Consider the Lagrangian of a massive vector field. Compute the conjugate momenta of A µ and show that Π 0 = 0.

Quantum Field Theory

Set 18

Exercise 1: Equivalence of Hamiltonian and Lagrangian formalism for massive vectors

Consider the Lagrangian of a massive vector field. Compute the conjugate momenta of A µ and show that Π 0 = 0.

This means that A 0 is not a dynamical variable but can be expressed in terms of the other fields. Show that A 0 = − _M ¹

∂ i Π ⁱ . Then show that the Hamiltonian is:

H = Z

d ³ x 1

2 Π ⁱ Π ⁱ + 1

2M ² (∂ _i Π ⁱ ) ² + 1

4 F ^ij F ^ij + 1

2 M ² A ⁱ A ⁱ

.

Using the following commutation relations A i (~ x, t), Π ^j (~ y, t)

= iδ _i ^j δ ³ (~ x − ~ y), [A _i (~ x, t), A _j (~ y, t)] =

Π ⁱ (~ x, t), Π ^j (~ y, t)

=

A ₀ (~ x, t), Π ^j (~ y, t)

= 0, [A i (~ x, t), A 0 (~ y, t)] = − 1

M ² [A i (~ x, t), ∂ m Π ^m (~ y, t)] = i

M ² ∂ _i ^(x) δ ³ (~ x − ~ y), show that the Hamilton equations of motion are equivalent to the Lagrange equations of motion.

Exercise 2: Generator of rotations

Consider the Lagrangian of a massive vector field. Show that the Noether current associated to Lorentz transfor- mation is

J _αβ ^µ = −F ^µρ (η ρα A β + x α ∂ β A ρ ) + F ^µρ (η ρβ A α + x β ∂ α A ρ ) +

δ ^µ _α x β − δ _β ^µ x α

L.

Consider the component J _ij ⁰ and thus find the Noether charge associated to rotations. Show that

J k = 1 2 ijk

Z

J _ij ⁰ d ³ x = ijk

Z

d ³ x (Π i A j − Π m x i ∂ j A m ) .

Substitute the expansion of Π i (x) and A j (x) in terms of a i (k) and a i (k) ^† and show that the generator of rotations can be written as

J k = i ijk

Z dΩ ~ k

a i ( ~ k)a ^† _j ( ~ k) − a m ( ~ k)

k i

∂

∂k ^j

a ^† _m ( ~ k)

.