Classical Electrodynamics · Problem 11.11

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · lorentz-transformation-commutator-expansion

Problem

Two Lorentz transformations differ by an infinitesimal amount. In the notation of Section 11.7 they are represented by

A_1 = e^L

A_2 = e^{L+\delta L}

. Without using explicit matrix representations show that to first order in

\delta L

the Lorentz transformation

A = A_2 A_1^{-1}

can be written as

A = I + \delta L + \frac{1}{2!}[L, \delta L] + \frac{1}{3!}[L, [L, \delta L]] + \frac{1}{4!}[L, [L, [L, \delta L]]] + \cdots

Hint: The early terms can be found by brute force, but alternatively consider the Taylor series expansion in

\lambda

of the operator

A(\lambda) = e^{\lambda(L+\delta L)}e^{-\lambda L}

and then put

\lambda = 1