1. No category

Quantum Field Theory
Set 8
Exercise 1: Noether’s currents in classical mechanics
Consider the general Lagrangian of a classical system
X ma
X
L=
(~q˙a )2 −
V (|~qa − ~qb |) .
2
a
a6=b
Using the Noether procedure compute the Noether current associated to the following transformations
• time translations:
t0 = t − α,
~q 0a (t0 ) = ~qa (t)
• coordinate translations:
t0 = t,
~q 0a (t) = ~qa (t) + α
~
t0 = t,
q 0 ia (t) = Rij qaj (t)
• coordinate rotations:
Exercise 2: Noether’s current for U (1) global symmetry
Consider the Lagrangian of a free complex massive scalar field φ(x) = φ1 (x) + iφ2 (x):
L = ∂µ φ † ∂ µ φ − m2 φ † φ .
Show that the Lagrangian density is invariant under the following transformation
xµ −→ xµ ,
φ(x) −→ eiα φ(x),
where α does not depend on x.
Compute the Noether current associated to this symmetry and show that is conserved only if one uses the equations
of motion.
Exercise 3: Casimirs of Poincar´
e algebra
The physical states in a quantum field theory belong to irreducible, unitary representations of the Poincar´e group.
In order to construct such representations, it is useful to find the Casimir operators of the Poincar`e algebra, i.e.
those which commute with all its generators. For instance, the operator P 2 is a Casimir and its value determines
the mass of the physical state. After recalling the structure of the Poincar`e algebra:
[J µν , J ρσ ] = i(η νρ J µσ − η µρ J νσ − η νσ J µρ + η µσ J νρ )
[P µ , J ρσ ] = i(η µρ P σ − η µσ P ρ )
[P µ , P ν ] = 0
• Find an operator W µ commuting with P ν and transforming as a vector under the action of J µν
• Check that W 2 is a Casimir
• Consider the subspace of the representation for a massive particle at rest, P µ = (M, 0, 0, 0). What does W µ
correspond to?
Exercise 4: Noether’s charge as generator of transformations
Given a Lagrangian density L(φa , ∂µ φb ) consider the symmetry transformation defined by
xµ −→ x0µ = f (x) ' xµ − µi αi ,
φa (x) −→ φ0a (x0 ) = D[φ]a (f −1 (x0 )) ' φa (x0 ) + αi ∆ai (φ(x0 )),
• Show that charge Qi built starting from the Noether’s current is the generator of the transformation:
δα φa (x) ≡ φ0a (x) − φa (x) = αi {Qi , φa (x)} = αi ∆ai (x).
• Compute explicitly the charges associated to spacetime translation and Lorentz transformations in a scalar
field theory.
• Using the formalism of the Poisson brackets check that the charges related to translations and spatial
rotations generate the respective infinitesimal transformations.
2