By Laurent Schwartz

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Regularize the integral I= d4 k using Pauli–Villars regularization. 1 1 , k 2 (k + p)2 − m2 Chapter 11. 5. Compute kα kβ kµ kν kρ kσ . (k 2 )n Also, ﬁnd the divergent part of the previous integral for n = 5. Apply the dimensional regularization. 6. Consider the interacting theory of two scalar ﬁelds φ and χ: L= 1 1 1 1 (∂φ)2 − m2 φ2 + (∂χ)2 − M 2 χ2 − gφ2 χ . 2 2 2 2 (a) Find the self–energy of the χ particle, −iΠ(p2 ). (b) Calculate the decay rate of the χ particle into two φ particles. (c) Prove that Im Π(M 2 ) = −M Γ.

This is the Gupta–Bleuler method of quantization. H) Chapter 9. Canonical quantization of the electromagnetic ﬁeld 51 while A0 = 0. I) [π i (t, x), π j (t, y)] = 0 , (3) where π = E and δ⊥ij (x − y) is the transversal delta function given by (3) δ⊥ij (x − y) = 1 (2π)3 d3 keik·(x−y) δij − ki kj k2 . J) [a†λ (k), a†λ (q)] = 0 . • The Feynman propagator for the electromagnetic ﬁeld is given by iDFµν (x − y) = 0| T (Aµ (x)Aν (y)) |0 . 1. G) prove that [Aµ (t, x), A˙ ν (t, y)] = −ig µν δ (3) (x − y) .

4! Find the expression for the self–energy and the mass shift δm. 8. The Lagrangian density is given by L= 1 1 m2 2 λ (∂µ σ)2 + (∂µ π)2 − σ − λvσ 3 − λvσπ 2 − (σ 2 + π 2 )2 , 2 2 2 4 2 where σ and π are scalar ﬁelds, and v 2 = m 2λ is constant. Classically, π ﬁeld is massless. Show that it also remains massless when the one–loop corrections are included. 9. Find the divergent part of the diagram Prove that this diagram cancels with the diagram of the reverse orientation inside the fermion loop.