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 gauge for this circle bundle, i.e. selecting orientation
- 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
- preferred oriented edge