Exercise 2: Noether charges and ladder operators

Quantum Field Theory

Set 10

Exercise 1: Heisenberg’s representation

Write the expansion of a free massive real field in the Heisenberg’s representation in terms of the time dependent ladder operators a( ~ k, t) and a ^† ( ~ k, t), and show that it satisfies the classical equations of motion.

Exercise 2: Noether charges and ladder operators

Starting from the expression of a free massive real scalar field and its conjugate momentum φ(~ x, t) =

Z d ³ k (2π) ³ 2k 0

h

a( ~ k, t) + a ^† (− ~ k, t) i e ^{i~ k·~ x} , π(~ x, t) =

Z d ³ k

(2π) ³ 2k ₀ (−ik 0 ) h

a( ~ k, t) − a ^† (− ~ k, t) i e ^{i~ k·~ x} ,

where a( ~ k, t) = e ^−ik

^t a( ~ k), write in terms of ladder operators a and a ^† the Noether’s charges associated to:

• Translations: P µ = R

d ³ x T 0µ ,

• Boosts K i = R

d ³ x (x 0 T 0i − x i T 00 ),

• Rotations J ij = R

d ³ x (x i T 0j − x j T 0i ).

Show that Q _i (t) = Q _i (0) and therefore the charges do not depend on time, as expected.

Exercise 3: Noether’s charges as generators

Given the canonical commutation rules at equal time [φ(~ x, t), π(~ y, t)] = iδ ³ (~ x − ~ y),

• show that the commutator of

J ^ij , φ(~ x, t)

is a generator of an infinitesimal transformation, namely that J ^ij , φ(~ x, t)

= i∆ ^ij (~ x, t), where ∆ is the variation of the field (at fixed coordinate) induced by the symmetry transformation of which J ^ij is the Noether’s charge.

• Repeat for

K ⁱ , φ(~ x, t) and

P ⁱ , φ(~ x, t) .

• From these results deduce

J ^ij , P ^k

, φ(~ x, t)

.