8 1. PROLOGUE

neutron, and in quark-antiquark pairs to make mesons, of which the most im-

portant are the pions; baryons are Fermions, whereas mesons are Bosons. The

leptons comprise electrons,

muons,2

tauons, and their associated neutrinos. The

fundamental Bosons are the mediating particles of the various interactions: pho-

tons (electromagnetism), W

±

and Z particles (weak interaction), gluons (strong

interaction), and the Higgs Boson (the field that generates particle masses). There

is also presumably a graviton for gravity, but it has not been observed as of this

writing. For each of the particles mentioned above there is also an antiparticle; in

some cases (photons, gravitons, Higgs Bosons, Z particles, and some gluons and

mesons) the particle and antiparticle coincide. Griﬃths [59] is a good reference for

the physics of elementary particles.

The rest masses of the particles that we will encounter most frequently in this

book are as follows:

mγ = 0, me = 0.511 MeV, mp = 938.280 MeV, mn = 939.573 MeV,

mμ = 105.659 MeV, mπ± = 139.569 MeV, mπ0 = 134.964 MeV,

where the subscripts denote photons (γ), electrons, protons, neutrons, muons,

charged pions, and neutral pions. The mass of a proton in grams is

1.672635×10−24;

the reciprocal of this number, approximately

6×1023,

is essentially Avogadro’s num-

ber, which to this one-significant-figure accuracy can be considered as the number

of nucleons (protons or neutrons) in one gram of matter.3

A few characteristic lengths: The diameter of a proton is about 10−13 cm. Di-

ameters of atoms are in the range 1–5×10−8 cm. (Heavy atoms are about the same

size as light ones, because although the former have more electrons, the increased

charge of the nucleus causes them to be packed in more tightly.) Electrons, as far

as we know, are point particles, but the Compton radius or classical radius of an

electron is the number r0 such that the mass of the electron is equal to the electro-

static energy of a solid ball of radius r0 with a uniform charge distribution of total

charge e, which in natural units is

e2/4πme:

(1.7) r0 =

e2

4πme

= 2.8 ×

10−13

cm.

1.3. Mathematical prologue: some Lie groups and Lie algebras

In this section we review the basic facts and terminology concerning the Lie

groups and Lie algebras that play a central role in quantum mechanics and relativity.

The Lorentz and orthogonal groups and their Lie algebras. The Lorentz

group O(1, 3) is the group of all linear transformations of

R4

(or 4×4 real matrices)

that preserve the Lorentz inner

product:4

A ∈ O(1, 3) ⇐⇒

(Ax)μ(Ay)μ

=

xμyμ

for all x, y ⇐⇒

A†gA

= g.

We employ the notation discussed in §1.1; in particular, g is the matrix (1.1).

2Muons were originally called “μ-mesons.” This is a misuse of the word “meson” according

to modern usage.

3The

actual mass of an atom is generally less than the sum of the masses of its nucleons, even

after adding in the mass of its electrons. One has to subtract the binding energy of the nucleus.

4O(1,

3) is more commonly called O(3, 1), but I prefer to keep the arguments of O(·, ·) in the

same order as the coordinates on

R4.