# Symmetries of Particles and Fields

## Table of Contents

- is the symmetries of
*electrodynamics* - is the symmetries of the
*weak interactions* - is the symmetries of the
*color (quarks)*

## Metric tensor

has real eigenvalues :

Rescale sucht that keeping fixed. Rescaling follows

which we can do

*separately*for each , hencewhich we call the

**canonical basis**.

### Example: space time

such that

Usually, we use the opposite signature metric, using and $x_{4} = x_{0}:

which gives

#### Raising and lowering indices using the metric tensor

if and only if is *invertible*.

Can use metric to raise and lower indices

Can raise indices of metric

were we've contracted over . Then

which implies is the /inverse of .

##### Same with Levi-Civita symbols

and derivatives wrt. $x^{μ}

##### If metric is definite

If metris is definite, the we can choose basis such that , i.e. can ignore distinction between upper and lower indices!

## Matrix groups

### Lorentz group

is called which has .

## 1D groups

All 1D Lie groups are such that the parameter can be chosen such that

Furthermore, all 1D Lie groups are Abelian.

and are *isomorphic*.

Let be a dimensional *compact Abelian* Lie group. Then

If is compact, then for every representation there exists an equivalent *unitary* representation.

## Lie algebras

### Generators

Consider group where we choose with such that

Taylors thm expand about , so we can write

for "small" .

Define

Matrices are the **generators** of the group.

The is put in to make *hermitian* (in a unitary representation).

#### Examples

##### Example:

which implies

Thus, the **generators** are *hermitian* and *traceless*.

##### Example:

We generally choose Pauli matrices as generators:

##### Example:

Generators are Gell-Mann

##### Example:

- Don't have constraint .
- Generators of , are also the part a subset of the generators of , and "including the trace" we have as the generators for

##### Example:

for , is *real*, where we generally omit the in the def. of generators.

Taking Taylor expansion, as for , we have

i.e. are *anti-symmetric*.

##### Example: and

For , generally choose:

For , generally choose

Additional constraint is automatically satisfied (why?).

#### Commutations

Consider with .

Expanding each of these using their Taylor expansion up to and *including* 2nd order terms, then take the commutation relation between and , the expansion of and , respectively, we have

For this to be true we must have

for some numbers .

Which implies

where are called the structure constants.

#### Exponentiation

Consider 1D subgroup , labeled by the single parameter .

Chose such that

Diff. wrt.

and set , which gives us a differential equation for all :

which has the solution

Or, if you consider this in the familiar notation used in Diff. Geom. for the exponential map, we can write this as

where , s.t.

where , i.e. maps from the Lie algebra to the Lie group.

Going back to the earier notation of , we can write

where

which implies

### Adjoint representation

Can define an **adjoint representation**:

then

##### Examples

- and

take generators , with

and so the structure constants are .

Consider

Then

So if we write , then we can write

which is just

*rotation*in !Therefore the

*adjoint representation*of is the*defining*or*fundamental*representation of . hencefor elements "close to the identity".

In general, the adjoint representation is

*not*the fundamental representation of some other group…

### Killing form

The **Killing form**

real symmetric matrix.

Usually we'll choose diagonal basis such that

#### Examples

So we rescale generators such that

#### Application of killing form

- Can use Killing form to lower and raise indices
E.g. kan define structure consts with all lower indices

This is an invariant tensor → can be written as a product of generators

## Invariant Tensors

Let be a Lie group.

An **invariant tensor**

is a tensor that remains invariant under the actions of group , i.e.