Quantum Field Theory and the Standard Model by Schwartz M.D. z-lib.org · Problem 2.1

Lorentz invariance and second quantization — Schwartz M.D. z-lib.org

Physics — QFT · lorentz-transformations-perturbation-theory

Problem

Derive the transformations

x \to \frac{x+vt}{\sqrt{1-v^2}}

and

t \to \frac{t+vx}{\sqrt{1-v^2}}

in perturbation theory. Start with the Galilean transformation

x \to x + vt

. Add a transformation

t \to t + \delta t

and solve for

\delta t

assuming it is linear in

x

and

t

and preserves

t^2 - x^2

\mathcal{O}\left(v^2\right)

. Repeat for

\delta t

and

\delta x

to second order in

v

and show that the result agrees with the second-order expansion of the full transformations.