinors transform under Lorentz transformations according to the law
where are gamma matrices and is an antisymmetric 4x4 matrix parametrizing the transformation. It can be shown that the scalar product
is preserved. It is not, however, positive definite, so the representation is not unitary.