The text computed and quoted the other commutators of the Lorentz algebra

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Relativity & Gravitation · Special relativity: spacetime and kinematics

Invariance, Ch Lie groups and Lie algebras: rotations, SU(2) and the double cover

Problem

The text computed

[K_x,K_y]

and quoted the other commutators of the Lorentz algebra. Work in the coordinates

(ct,x,y,z)

with the active boosts

\Lambda_x(\phi)=e^{-i\phi K_x}

and

\Lambda_y(\phi)=e^{-i\phi K_y}

, where

K_y

has

i

in the

(ct,y)

and

(y,ct)

entries and zeros elsewhere, and let

J_x

be the

3\times3

rotation generator placed in the spatial block. ((a)) Multiply the

4\times4

matrices to find

[J_x,K_y]

, and show that it equals

iK_z

. ((b)) Let

A=-i\alpha K_x

and

B=-i\beta K_y

, with rapidities

\alpha

and

\beta

. Use

e^Ae^B=e^{A+B+\frac12[A,B]+\cdots}

to write

\Lambda_x(\alpha)\Lambda_y(\beta)

and

\Lambda_y(\beta)\Lambda_x(\alpha)

each as a single exponential, correct to second order in the rapidities. Show that the two exponents differ only by a multiple of

J_z

, and find it. ((c)) Take as given that

e^{-i(\alpha K_x+\beta K_y)}

is a single boost, and that

e^{X+Y}=e^Xe^Y

to second order when

X

is of first order and

Y

of second order in the rapidities. What does

\Lambda_x(\alpha)\Lambda_y(\beta)

do to a particle at rest, besides setting it moving? Give the angle and sense of the effect for

\alpha=\beta=0.20

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