199 Appendix D Formation Time and Brick Estimates D.1 γ$_{c}$ for Bethe-Heitler to LPM Regime D.1.1 Formation Physics By the Heisenberg Uncertainty Principle, the formation time, τ ′ $_{f}$, for a gluon (or photon) of energy k 0 ′ separating from its parent quark (or gluon)—without loss of generality taken as a quark from now on—in the rest frame of the quark, is where the primed coordinates are in the rest frame of the quark and unprimed coordinates will be in the lab frame. To find τ$_{f}$ in the lab frame, one must first boost the gluon energy in the lab frame into the rest frame of the quark, then boost back to the lab frame. Take a quark with 4-momentum p = ( p $^{0}$, p $^{z}$, p ) and gluon of momentum k = ( k $^{0}$, k $^{z}$, k ). The velocity of the quark is β = p $^{z}$/p $^{0}$. τ ′ f ∼ 1 k 0 $_{′}$, (D.1)