3Discrete symmetries

III The Standard Model



3 Discrete symmetries
We are familiar with the fact that physics is invariant under Lorentz transfor-
mations and translations. These were relatively easy to understand, because
they are “continuous symmetries”. It is possible to “deform” any such transfor-
mation continuously (and even smoothly) to the identity transformation, and
thus to understand these transformations, it often suffices to understand them
“infinitesimally”.
There is also a “trivial” reason why these are easy to understand the
“types” of fields we have, namely vector fields, scalar fields etc. are defined by
how they transform under change of coordinates. Consequently, by definition,
we know how vector fields transform under Lorentz transformations.
In this chapter, we are going to study discrete symmetries. These cannot be
understood by such means, and we need to do a bit more work to understand
them. It is important to note that these discrete “symmetries” aren’t actually
symmetries of the universe. Physics is not invariant under these transformations.
However, it is still important to understand them, and in particular understand
how they fail to be symmetries.
We can briefly summarize the three discrete symmetries we are interested in
as follows:
Parity (P): (t, x) 7→ (t, x)
Time-reversal (T ): (t, x) 7→ (t, x)
Charge conjugation (C ): This sends particles to anti-particles and vice
versa.
Of course, we can also perform combinations of these. For example, CP corre-
sponds to first applying the parity transformation, and then applying charge
conjugation. It turns out none of these are symmetries of the universe. Even
worse, any combination of two transformations is not a symmetry. We will
discuss these violations later on as we develop our theory.
Fortunately, the combination of all three, namely CPT, is a symmetry of a
universe. This is not (just) an experimental observation. It is possible to prove
(in some sense) that any (sensible) quantum field theory must be invariant under
CPT, and this is known as the CPT theorem.
Nevertheless, for the purposes of this chapter, we will assume that C, P, T
are indeed symmetries, and try to derive some consequences assuming this were
the case.
The above description of P and T tells us how the universe transforms
under the transformations, and the description of the charge conjugation is just
some vague words. The goal of this chapter is to figure out what exactly these
transformations do to our fields, and, ultimately, what they do to the
S
-matrix.
Before we begin, it is convenient to rephrase the definition of P and T as
follows. A general Poincar´e transformation can be written as a map
x
µ
7→ x
0µ
= Λ
µ
ν
x
ν
+ a
µ
.
A proper Lorentz transform has
det
Λ = +1. The transforms given by parity and
time reversal are improper transformations, and are given by
Definition (Parity transform). The parity transform is given by
Λ
µ
ν
= P
µ
ν
=
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
.
Definition
(Time reversal transform)
.
The time reversal transform is given by
T
µ
ν
=
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
.

Contents