Types of quantum field thories
This includes notable the fields that carry the three fundamental forces of the standard model of particle physics:
Other examples include formal physical models.
The group in these examples is called the gauge group of the theory.
The above examples of gauge fields consisted of cocycles in degree- differential cohomology.
More generally, a higher gauge theory is a quantum field theory whose field configurations are cocycles in more general differential cohomology, for instance higher degree Deligne cocycles or more generally cocycles in other differential refinements, such as in differential K-theory.
This generalization does contain experimentally visible physics such as
But a whole tower of higher and generalized gauge theories became visible with the study of higher supergravity theories,
There are various models that realize gravity also as a gauge theory.
Sometimes one see the view expressed that gauge symmetry is “just a redundancy” in the description of a theory of physics, for instance in that among observables it is only the gauge invariant ones which are physically meaningful.
This statement however
In the presence of magnetic charge (and then even in the absence of chiral fermion anomalies?) the standard would-be action functional for higher gauge theories may be ill-defined. The Green-Schwarz mechanism is a famous phenomenon in differential cohomology by which such a quantum anomaly cancels against that given by chiral fermions.
The following tries to give an overview of some collection of gauge fields in physics, their models by differential cohomology and further details.
cocycle in lowest degree nonabelian differential cohomology
- electromagnetism (see below)
- electroweak force field strength
- strong nuclear force field
field strength: the electric field and magnetic field , locally at a point
on : underlying class in integral cohomology is the magnetic charge
gauge field: models and components
|physics||differential geometry||differential cohomology|
|gauge field||connection on a bundle||cocycle in differential cohomology|
|instanton/charge sector||principal bundle||cocycle in underlying cohomology|
|gauge potential||local connection differential form||local connection differential form|
|field strength||curvature||underlying cocycle in de Rham cohomology|
|minimal coupling||covariant derivative||twisted cohomology|
|BRST complex||Lie algebroid of moduli stack||Lie algebroid of moduli stack|
|extended Lagrangian||universal Chern-Simons n-bundle||universal characteristic map|
An introduction to concepts in the quantization of gauge theories is in
A standard textbook on the BV-BRST formalism for the quantization of gauge systems is in
Discussion of abelian higher gauge theory in terms of differential cohomology is in
Tohru Eguchi, Peter Gilkey, Andrew Hanson, Gravitation, gauge theories and differential geometry, Physics Reports 66:6 (1980) 213—393 pdf
This was established in
An exposition of the relation to geometric Langlands duality is in
Hermann Weyl, Raum, Zeit, Materie: Vorlesungen über die Allgemeine Relativitätstheorie, Springer Berlin Heidelberg 1923
The manuscript of Weyl’s first book on mathematical physics, Space – Time – Matter (STM) (Raum – Zeit – Materie), delivered to the publishing house (Springer) Easter 1918, did not contain Weyl’s new geometry and proposal for a UFT. It was prepared from the lecture notes of a course given in the Summer semester of 1917 at the Polytechnical Institute (ETH) Zürich. Weyl included his recent findings only in the 3rd edition (1919) of the book. The English and French versions (Weyl 1922b, Weyl 1922a), translated from the fourth revised edition (1921), contained a short exposition of Weyl’s generalized metric and the idea for a scale gauge theory of electromagnetism. (Scholz)
Quick reviews include
More comprehensive historical accounts include
Lochlainn O'Raifeartaigh, The Dawning of Gauge Theory, Princeton University Press (1997)
Lochlainn O'Raifeartaigh, Norbert Straumann, Gauge Theory: Historical Origins and Some Modern Developments Rev. Mod. Phys. 72, 1-23 (2000).