By Rau J., Mueller B.

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4. 4. Prove that the modes uk = 1 e−iωk t+ik·x , 2(2π)3 ωk are orthonormal with respect to the scalar product f |g = −i d3 x[f (x)∂0 g ∗ (x) − g ∗ (x)∂0 f (x)] . 5. Show that the vacuum expectation value of the scalar ﬁeld Hamiltonian is given by 1 0| H |0 = − πm4 δ (3) (0)Γ (−2) . 4 As one can see, this expression is the product of two divergent terms. Note that normal ordering gets rid of this c–number divergent term. 6. Calculate the following commutators: (Assume that the scalar ﬁeld is a real one except for case (d)) (a) [P μ , φ(x)] , (b) [P μ , F (φ(x), π(x))], where F is an arbitrary polynomial function of ﬁelds and momenta, (c) [H, a† (k)a(q)] , (d) [Q, P μ ] , (e) [N, H], where N = d3 ka† (k)a(k) is the particle number operator, (f) d3 x[H, φ(x)]e−ip·x .

I) Properties of the matrix T , are given in Chapter 4. One should not forget that time reversal includes complex conjugation: τ (c . )τ −1 = c∗ τ . . τ −1 . Chapter 8. Canonical quantization of the Dirac ﬁeld 45 • The operator C generates charge conjugation in the space of spinors: Cψa (x)C −1 = (Cγ0T )ab ψb† (x) . J) Properties of the matrix C are given in Chapter 4. The charge conjugation transforms a particle into an antiparticle and vice–versa. • In this chapter we will very often use the identities: [AB, C] = A[B, C] + [A, C]B , [AB, C] = A{B, C} − {A, C}B .

B) Show that Φ| Aμ (x) |Φ is a pure gauge. 8. Let P μν = g μν − and P⊥μν = k μ k¯ν + k ν k¯μ , k · k¯ k μ k¯ν + k ν k¯μ , k · k¯ where k¯ μ = (k 0 , −k). ⊥ ⊥ Calculate: P μν Pνσ , P⊥μν Pνσ , P μν + P⊥μν , g μν Pμν , g μν Pμν , P μ ν P⊥νσ , if k 2 = 0. 9. 18. 1 lij M ij , 2 (a) Express J in terms of the potentials in the Coulomb gauge. (b) Express the spin part of the angular momentum in terms of aλ (k), a†λ (k) and diagonalize it. (c) Show that the states 1 a†± (q) |0 = √ (a†1 (q) ± ia†2 (q)) |0 , 2 are the eigenstates of the helicity operator with the eigenvalues ±1.

### From reversible quantum microdynamics to irreversible quantum transport by Rau J., Mueller B.

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