1Introduction

III Symmetries, Fields and Particles

1 Introduction
In this course, we are, unsurprisingly, going to talk about symmetries. Unlike
what the course name suggests, there will be relatively little discussion of fields
or particles.
So what is a symmetry? There are many possible definitions, but we are
going to pick one that is relevant to physics.
Definition
(Symmetry)
.
A symmetry of a physical system is a transformation
of the dynamical variables which leaves the physical laws invariant.
Symmetries are very important. As Noether’s theorem tells us, every sym-
metry gives rise to a conserved current. But there is more to symmetry than
this. It seems that the whole physical universe is governed by symmetries. We
believe that the forces in the universe are given by gauge fields, and a gauge
field is uniquely specified by the gauge symmetry group it works with.
In general, the collection of all symmetries will form a group.
Definition
(Group)
.
A group is a set
G
of elements with a multiplication rule,
obeying the axioms
(i) For all g
1
, g
2
G, we have g
1
g
2
G. (closure)
(ii)
There is a (necessarily unique) element
e G
such that for all
g G
, we
have eg = ge = g. (identity)
(iii)
For every
g G
, there exists some (necessarily unique)
g
1
G
such that
gg
1
= g
1
g = e. (inverse)
(iv) For every g
1
, g
2
, g
3
G, we have g
1
(g
2
g
3
) = (g
1
g
2
)g
3
. (associativity)
Physically, these mean
(i) The composition of two symmetries is also a symmetry.
(ii) “Doing nothing” is a symmetry.
(iii) A symmetry can be “undone”.
(iv) Composing functions is always associative.
Note that the set of elements G may be finite or infinite.
Definition
(Commutative/abelian group)
.
A group is abelian or commutative
if g
1
g
2
= g
2
g
1
for all g
1
, g
2
G. A group is non-abelian if it is not abelian.
In this course, we are going to focus on smooth symmetries. These are
symmetries that “vary smoothly” with some parameters. One familiar example
is rotation, which can be described by the rotation axis and the angle of rotation.
These are the symmetries that can go into Noether’s theorem. These smooth
symmetries form a special kind of groups known as Lie groups.
One nice thing about these smooth symmetries is that they can be studied by
looking at the “infinitesimal” symmetries. They form a vector space, known as
a Lie algebra. This is a much simpler mathematical structure, and by reducing
the study of a Lie group to its Lie algebra, we make our lives much easier. In
particular, one thing we can do is to classify all (simple) Lie algebras. It turns
out this isn’t too hard, as the notion of (simple) Lie algebra is very restrictive.
After understanding Lie algebras very well, we will move on to study gauge
theories. These are theories obtained when we require that our theories obey
some sort of symmetry condition, and then magically all the interactions come
up automatically.