Quantum Mechanics for Mathematicians

Problem 4:
The Pauli-Lubanski operator is the four-component operator

W 0 =−P·L, W=−P 0 L+P×K

(same notation as in problem 1) Show that

W^2 =−W 0 W 0 +W 1 W 1 +W 2 W 2 +W 3 W 3

commutes with the energy-momentum operator (P 0 ,P)
Show thatW^2 is a Casimir operator for the Poincar ́e Lie algebra.

B.19 Chapters 43 and 44

Problem 1:
Show that, assuming the standard Poisson brackets

{φ(x),π(x′)}=δ(x−x′), {φ(x),φ(x′)}={π(x),π(x′)}= 0

Hamilton’s equations for the Hamiltonian

h=

∫

R^3

1

2

(π^2 + (∇φ)^2 +m^2 φ^2 )d^3 x

are equivalent to the Klein-Gordon equation for a classical fieldφ.

Problem 2:
In section 44.1.2 we studied the theory of a relativistic complex scalar field,
with aU(1) symmetry, and found the charge operatorQ̂that gives the action
of the Lie algebra ofU(1) on the state space of this theory.

• Show that the charge operatorQ̂has the following commutators with the
  fields
  [Q,̂φ̂] =−φ,̂ [Q,̂φ̂†] =φ̂†

and thus thatφ̂on charge eigenstates reduces the charge eigenvalue by 1,

whereasφ̂†increases the charge eigenvalue by 1.

• Consider the theory of two identical complex free scalar fields, and show
  that this theory has aU(2) symmetry. Find the four operators that give
  the Lie algebra action for this symmetry on the state space, in terms of a
  basis for the Lie algebra ofU(2).
  Note that this is the field content and symmetry of the Higgs sector of
  the standard model (where the difference is that the theory is not free,
  but interacting, and has a lowest energy state not invariant under the
  symmetry).