Gauge Theory

Table of Contents

Mathematical formalism

Gauge theory involves a fibre bundle in which the fibre at each point, gauge_theory_0e03351d36fc1c219ce0fe604a4b773183b04f2e.png, of the base space gauge_theory_fbe98aaf8359e7eed3ba031caeaf6a3c13ae8690.png 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 gauge_theory_7639a9a7b0bbf7b62753bc1eebb76f333edfa152.png identifies some geometric object gauge_theory_8939ac39ed16887f3b3c033a47d4ab60d120251a.png with standard object gauge_theory_2de98136973021abb46a5a3fc1e4318bafb84264.png
  • gauge_theory_d186190e5b44416df9084e7b9f1f050068c5659e.png is an isomorphism of that standard object
  • new coordinate system gauge_theory_af5f3ce088ab22ce617f22abb062c439424f6fd4.png
  • gauge_theory_7c16fafc1839bbfe8eede62bcafae99154358acd.png is a family (fibre bundle) of geometric (or combinatorial) objects (fibres) parametrized by some base point gauge_theory_9e47faa96d468cca71dc58fc8a06f4b5200df197.png, where gauge_theory_0207be880056b9a69e22e729dd37bced29cd174a.png is the base space

Definitions

A homogenous space for a group gauge_theory_2de98136973021abb46a5a3fc1e4318bafb84264.png is a non-empty manifold or topological space gauge_theory_0207be880056b9a69e22e729dd37bced29cd174a.png on which gauge_theory_2de98136973021abb46a5a3fc1e4318bafb84264.png acts transitively.

The elements of gauge_theory_2de98136973021abb46a5a3fc1e4318bafb84264.png are called the symmetries of gauge_theory_0207be880056b9a69e22e729dd37bced29cd174a.png.

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 gauge_theory_9e47faa96d468cca71dc58fc8a06f4b5200df197.png

Examples

Circle bundle of the sphere

  • Space of directions in a plane gauge_theory_4c9f35fdc4fb21e899dc415b1e151e20c59ca82b.png (which can be viewed as the circle of unit vectors) can be identified with the standard circle gauge_theory_3fd99b9e5e831fa78d04a21863b6b230a7efffe9.png after picking:
    • orientation
    • reference direction
  • Consider the sphere instead on the surface gauge_theory_0207be880056b9a69e22e729dd37bced29cd174a.png of the earth
  • Each point gauge_theory_9e47faa96d468cca71dc58fc8a06f4b5200df197.png on surface, there is a circle gauge_theory_0eeb8e1b919d97f0b3dff9ae12afa1b6a7b7c5f3.png of directions that one can travel along
  • Defines the collection

    gauge_theory_e652cbb7b5e860de51e22d29e126b01b16aa8e6b.png

    of all such circles is then a circle bundle with base space gauge_theory_0207be880056b9a69e22e729dd37bced29cd174a.png (know as the circle bundle)

  • Structure group of this bundle is the circle group

    gauge_theory_61a24030ea16d33515c019207dd66b7b43059645.png

    if one preserves orientation

  • Suppose every point gauge_theory_3c314f80373742988ad542a6d4ce66a111b17847.png on the earth gauge_theory_0207be880056b9a69e22e729dd37bced29cd174a.png, we have wind gauge_theory_f0d7aacc34f0cbdab7fd659a4b5904e3f982afd5.png (ignoring the hairy ball theorem)
  • Wind direction is collection gauge_theory_11ce9fc52358d9700c7b5a774fa989cbf3c34e6a.png of representatives from the fibres of teh fibre bundle gauge_theory_d8a42c558043dfbef369a0c8a9d420750ff6f4ae.png
    • Such a collection is known as a section of the fibre fundle
  • Can define function

    gauge_theory_5f0f7a983dca9e448e6266a0f38e45f0f51632cd.png

    i.e. a function which "converts" the collection of winds gauge_theory_fe480f547555026d210b55b5d4ef758235f32832.png into points on the earth

    • Requires choosing a gauge for this circle bundle, i.e. selecting orientation gauge_theory_2ce87375afa669e4d926c0e2dd79315739b94460.png and reference direction gauge_theory_b1203b557919ab6b762b115124624ce74614fcc7.png for each point gauge_theory_9e47faa96d468cca71dc58fc8a06f4b5200df197.png
  • 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 gauge_theory_a5177eee309f705d90af4a252c1cb0e6a1e3a05d.png with gauge_theory_79dc7b66c784a82e557ef3df794e82b67a292b17.png being the space of vertices and gauge_theory_fbe98aaf8359e7eed3ba031caeaf6a3c13ae8690.png the space of edges
  • gauge_theory_4ece8678389dae5d48f31dcb3f9f7157d43d989c.png (not the space of vertices!)
  • gauge_theory_f7122caae385461c6e5aa659ff6ac59f3bf56747.png can be oriented / directed in two different ways; let gauge_theory_948a3d15222535f5263ae206895631f1592d78d2.png be the pair of directed edges of gauge_theory_eaac7646cb022aa963b396c0e936a271d8f5e34f.png arising in this manner
  • gauge_theory_4a577de8d60e8a6ed2cf61abc1f227c70b4d2407.png is a fibre bundle with base space gauge_theory_0207be880056b9a69e22e729dd37bced29cd174a.png and with each fibre isomorphic to the standard two-element set gauge_theory_6cb166a95055463f0e14f11956d984e80b8cd9a5.png with structure group gauge_theory_ebe1d0ca1ce56ea49ee3e093e26e807caa4507c9.png.
  • We can choose orientations for each edge, thus creating a gauge (or a section)

    gauge_theory_38f5341dd981bad3b07347cc4dc967cf97cd382b.png

    of the bundle gauge_theory_4a577de8d60e8a6ed2cf61abc1f227c70b4d2407.png

  • Identify the bundle gauge_theory_4a577de8d60e8a6ed2cf61abc1f227c70b4d2407.png with the trivial bundle gauge_theory_cff34cc24cd2cc2d0c8acab1f299dcee47f51c07.png:
    • preferred oriented edge gauge_theory_f083f10d08996d863d0718e70d7182c46957b112.png (or preferred orientation) of each gauge_theory_f7122caae385461c6e5aa659ff6ac59f3bf56747.png is assigned gauge_theory_4cf7d77a3c0863ef649cb0dae7d07dbc2427bdc1.png
    • other oriented edge of gauge_theory_f7122caae385461c6e5aa659ff6ac59f3bf56747.png is assigned gauge_theory_052816dcdf0e697dfad5c09e15301b2294db4e2d.png