Gauge Theory - Overview

Summary

A gauge theory is a field theory whose Lagrangian is invariant under a continuous group of local (spacetime-dependent) transformations — the gauge group. This local symmetry forces the existence of new fields (gauge fields) and mediating particles (gauge bosons). Gauge theories describe all fundamental interactions: QED (U(1)), electroweak (SU(2)×U(1)), and QCD (SU(3)).

Overview

The central idea of gauge theory: local symmetry generates interactions. Starting from a free-field Lagrangian with only a global symmetry, demanding that the symmetry hold locally (at each spacetime point independently) forces the introduction of gauge fields. These gauge fields then mediate interactions between matter particles.

Key examples:

• QED: U(1) gauge symmetry → photon as gauge boson → electromagnetic force
• Electroweak: SU(2)×U(1) → W${}^{\pm }$, Z, photon → electroweak force
• QCD: SU(3) → 8 gluons → strong force

Global vs. Local Symmetry

Global Symmetry

A Lagrangian has global symmetry if it is invariant under a transformation that is performed identically at every point in spacetime. The parameters of the transformation are constants.

Example: rotating all scalar fields by the same O(n) rotation $\Phi \mapsto G\Phi$ with $G$ constant.

Local Symmetry (Gauge Symmetry)

A Lagrangian has local (gauge) symmetry if it is invariant under transformations where the parameters can vary independently at each spacetime point: $\Phi (x)\mapsto G(x)\Phi (x)$.

Global symmetry is a special case of local symmetry where $G(x)=\text{const}$.

The problem with making symmetry local: When $G$ depends on $x$, ordinary derivatives fail to transform covariantly:

$${\partial }_{\mu }(G\Phi )\ne G({\partial }_{\mu }\Phi )\text{if }G=G(x)$$

This spoils the invariance of the Lagrangian. The solution is to introduce a gauge field.

Gauge Fields and Covariant Derivatives

Gauge Field and Covariant Derivative

To restore local gauge invariance, replace ordinary derivatives with gauge covariant derivatives:

$$D_{\mu }={\partial }_{\mu }-ig{A}_{\mu }$$

where:

• $g$ = coupling constant (interaction strength)
• $A_{\mu }(x)$ = gauge field (Lie algebra-valued connection)

The gauge field must transform as:

$$A_{\mu }^{\prime }=G{A}_{\mu }{G}^{-1}-\frac{i}{g}({\partial }_{\mu }G)G^{-1}$$

to ensure $(D_{\mu }\Phi {)}^{\prime }=G(D_{\mu }\Phi )$.

The gauge field $A_{\mu }$ can be expanded in terms of Lie algebra generators $T^a$:

$$A_{\mu }=\sum _a{A}_{\mu }^a{T}^a$$

There is one gauge field component per generator of the Lie algebra.

Gauge Boson

When the gauge field theory is quantized, the quanta of the gauge field $A_{\mu }$ are the gauge bosons — the force carriers.

• U(1) gauge theory → 1 gauge boson: the photon
• SU(2) gauge theory → 3 gauge bosons (one per generator): $W^{+}$, $W^{-}$, $W^0$
• SU(3) gauge theory → 8 gauge bosons: the gluons

Yang-Mills Lagrangian

Yang-Mills Action

The Lagrangian for the gauge field itself (which gives gauge bosons kinetic energy and allows them to propagate) is:

$${\mathcal{L}}_{\text{gf}}=-\frac{1}{4}{F}^{a\mu \nu }{F}_{\mu \nu }^a$$

where the field strength tensor $F_{\mu \nu }^a$ is:

$$F_{\mu \nu }^a={\partial }_{\mu }{A}_{\nu }^a-{\partial }_{\nu }{A}_{\mu }^a+g\sum _{b,c}{f}^{abc}{A}_{\mu }^b{A}_{\nu }^c$$

and $f^{abc}$ are the structure constants of the Lie algebra.

For abelian (U(1)) gauge theory, the $f^{abc}$ terms vanish and this reduces to the familiar electromagnetic field strength $F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }$.

For non-abelian gauge theories (SU(2), SU(3)), the gauge bosons self-interact — unlike photons, gluons carry color charge and interact with each other.

