The aim of this chapter is to look at the structures of Lie groups related to space and spacetime. We look very briefly at SO (3) which we have studied earlier quite extensively, then the Euclidean group E (3) which acts on physical space; the Galilei group on spacetime underlying nonrelativistic Galilean–Newtonian mechanics; the homogeneous Lorentz group SO (3, 1) and its double cover SL (2,ℂ); and finally the Poincaré group P basic to special relativity. In each case we look at the defining representation resulting from action on spacetime; useful descriptions of the group, the composition law and inverses; the structure of the Lie algebra and possible neutral elements which are permitted; and then a study of the UIR's of the concerned group. This involves constructing Casimir invariants, physical interpretation, etc. In the SO (3, 1) and SL (2,ℂ) cases, we also look at all their finite dimensional nonunitary representations. We will see similarities to the discussion of induced group representations in Chapter 7, in connection with representations of the group E(3) and the Poincaré group.

SO (3) andSU (2)

We studied these two groups and their UIR's in some detail in Chapter 3. We saw that SU (2) is a two-fold covering of SO (3). Here we first deal with the possible presence of neutral elements in a hermitian representation of the Lie algebra generators Jj . As we have seen in Chapter 9, Eq. (9.76), to handle ray representations of SO (3) and SU (2) in quantum mechanics we must allow for the presence of neutral elements djk in the basic angular momentum commutation relations:

But these are immediately and easily eliminated: antisymmetry djk = ∈dkj implies djk = ∈jkldl for some real dl ; then if we redefine J′jj = Jj + dj we get the standard commutation relations without any neutral elements:

This happens since physical space is three dimensional, there being no need to invoke the Jacobi identity explicitly. At the same time, no further shifts in J′l are permitted.