Gauge Theory - Overview

Summary

A gauge theory is a field theory whose Lagrangian is invariant under a continuous group of local (spacetime-dependent) transformations forming a Lie group — the gauge group. Maintaining this invariance when the symmetry parameter varies in spacetime forces the introduction of new fields (gauge fields), which become the mediators of the fundamental forces; when quantized, they give rise to gauge bosons. All fundamental interactions except gravity are described by gauge theories: QED (U(1)), electroweak (SU(2)×U(1)), and QCD (SU(3)).

Overview

The central idea of gauge theory: local symmetry generates (indeed demands) interactions. Start from a free-field Lagrangian with only a global symmetry. If you require that the theory be invariant not just under global transformations (the same everywhere) but under local ones (chosen independently at each spacetime point), the derivative terms in the Lagrangian break invariance. To restore it, one must introduce a new dynamical field — the gauge field — which then mediates interactions between the matter particles. The photon, $W^{\pm }$, $Z$ bosons, and gluons are all gauge bosons arising from this mechanism.

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

Main Content

Global vs. Local Symmetry

Global Symmetry

A theory has a global symmetry under a group $G$ if its Lagrangian is invariant under a transformation that is performed identically at every point in spacetime — all fields are transformed simultaneously by a constant group element, so 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, i.e. rotating all field values by the same angle everywhere in spacetime.

Local Symmetry (Gauge Symmetry)

A theory has a local (gauge) symmetry if its Lagrangian is invariant under transformations whose parameters can vary independently (and continuously) from point to point in spacetime: $\Phi (x)\mapsto G(x)\Phi (x)$.

Local symmetry is a stronger constraint — global symmetry is the special case where $G(x)=\text{const}$.

The Problem with Local Transformations

Consider the O($n$) global symmetry of $n$ scalar fields:

$$\mathcal{L}=\frac{1}{2}({\partial }_{\mu }\Phi {)}^{\mathsf{T}}{\partial }^{\mu }\Phi -\frac{1}{2}{m}^2{\Phi }^{\mathsf{T}}\Phi$$

Under a global rotation $\Phi \mapsto G\Phi$ (constant $G\in O(n)$), both $\Phi$ and ${\partial }_{\mu }\Phi$ transform identically — invariance holds.

Under a local transformation $\Phi \mapsto G(x)\Phi$, ordinary derivatives fail to transform covariantly:

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

The extra term $({\partial }_{\mu }G)\Phi$ spoils the invariance of the Lagrangian. The solution is to introduce a gauge field.

Gauge Fields and Covariant Derivatives

Gauge Covariant Derivative

To restore local gauge invariance, replace the ordinary derivative ${\partial }_{\mu }$ with the gauge covariant derivative:

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

where:

• $g$ = coupling constant (interaction strength)
• $A_{\mu }(x)$ = gauge field (a Lie-algebra-valued 1-form / connection)

By construction, $D_{\mu }\Phi$ transforms as $\Phi$ itself: $(D_{\mu }\Phi {)}^{\prime }=G(x)D_{\mu }\Phi$.

Gauge Field

To ensure $(D_{\mu }\Phi {)}^{\prime }=G(D_{\mu }\Phi )$, the gauge field $A_{\mu }$ must transform as:

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

The gauge field can be expanded in terms of the 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 — as many gauge fields as there are generators of the symmetry group.

Gauge Transformation

A gauge transformation changes the choice of local coordinate basis (section of the fiber bundle). Two field configurations related by a gauge transformation represent the same physical situation. Gauge invariance is a redundancy, not a symmetry with physical consequences.

Gauge Bosons: Gauge Fields as Force Mediators

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

Gauge Theory	Gauge Group	Gauge Boson(s)
QED	U(1)	Photon ($\gamma$)
Weak force	SU(2)	$W^{\pm }$, $Z^0$
QCD	SU(3)	8 gluons
Standard Model	U(1)×SU(2)×SU(3)	All of the above
General Relativity	Diffeomorphisms	Graviton (proposed)

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.

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

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 $J^{\mu }=\frac{e}{\hslash }\bar{\psi }{\gamma }^{\mu }\psi$, and the conserved charge is the electric charge.

Geometric Interpretation (Mathematical Formalism)

In differential geometry, gauge theory is the theory of connections on principal fiber bundles. A gauge is a choice of local section of a principal bundle $P$ with structure group $G$:

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

The curvature is

$$\mathbf{F}=d\mathbf{A}+\mathbf{A}\wedge \mathbf{A}$$

where $d$ is the exterior derivative and $\wedge$ is the wedge product. For an abelian group (e.g., U(1)), $\mathbf{A}\wedge \mathbf{A}=0$ and $F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }$ is the electromagnetic field tensor: 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 is pure gauge).

Historical Development

Year	Event
1864–65	Maxwell’s formulation of electrodynamics already contains the original (classical) gauge invariance of the potentials — unnoticed at the time
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

Examples

Classical Gauge Freedom of the Electromagnetic Potentials

The potentials can be shifted by

$$V\mapsto V-\frac{\partial f}{\partial t},\mathbf{A}\mapsto \mathbf{A}+\nabla f$$

for any twice-differentiable $f(x,t)$, leaving $\mathbf{E}$ and $\mathbf{B}$ unchanged. This is the original gauge invariance of classical electrodynamics (Maxwell, 1864–65).

Deriving the Electromagnetic Interaction from U(1) Gauge Symmetry

Setup: Start with the free Dirac action for an electron field $\psi$:

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

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

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

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

Identification: $A_{\mu }(x)$ is the electromagnetic four-potential; $e$ is the electric charge.

Interaction Lagrangian:

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

where $J^{\mu }=\frac{e}{\hslash }\bar{\psi }{\gamma }^{\mu }\psi$ is the electric four-current.

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

Electromagnetism: U(1) Gauge Theory and the QED Lagrangian

Adding the gauge-field (Yang-Mills) term for the abelian field strength $F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }$ to the locally invariant Dirac Lagrangian of the previous example gives the full QED Lagrangian:

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

Conclusion: The entire electromagnetic interaction arises from demanding U(1) local invariance of the free Dirac Lagrangian.

Connections

• Quantum Field Theory - Overview — Gauge theory is the organizing principle of QFT: all known fundamental interactions (except gravity) are gauge theories within the QFT framework.
• QED and Renormalization — QED is the U(1) gauge theory; renormalization handles its divergences.
• Standard Model and Gauge Groups — The full Standard Model: SU(3)×SU(2)×U(1).
• Yang-Mills Theory and Gauge Fields — Non-abelian (SU($n$), $n>1$) gauge theories; the field strength tensor is non-linear.
• Renormalization — Ward identities arising from gauge invariance constrain renormalization and ensure the photon remains massless.
• Canonical Quantization of Fields — Quantizing gauge theories requires gauge fixing (Faddeev–Popov ghosts) due to the redundancy.
• 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
• Yang-Mills Theory and Gauge Fields — non-abelian gauge theories and the Standard Model
• Quantum Field Theory - Overview — gauge theory in the broader QFT context
• Renormalization — gauge invariance constrains renormalization