the article Galilean group for the central extension of its Lie algebra.
In 3 + 1 dimensions, a third Casimir invariant is W2, where somewhat analogous to the Pauli–Lubanski pseudovector of relativistic mechanics.
Using Schur's lemma, in an irreducible unitary representation, all these Casimir invariants are multiples of the identity.
In 3 + 1 dimensions, when In m > 0, we can write, w = ms for the third invariant, where s represents the spin, or intrinsic angular momentum.
More generally, in n + 1 dimensions, the generators L and C will be related, respectively, to the total angular momentum and center-of-mass moment by From a purely representation-theoretic point of view, one would have to study all of the representations; but, here, we are only interested in applications to quantum mechanics.
In fact, treating the energy E as the Hamiltonian, differentiating with respect to P, and applying Hamilton's equations, we obtain the mass-velocity relation m v→ = P→.
Because of transitivity, we know the unitary irrep contains a nontrivial linear subspace with these energy-momentum eigenvalues.
(This subspace only exists in a rigged Hilbert space, because the momentum spectrum is continuous.)
As far as we can tell, only the case which transforms trivially under the little group has any physical interpretation, and it corresponds to the no-particle state, the vacuum.
They represent an instantaneous transfer of non-zero momentum across a (possibly large) distance.
In the 3 + 1-dimensional Galilei group, the boost generator may be decomposed into with W→ playing a role analogous to helicity.