Quantum Field Theory and the Standard Model by Schwartz M.D. z-lib.org · Problem 2.6

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

Physics — QFT · lorentz-invariant-integration-measure

Problem

Lorentz invariance. (a) Show that

\int_{-\infty}^{\infty} dk^0 \, \delta(k^2 - m^2) \theta(k^0) = \frac{1}{2\omega_k}, \tag{2.103}

where

\theta(x)

is the unit step function and

\omega_k \equiv \sqrt{\vec{k}^2 + m^2}

. (b) Show that the integration measure

d^4k

is Lorentz invariant. (c) Finally, show that

\int \frac{d^3k}{2\omega_k} \tag{2.104}

is Lorentz invariant.