6 1. PROLOGUE

Quantity Dimensions Defining relation

Momentum

[mlt−1]

p=mv

Angular momentum [ml2t−1] l = x × p

Force [mlt−2] F = m(dv/dt)

Energy [ml2t−2] E =

1

2

m|v|2 or F · dx or mc2 or . . .

Action [ml2t−1] S = (Ekinetic − Epotential) dt

Table 1.1. Some basic quantities of mechanics.

mechanical units. However, for purposes of fundamental physics it is better to take

electric charge as the basic quantity. Naively, the simplest procedure (“Gaussian

units”) is to define the unit of charge so that the constant in Coulomb’s law is equal

to one. That is, having chosen a system of units for mechanics, one sets the unit

of charge so that the force between two point charges is equal in magnitude to the

product of the charges divided by the square of the distance between them. From a

slightly more sophisticated point of view, however, a better procedure (“Heaviside-

Lorentz units”) is to take the constant in Coulomb’s law to be 1/4π, essentially

because −1/4π|x| is the fundamental solution of the Laplacian (∇2[−1/4π|x|] = δ),

and this is the procedure we shall follow. Its practical advantage is that it makes

all the 4π’s disappear from Maxwell’s equations. Either way, the dimension [q]

of charge is related to mechanical dimensions by [q2l−2] = [force] = [mlt−2], or

[q] = [m1/2l3/2t−1].

The idea of setting proportionality constants equal to one can be carried further.

In relativistic physics it is natural to relate the units of length and time so that the

speed of light c is equal to one; doing this makes [l] equivalent to [t]. Moreover,

in quantum mechanics it is natural to choose units so that Planck’s constant is

equal to one. Since has dimensions [ml2t−1], in conjunction with the condition

c = 1 this makes [m] equivalent to [l−1] or [t−1]. We have c = 299792458 m/s

(an exact equality according to the current oﬃcial definition of the meter) and

= 1.054589 ×

10−34

kg ·

m2/s,

so the equivalences of the basic MKS units are as

follows:

1 second

∼

= 299792458 meter

∼

= 8.522668 ×

1050 kilogram−1.

These large numbers make the MKS system awkward for the world of particle

physics. Seconds and centimeters (accompanied by large negative powers of 10)

are still generally used for times and lengths, but the other commonly used unit is

the electronvolt (eV), which is the amount of energy gained by an electron when

passing through an electrostatic potential of one volt, or its larger relatives the

mega-electronvolt (MeV) or giga-electronvolt (GeV):

1 eV = 1.602176 ×

10−19

kg ·

m2/s2,

1 MeV =

106

eV, 1 GeV =

109

eV.

The eV is a unit of energy, but setting c = 1 makes Einstein’s E = mc2 into an

equality of mass and energy, and in the subatomic world this is not just a formal

equivalence but an everyday fact of life. The eV, or more commonly the MeV or

GeV, is therefore used as a unit of mass:

1 MeV = 1.782661 ×

10−27

gram.