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Check the properties listed in Subsect

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Exercice 1. Letαi, β∈L(V) the operators defined in Sect. 5.4 of the notes.

1)Show that αi andβ are selfadjoint for (·|·).

2) Show that fork∈Rdσ(αiki+mβ) ={(k),−(k)}for(k) = (k2+m2)1/2. Exercice 2. Check the properties listed in Subsect. 5.5.1 of the map:

U :

( Solsc(KG)∼C0∞(Rd)⊕C0∞(Rd)→(2)12h (π, ϕ)7→π+ iϕ.

Exercice 3. Check that the Klein-Gordon field Φ(x) defined in Subsect. 5.5.2 satisfies the Klein-Gordon equation:

(−2+m2)Φ(x) = 0, in distribution sense.

Same question for the Dirac field Ψ(x) defined in Subsect. 5.6.2.

Exercice 4. We use the notation in Subsect. 5.6.1. On Γa(Z) we define thetotal charge operator by:

Q:= dΓ(1lh+⊕ −1lh−).

Show that

spec(Q) =Z. Check that

eiθQψ(∗)(h)e−iθQ=ψ(∗)(eiθh), θ∈R, h∈h.

Deduce from this fact that:

Qψ(h) =ψ(h)(Q−1), Qψ∗(h) =ψ∗(h)(Q+ 1),

i.e. that ψ(h) resp. ψ∗(h) decreases , resp. increases the total charge by 1.

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