Understanding the Relative Velocity of Particle Beams

[guest1234]

How come relative velocity of the beams can be expressed by
v_{12} = \left| \vec{v}_1 - \vec{v}_2 \right| = \left|\frac{\vec{p}_1}{E_1} - \frac{\vec{p}_2}{E_2}\right|
where \vec{p}_{1,2} and E_{1,2} is the momenta and energies of incoming particles, respectively?
Similar equation is in Peskin & Schroeder, p 105. The dimensions don't add up ...

 

[WannabeNewton]

Velocity is dimensionless; $p$ and $E$ both have dimension 1. The desired expression follows essentially from the definition of $p$ relative to a Lorentz frame.

 

[guest1234]

doh... natural units... thx