Classical Electrodynamics — PHY5347

HOMEWORK 14

(January 14, 2013)

Due on Thursday, January 31, 2013

PROBLEM 40

The following six 4 × 4 matrices are a representation of the infinitesimal generatorsof the Lorentz group.

S1 =

0 0 0 00 0 0 00 0 0 −10 0 1 0

; S2 =

0 0 0 00 0 0 10 0 0 00 −1 0 0

; S3 =

0 0 0 00 0 −1 00 1 0 00 0 0 0

;

K1 =

0 1 0 01 0 0 00 0 0 00 0 0 0

; K2 =

0 0 1 00 0 0 01 0 0 00 0 0 0

; K3 =

0 0 0 10 0 0 00 0 0 01 0 0 0

.

The first three (orthogonal) matrices (S1,S2,S3) are the generators of infinitesimal

rotations, whereas the last three (symmetric) matrices (K1,K2,K3) are the generatorsof infinitesimal Lorentz boosts. In this problem you will show that the generators ofthe Lorentz group satisfy some fundamental commutation relations. Note that herea symbolic manipulator (such as Maple or Mathematica) might come very handy.

(a) Show that the first three matrices satisfy the fundamental commutation relationsfor angular momentum. That is,

[Si, Sj] = εijkSk .

This last expression suggests that the difference of two infinitesimal rotationsis equivalent to a single infinitesimal rotation.

(b) Show that K transforms as a vector under rotations. That is,

[Si, Kj] = εijkKk .

(b) Show that boosts (as rotations) in general do not commute. That is,

[Ki, Kj] = −εijkSk .

This last expression suggests that the difference of two infinitesimal Lorentzboosts is equivalent to a single infinitesimal rotation.

PROBLEM 41

Particle accelerators use low mass stable particles to produce unstable particles withlarger masses. Take the kaon-photoproduction reaction as an example: γ+p→ K+Λ.Here a massless photon collides with a stationary proton to produce two “strange”particles: the kaon and the lambda hyperon. Compute the lowest possible energythat the photon must have in the laboratory frame (Elab

γ ) for the reaction to proceed.Follow closely the next steps (you will not regret it ...).

p’p

k k’K

p

!

"

(a) Use energy-momentum conservation to show that si=sf≡s, where

si=(k + p)2 and sf =(k′ + p′)2 .

Note that k, p, k′, and p′ are all relativistic four-vectors and k2 ≡ kµkµ. Alsonote that the quantity s is known as the “Mandelstam s-variable”.

(b) Compute si≡ slab in the laboratory frame. That is, in the frame in which theproton is at rest.

(c) Compute sf ≡ scm in the center-of-momentum frame. Note that in this framethe kaon and the lambda will be produced at rest.

(d) Argue that the Mandelstam variable s is a Lorentz scalar (namely, s remainsinvariant under a change of frame) to compute Elab

γ .

PROBLEM 42

Imagine a hypothetical space shuttle that is launched from the Kennedy Space Centerat a constant velocity. An astronaut aboard the space shuttle records the separationof the two Solid Rocket Boosters (SRBs) at exactly two minutes after launch. In con-trast, the same “space-time event” (namely, the separation of the SRBs) is recordedat two and a half minutes (150 seconds) by a NASA operator at the Kennedy SpaceCenter. At that precise moment the NASA operator sends a violet pulse of light (witha wavelength of 375 nm) to the shuttle.

(a) At what constant velocity is the shuttle flying?

(b) At what altitude (as measured by the NASA operator) do the two SRBs separate?

(c) At what time (as measured by an astronaut) does the light pulse arrive?

(d) What is the color (i.e., wavelength) of the light pulse as measured by an astro-naut?