Classical Example: Electrodynamics

Deriving QED from U(1) Gauge Symmetry

Start with the free Dirac Lagrangian for an electron:

$$\mathcal{S}=\int \bar{\psi }(i\hslash c{\gamma }^{\mu }{\partial }_{\mu }-m{c}^2)\psi d^{4}x$$

This has the global U(1) symmetry: $\psi \mapsto e^{i\theta }\psi$ for constant $\theta$.

Demanding local U(1) symmetry: $\psi (x)\mapsto e^{i\theta (x)}\psi (x)$ requires the covariant derivative:

$$D_{\mu }={\partial }_{\mu }-i\frac{e}{\hslash }{A}_{\mu }$$

The resulting interaction term is:

$${\mathcal{L}}_{\text{int}}=\frac{e}{\hslash }\bar{\psi }(x){\gamma }^{\mu }\psi (x)A_{\mu }(x)=J^{\mu }(x)A_{\mu }(x)$$

This is exactly the minimal coupling of electromagnetism! The electromagnetic field $A_{\mu }$ is forced into existence by demanding local phase invariance. The full QED Lagrangian is:

$${\mathcal{L}}_{\text{QED}}=\bar{\psi }(i\hslash c{\gamma }^{\mu }{D}_{\mu }-m{c}^2)\psi -\frac{1}{4{\mu }_0}{F}_{\mu \nu }{F}^{\mu \nu }$$

Historical Development

Year	Event
1918	Weyl proposes Eichinvarianz (scale invariance) as a local symmetry of general relativity
1929	Weyl, Fock, London: replace scale factor with complex phase → U(1) gauge symmetry
1929	Weyl’s paper establishes modern gauge invariance concept
1941	Pauli’s review popularizes gauge invariance
1954	Yang and Mills: non-abelian SU(2) gauge theory (Yang-Mills theory)
1960s	Glashow, Salam, Ward: electroweak unification via gauge theory
1967	Weinberg: electroweak theory with Higgs mechanism
1971	’t Hooft proves non-abelian gauge theories are renormalizable
1973	Fritzsch, Gell-Mann, Leutwyler: QCD as SU(3) gauge theory

Geometric Interpretation

In differential geometry, gauge theory is the theory of connections on principal fiber bundles:

• Base space: spacetime $M$
• Fiber: the gauge group $G$ at each point
• Gauge field $A_{\mu }$: a Lie-algebra-valued 1-form (connection form)
• Field strength $F_{\mu \nu }$: the curvature of the connection, $\mathbf{F}=d\mathbf{A}+\mathbf{A}\wedge \mathbf{A}$
• Gauge transformation: change of local section of the principal bundle

The physical electromagnetic field is the curvature of a U(1) connection. The condition “zero curvature everywhere” means the gauge field can be removed by a gauge transformation (it’s pure gauge).

Noether’s Theorem and Conservation Laws

Gauge Symmetry → Conserved Currents

By Noether’s theorem, every continuous global symmetry of a Lagrangian gives rise to a conserved current. For O(n) global symmetry:

$$J_{\mu }^a=i{\partial }_{\mu }{\Phi }^{\mathsf{T}}{T}^a\Phi$$

with one conserved current per generator.

For U(1): the single conserved current is the electric current $J^{\mu }=\bar{\psi }{\gamma }^{\mu }\psi$, and the conserved charge is the electric charge.

Non-Abelian Gauge Theories

When the gauge group is non-abelian (e.g., SU(2), SU(3)):

• The gauge bosons themselves carry “charge” (color charge for gluons, weak isospin for W/Z)
• Gauge bosons self-interact (3- and 4-boson vertices)
• The field strength tensor has an extra non-linear term: $F_{\mu \nu }^a={\partial }_{\mu }{A}_{\nu }^a-{\partial }_{\nu }{A}_{\mu }^a+g{f}^{abc}{A}_{\mu }^b{A}_{\nu }^c$
• This leads to asymptotic freedom in QCD: strong coupling decreases at high energies

Connections

• Quantum Field Theory - Overview — Gauge theory is the organizing principle of QFT
• QED and Renormalization — QED is the U(1) gauge theory; renormalization handles divergences
• Standard Model and Gauge Groups — Full Standard Model: SU(3)×SU(2)×U(1)
• Quantum Mechanics - Mathematical Formalism — Noether’s theorem, symmetry generators

See Also

• Standard Model and Gauge Groups — The particle physics application of gauge theory
• QED and Renormalization — The simplest example: U(1) gauge theory