Gauge theory

Gauge theory
Joel Holdsworth (Joelholdsworth) · Public domain · source
Name	Gauge theory
Field	Mathematical physics, Theoretical physics
Authors	Hermann Weyl, James Clerk Maxwell
Introduced	20th century

Gauge theory is a class of field theories characterized by local symmetries under continuous transformations associated with Lie groups. It underpins modern descriptions of fundamental interactions in Particle physics and provides deep connections between Differential geometry, Topology, and Mathematical physics. Central figures in its development include Hermann Weyl, Paul Dirac, Chen-Ning Yang, and Robert Mills.

Introduction

Gauge theory originated from attempts to generalize James Clerk Maxwell's formulation of electromagnetism and to understand phase invariance in quantum mechanics associated with Paul Dirac. Early conceptual advances by Hermann Weyl and later formalization by Yang and Mills led to the identification of local symmetry groups such as U(1), SU(2), and SU(3). It became the organizing principle for the Standard Model of Particle physics through the work of Sheldon Glashow, Steven Weinberg, and Abdus Salam.

Mathematical formulation

A gauge theory is defined by a principal bundle with structure group given by a Lie group such as U(1), SU(2), SU(3), or more general compact groups studied in Lie group theory. Connections on principal bundles correspond to gauge fields and are locally represented by Lie algebra-valued one-forms as in the Cartan formalism developed in Élie Cartan's work. Curvature two-forms (field strengths) satisfy Bianchi identities that reflect covariance under gauge transformations related to results by Élie Cartan and Henri Poincaré. Action functionals are constructed from invariant polynomials like the Yang–Mills action influenced by concepts in Calculus of variations and methods used by Richard Feynman and Julian Schwinger in quantum field theory. Global issues involve transition functions examined in Fiber bundle theory and classification results connected to the Atiyah–Singer index theorem by Michael Atiyah and Isadore Singer.

Examples and physical applications

Abelian gauge theory exemplified by Electromagnetism uses the group U(1) and reproduces Maxwell's equations historically derived by James Clerk Maxwell; quantum electrodynamics (QED) was developed by Richard Feynman, Julian Schwinger, and Sin-Itiro Tomonaga. Non-abelian gauge theories include Quantum chromodynamics (QCD) with group SU(3) describing the work of Murray Gell-Mann and Heinrich Leutwyler, and the electroweak theory combining SU(2)×U(1) as formulated by Sheldon Glashow, Steven Weinberg, and Abdus Salam. Gauge principles also appear in Condensed matter physics in descriptions of Superconductivity influenced by Lev Landau and Vitaly Ginzburg, and in topological phases related to studies by Frank Wilczek and Xiao-Gang Wen. Classical solutions such as instantons and monopoles were analyzed by Alexander Belavin, George 't Hooft, and Ernst Stueckelberg with implications in Cosmology considered by Alan Guth and Andrei Linde.

Quantization and quantum field theory

Quantization of gauge theories employs methods like path integral quantization introduced by Richard Feynman and canonical quantization techniques elaborated by Paul Dirac. Gauge fixing procedures such as the Faddeev–Popov method and BRST symmetry were developed by Ludvig Faddeev, Victor Popov, and Igor Tyutin with algebraic structure linked to work by Becchi, Rouet, Stora and Claude Itzykson. Renormalization of non-abelian theories built on the proofs by Gerard 't Hooft and Martinus Veltman established the perturbative consistency of the Standard Model. Lattice gauge theory as a non-perturbative regularization was pioneered by Kenneth Wilson and later computationally advanced by collaborations such as those at CERN and national laboratories.

Topological and geometric aspects

Topological invariants like Chern classes and Pontryagin classes enter gauge theory through characteristic classes studied by Shiing-Shen Chern and Lev Pontryagin. The moduli spaces of solutions to gauge-theoretic equations, including instantons and monopoles, were examined in foundational work by Michael Atiyah, Nigel Hitchin, and Simon Donaldson, connecting to four-manifold topology investigated by Donaldson and Clifford Taubes. The interplay with String theory and dualities such as S-duality and T-duality was developed in contributions from Edward Witten, Juan Maldacena, and Cumrun Vafa, linking gauge theory to mirror symmetry and geometric representation theory studied by Pierre Deligne and George Lusztig.

Experimental tests and phenomenology

Experimental confirmation of gauge-theoretic predictions includes precision tests of QED by groups at institutions like CERN and SLAC National Accelerator Laboratory, discovery of weak neutral currents in experiments tied to CERN and Fermilab that supported electroweak unification by Sheldon Glashow et al., and observation of jet structure and asymptotic freedom in QCD through experiments at DESY and Brookhaven National Laboratory. The discovery of the Higgs boson at CERN's Large Hadron Collider provided direct evidence for spontaneous symmetry breaking mechanisms embedded in gauge frameworks developed by Peter Higgs, François Englert, and Robert Brout. Ongoing tests of gauge extensions and searches for phenomena such as magnetic monopoles involve collaborations at Fermilab, CERN, and international detectors like IceCube.

Category:Mathematical physics