# Gauge Theory

## Table of Contents

## Mathematical formalism

*Gauge theory* involves a fibre bundle in which the fibre at each point, , of the base space consists of possible coordinate bases for use when describing the values of the objects at that point.

One chooses a particular coordinate basis at each point (a *local section* of the fibre bundle), and express the values of the objects of the theory (usually "fields" in the physicst's sense) using this basis.

Two such configurations are equivalent if they are related by a transformation of this abstract coordinate basis (a change of local section, or a **gauge transformation**).

A **gauge** is just a choice of (local) section of some principal bundle.

A **gauge transformation** is just a transformation between two such sections / "choices".

## What is gauge? - Terry Tao

### Notation

- coordinate system identifies some geometric object with standard object
- is an
*isomorphism*of that standard object - new coordinate system
- is a family (fibre bundle) of geometric (or combinatorial) objects (fibres) parametrized by some
*base point*, where is the base space

### Definitions

A **homogenous space** for a group is a non-empty manifold or topological space on which acts transitively.

The elements of are called the **symmetries** of .

In physics, **gauge fixing** (also called **choice of gauge**) denotes a procedure for coping with redundant degrees of freedom in field variables.

A **gauge theory** represents each physically distinct configuration of the system as an *equivalence class* of detailed local field configurations.

Any two detailed configurations in the same equivalence class are related by a **gauge transformation**.

### Stuff

- Gauge
- "coordinate system" that varies depending on one's "location" wrt. some base space or "parameter space"
- Gauge transform
- change of coordinates applied to
*each*such location - Gauge theory
- model for some physical or mathematical system to which
*gauge transforms*can be applied (and is typically*gauge invariant*, in that all physically meaningful quantities are left unchanged under gauge transformations) - (no term)
- Dimensional analysis is nothing more than the analysis of the scaling symmetries in one's coordinate systems.
- (no term)
- Consider general case where we have a
*family*(or fibre bundle) of geometric (or combinatorial)*objects*(or fibres) parametrised by some*base point*

### Examples

#### Circle bundle of the sphere

- Space of directions in a plane (which can be viewed as the circle of unit vectors) can be identified with the standard circle after picking:
- orientation
- reference direction

- Consider the sphere instead on the surface of the earth
- Each point on surface, there is a circle of directions that one can travel along
Defines the collection

of all such circles is then a

*circle bundle*with*base space*(know as the**circle bundle**)Structure group of this bundle is the circle group

if one

*preserves orientation*- Suppose every point on the earth , we have wind (ignoring the hairy ball theorem)
- Wind direction is collection of representatives from the fibres of teh fibre bundle
- Such a collection is known as a section of the fibre fundle

Can define function

i.e. a function which "converts" the collection of winds into points on the earth

- Requires choosing a
**gauge**for this*circle bundle*, i.e. selecting orientation and reference direction for each point

- Requires choosing a
- Thus, we have numerical representation of the "winds", which allows usage of analytical tools (e.g. differentiation, integration, Fourier transforms, etc.)

#### Orienting / directing an undirected graph

- Undirected graph with being the space of
*vertices*and the space of*edges* - (
*not*the space of vertices!) - can be oriented / directed in two different ways; let be the pair of directed edges of arising in this manner
- is a fibre bundle with base space and with each fibre isomorphic to the standard two-element set with structure group .
We can

*choose*orientations for each edge, thus creating a gauge (or a section)of the bundle

- Identify the bundle with the trivial bundle :
- preferred oriented edge (or preferred orientation) of each is assigned
- other oriented edge of is assigned