8Lie groups in nature

III Symmetries, Fields and Particles

8.1 Spacetime symmetry

In special relativity, we have a metric

ds

2

= −dt

2

+ dx

2

+ dy

2

+ dz

2

.

The group of (orientation-preserving and) metric-preserving symmetries gives

us the Lorentz group

SO

(3

,

1). However, in certain cases, we can get something

more (or perhaps less) interesting. Sometimes it makes sense to substitute

τ

=

it

,

so that

ds

2

= dτ

2

+ dx

2

+ dy

2

+ dz

2

.

This technique is known as Wick rotation. If we put it this way, we now have a

symmetry group of SO(4) instead.

It happens that when we complexify this, this doesn’t really matter. The

resulting Lie algebra is

L

C

(SO(3, 1)) = L

C

(SO(4)) = D

2

.

Its Dynkin diagram is

This is not simple, and we have

D

2

= A

1

⊕ A

1

.

In other words, we have

so(4) = su(2) ⊕ su(2).

At the Lie group level,

SO

(4) does not decompose as

SU

(2)

× SU

(2). Instead,

SU(2) × SU(2) is a double cover of SO(4), and we have

SO(4) =

SU(2) × SU(2)

Z

2

.

Now in general, our fields in physics transform when we change coordinates.

There is the boring case of a scalar, which never transforms, and the less boring

case of a vector, which transforms like a vector. More excitingly, we have objects

known as spinors. The weird thing is that spinors do have an

so

(3

,

1) action,

but this does not lift to an action of

SO

(3

,

1). Instead, we need to do something

funny with double covers, which we shall not go into.

In general, spinors decompose into “left-handed” and “right-handed” com-

ponents, known as Weyl fermions, and these correspond to two different rep-

resentations of

A

1

. We can summarize the things we have in the following

table:

Field so(3, 1) A

1

⊕ A

1

Scalar 1 (ρ

0

, ρ

0

)

Dirac Fermion LH ⊕ RH (ρ

1

, ρ

0

) ⊕ (ρ

0

, ρ

1

)

Vector 4 (ρ

1

, ρ

1

)

However, the Lorentz group

SO

(3

,

1) is not all the symmetries we can do to

Minkowski spacetime. These are just those symmetries that fix the origin. If we

add in the translations, then we obtain what is known as the Poincar´e group.

The Poincar´e algebra is not simple, as the translations form a non-trivial ideal.