DR. SVEN KRIPPENDORFHilary Term 2017

Supersymmetry: Example Sheet 1Due by lecture on February 1st

1. In the lectures we have written the N = 1 SUSY algebra:

{Qα, Qβ} = 2 (σµ)αβ Pµ ,

{Qα, Qβ} = {Qα, Qβ} = 0 ,

[Pµ, Qα] =[Pµ, Qα

]= 0 ,

[Qα,Mµν ] = (σµν)α

β Qβ .

Show that these commutation relations hold! It is useful to consider the Jacobi identity andproperties of Qα as a spinor and as an operator transforming under the Lorentz algebra.

2. Prove the following identities:

(σµ)αβ(σµ)γδ = 2δδαδγ

β,

Tr(σµσν) = 2ηµν ,

(σµσν) + (σν σµ)αβ = 2ηµνδα

β .

Using these, show that

V µ =1

2(σµ)ααVαα

and the decomposition (1/2, 0)⊗ (0, 1/2) = (1/2, 1/2) :

ψαχα =1

2(σµ)αα (ψσµχ) .

3. Show that σµν = i4(σµσν − σν σµ) satisfy the Lorentz algebra and are therefore the

generators of the Lorentz group in the spinor representations.

4. Prove the explicit map from SL(2,C)→ SO(3, 1), Λµν = 1

2Tr(σµNσνN

†).

5. Using the definition ψα ≡ εαβψβ, prove that under SL(2,C), ψα transforms as

ψα → ψβ(N−1)βα .

Find an expression for N−1 in terms of N.

6. Prove thatψχ ≡ ψαχα = −ψαχα ≡ χψ .

Check that both ψχ and ψχ are scalars under SL(2,C), and verify the Fierz identities:

(θψ)(χη) = −1

2(θσµη)(χσµψ) ,

(θσµθ)(θσν θ) =1

2ηµν(θθ)(θθ) .

7. Using the fact that(σµν)α

β(σµν)γδ = εαγε

βδ + δαδδγ

β ,

1

prove that (1/2, 0)⊗ (0, 1/2) = (0, 0)⊕ (1, 0), i.e.

ψαχβ =1

2εαβ(ψχ) +

1

2(σµνεT )αβ(ψσµνχ) .

8. Prove that the Pauli-Ljubanski vector Wµ ≡ 12εµνρσP

νMρσ satisfies the following commu-tation relations:

[Wµ, Pν ] = 0 ,

[Wµ,Wν ] = iεµνρσWρP σ ,

[W µ, Qα] = −i(σµν)αβQβPν

Show that W µWµ is a Casimir operator of the Poincare algebra, but not of the super-Poincaree algebra.

9. Write down the N = 1 supersymmetry algebra in four-component Dirac formalism.Assume the supersymmetry generators are Majorana spinors.

10. Write explicitly all components of the N = 1 supersymmetry multiplets correspondingto:

1. Massive particles of maximum spins 1/2 and 1.

2. Massless particles of maximum helicity 2 and 1.

Verify in each case that the number of bosons equals the numbers of fermions. Explainhow the Higgs mechanism works in supersymmetry (i.e. which massless multiplets turn intowhich massive multiplets?).

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