Transformations in Space—Time

Abstract

The review of translations, rotations, and the rotation group being completed, we are prepared to investigate their relativistic generalizations: translations, rotations, and velocity transformations in space—time, along with the homogeneous and inhomogeneous Lorentz groups. We shall again stress group closure properties and the resulting transformation laws and commutation relations. The new concept of a “boost” operation, a form of active Lorentz velocity transformation, is particularly useful for the construction of relativistic wave functions describing particles with spin. Helicity is another important notion which we shall develop. Spin−1/2 helicity spinors and spin−1 helicity polarization vectors are covariantly generalized from their rotation-group form. Massive-particle helicity states are contrasted with massless-particle helicity states with an eye to their use in later chapters.