Here,pis the momentum of the incoming particle, which is related tothe energy byE^2 =p^2 c^2 +
m^2 c^4. Eliminatingp^2 between these two equations gives a formula forEcmdirectly in terms ofE
andm,
Ecm=√
2 m^2 c^4 + 2mc^2 E (21.41)We see that for large energy (mc^2 ≪E), the center of mass energy in the collider experiment gros
likeE, while in the fixed target experiment, it grows only like
√
E. Thus, the collider provides
much more bang for the buck!
21.6 A physical application of time dilation
There are all the funny applications of time dilation in relativity related to space-travel and the
twin paradox. But there are also very important and directlyobservable applications to particle
physics. Here, we shall provide one such example in terms of the observed life-time of unstable
particles.
For definiteness, we consider the example of a muonμ−particle; it is very similar to the electron,
but it is 200 times heavier, namelymc^2 ∼ 100 MeV, and unstable against weak interactions, with a
life-time of approximately 2× 10 −^6 sec. Muons are observed in cosmic ray showers. How far can the
muons travel since they were first created? Without taking time-dilation into account, they can
travel at most a distance 2× 10 −^6 sec×c= 600m, certainly not enough to travel intergalactically.
But if the energy of the muon is actuallyE=mc^2 γin the frame of observation, then they can
travel a distanceγtimes than 600. For a muon of energyE∼ 106 GeV, this comes to approximately
a distance of 10^7 km, more in the range of intergalactic traveling distances.
21.7 Relativistic invariance of the wave equation
In the preceding sections, we have introduced scalars, vectors and tensors. Now we shall be inter-
ested inscalar fields,vector fields, and only rarely also intensor fields. A functionφ(x), where
xμ= (ct,x), is said to be a scalar field is it behaves as follows under a Lorentz transformation,
φ(x)→φ′(x′) =φ(x) x′μ= Λμνxν (21.42)In English, this means that the fieldφ′in the new frame with coordinatesx′equals to old fieldφ
in the old frame with coordinatesx.
A suitable Lorentz-invariant equation for the propagationof a scalar field is given by the wave
equation,
(
−
1
c^2∂^2
∂t^2+ ∆
)
φ(x) =∂μ∂μφ(x) = 0 (21.43)Note that there is no suitable Lorentz invarinat 1-st order differential equation for a scalar field,
since∂μφ= 0 would imply thatφis constant. An important generalization of the wave equation