Student Friendly Introduction to QFT · Problem 11

Looking Backward and Looking Forward: Book Summary and What's Next

Physics — QFT · relative-velocity-lorentz-frames

Problem

(Problem added to revision of 2nd^{\rm nd} edition.) Show that, in any Lorentz frame where two particles have collinear 3-momenta, that the magnitude of the relative velocity between them (where subscripts 1 and 2 refer to the first and second particles) is vrel=(p1p2)2−m12m22E1E2.v_{rel} = \frac{\sqrt{(p_1 p_2)^2 - m_1^2 m_2^2}}{E_1 E_2}. Hint: Start with (17-38) vrel=∣v1−v2∣=∣p1E1−p2E2∣v_{rel} = |\mathbf{v}_1 - \mathbf{v}_2| = \left|\dfrac{\mathbf{p}_1}{E_1} - \dfrac{\mathbf{p}_2}{E_2}\right|, solve for E1E2vrel=∣E2p1−E1p2∣=(E2p1−E1p2)⋅(E2p1−E1p2)E_1 E_2 v_{rel} = |E_2 \mathbf{p}_1 - E_1 \mathbf{p}_2| = \sqrt{(E_2 \mathbf{p}_1 - E_1 \mathbf{p}_2)\cdot(E_2 \mathbf{p}_1 - E_1 \mathbf{p}_2)}, and carry out the inner product in the latter expression. Then use (p1p2)2=(E1E2−p1⋅p2)2=E12E22+∣p1∣2∣p2∣2−2E1E2p1⋅p2(p_1 p_2)^2 = (E_1 E_2 - \mathbf{p}_1 \cdot \mathbf{p}_2)^2 = E_1^2 E_2^2 + |\mathbf{p}_1|^2|\mathbf{p}_2|^2 - 2E_1 E_2 \mathbf{p}_1 \cdot \mathbf{p}_2 (where the last part is valid when 3-momenta are co-linear) and crank the algebra. Then, show that the special cases for vrelv_{rel} for the COM frame and particle \#2 stationary are (17-39) and (17-40), respectively.

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