Introduction

By Forshaw J.

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Extra info for An introduction to QED and QCD

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Probability Density and Current Density Starting from the Schr¨odinger equation for the wave function ψ(x, t), show that the probability density ρ = ψ ∗ ψ satisfies the continuity equation ∂ρ +∇·J=0 ∂t where h ¯ ψ ∗ (∇ψ) − (∇ψ ∗ )ψ J= 2im What is the interpretation of J? Rotations and the Pauli Matrices Show that a 3-dimensional rotation can be represented by a 3 × 3 orthogonal matrix R with determinant +1 (Start with x ′ = R x, and impose x′ ·x′ = x·x). Such rotations form the special orthogonal group, SO(3).

1) above. 2) using x µ xµ , or otherwise, that x′µ = xν (Λ−1 )ν µ Vectors Aµ and Bµ that transform like xµ and xµ are sometimes called contravariant and covariant respectively. A simpler pair of names is vector and covector. A particularly important covector is obtained by letting ∂/∂x µ act on a scalar φ: ∂φ ≡ ∂µ φ ∂xµ Show that ∂µ does transform like xµ and not xµ . Electromagnetism The four Maxwell equations are: ∇·E = ρ ǫ0 ∇×E =− ∇·B=0 ∂B ∂t ∇ × B = µ0 J + µ0 ǫ0 ∂E ∂t Which physical laws are represented by each of these equations?

At the energies used at lep you have to remember the diagram with a Z replacing the photon. 5 Show that the cross section for e+ e− → µ+ µ− is equal to 4πα2 /(3s), neglecting the lepton masses. 4. 4 Feynman diagrams for Compton scattering. where λ represents the polarisation of the photon of momentum p. Since the photon is massless, the sum is over the two transverse polarisation states, and must vanish when contracted with pµ or pν . 53) proportional to pµ or pν does not contribute to the cross section.

### An introduction to QED and QCD by Forshaw J.

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