Chapter 42

The Poincar ́e Group and its

Representations

In chapter 19 we saw that the Euclidean groupE(3) has infinite dimensional
irreducible unitary representations on the state space of a quantum free particle.
The free particle Hamiltonian plays the role of a Casimir operator: to get irre-
ducible representations one fixes the eigenvalue of the Hamiltonian (the energy),
and then the representation is on the space of solutions to the Schr ̈odinger equa-
tion with this energy. There is also a second Casimir operator, with integral
eigenvalue the helicity, which further characterizes irreducible representations.

The case of helicity±^12 (which uses the double coverE ̃(3)) occurs for solutions
of the Pauli equation, see section 34.2.
For a relativistic analog, treating space and time on the same footing, we
will use instead the semi-direct product of space-time translations and Lorentz
transformations, called the Poincar ́e group. Irreducible representations of this
group will again be labeled by eigenvalues of two Casimir operators, giving
in the cases relevant to physics one continuous parameter (the mass) and a
discrete parameter (the spin or helicity). These representations can be realized
as spaces of solutions for relativistic wave equations, with such representations
corresponding to possible relativistic elementary particles.
For an element (a,Λ) of the Poincar ́e group, withaa space-time translation
and Λ an element of the Lorentz group, there are three different sorts of actions
of the group and Lie algebra to distinguish:

• The action
  x→Λx+a
  on a Minkowski space vectorx. This is an action on a real vector space,
  it is not a unitary representation.


• The action
  ψ→u(a,Λ)ψ(x) =S(Λ)ψ(Λ−^1 (x−a))