III Symmetries, Fields and Particles

1Introduction

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

of elements with a multiplication rule,

obeying the axioms

(i) For all g

, g

∈ G, we have g

∈ 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)

−1

∈ G

such that

−1

= g

−1

g = e. (inverse)

(iv) For every g

, g

∈ G, we have g

) = (g

. (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

= g

for all g

, g

∈ 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.