Subatomic Physics

1.3. Special Relativity, Feynman Diagrams 5
where
Momentum and velocity are connected by the relation
1
γ =
(1 − β2 v
, β = . (1.6)
) 1/2 c
p = mγv. (1.7)
Squaring this expression and using Eqs. (1.2) and (1.6) yields
β ≡ v
c
pc
= . (1.8)
E
As one application of the Lorentz transformation to subatomic physics, consider the
muon, a particle that we shall encounter often. It is basically a heavy electron with
a mass of 106 MeV/c 2 . While the electron is stable, the muon decays with a mean
life τ:
N(t) =N(0)e −t/τ ,
where N(t) is the number of muons present at time t. If N(t1) muons are present
at time t1, onlyN(t1)/e are still around at time t2 = t1 + τ. The mean life of a
muon at rest has been measured as 2.2µ sec. Now consider a muon produced at the
FNAL (Fermi National Accelerator Laboratory) accelerator with an energy of 100
GeV. If we observe this muon in the laboratory, what mean life τlab do we measure?
Nonrelativistic mechanics would say 2.2µ sec. To obtain the correct answer, the
Lorentz transformation must be used. In the muon’s rest frame (unprimed), the
mean life is the time interval between the two times t2 and t1 introduced above,
τ = t2 − t1. The corresponding times, t ′ 2 and t′ 1 , in the laboratory (primed) system
are obtained with Eq. (1.5) and the observed mean life τlab = t ′ 2 − t′ 1 becomes
τlab = γτ.
With Eqs. (1.6) and (1.8), the ratio of mean lives becomes
τlab
τ
E
= γ = . (1.9)
mc2 With E = 100 GeV, mc 2 = 106 MeV, τlab/τ ≈ 10 3 . The mean life of the muon
observed in the laboratory is about 1000 times longer than the one in the rest frame
(called proper mean life).
Although we will not use relativistic notation (e.g., four-vectors) very often, we
introduce it here for convenience. The quantity A ≡ Aµ =(A0, A) is called a
four-vector if it transforms under a Lorentz transformation like (ct, x). The time
component is A0. The scalar product of two four vectors A and B is defined as
A · B =
3�
gµ,νAµBν = A0B0 − A · B, (1.10)
µ,ν=0