Transformations in Space

Abstract

The invariance of the laws of physics under time translations, space translations, and rotations is naturally linked with the conservation laws of energy, linear momentum, and angular momentum, respectively. In Chapter 1 we have briefly described time translations in terms of the time development operator; spatial transformations, however, can be expressed most compactly and elegantly in the language of group theory. The rotation-group operators are the infinitesimal generators of angular momentum; the group closure property then leads to the familiar commutation relations of angular momentum. We shall also consider in detail the structure of vector and spinor wave functions, spin matrices, tensor operators, and the Wigner—Eckart theorem. Such a review of rotations will serve as a bridge between nonrelativistic and relativistic quantum theory. In particular it will serve as a basis for our treatment of Lorentz transformations in the next chapter.