Explore Gauge Symmetry, Yang-Mills Theory, and the Unified Framework of Modern Physics
The Canvas on the left shows real-time visualization of gauge fields. Colored flow lines represent the direction and strength of the gauge field A_μ. Test particles moving in the gauge field demonstrate the effect of the covariant derivative D_μ = ∂_μ + g A_μ.
The Wilson loop W(γ) = Tr P exp(ig ∮_γ A_μ dx^μ) is an important gauge-invariant observable. The visualization below shows parallel transport along a closed path.
| Property | U(1) | SU(2) | SU(3) |
|---|---|---|---|
| Group Type | Abelian (Commutative) | Non-Abelian | Non-Abelian |
| Number of Generators | 1 | 3 | 8 |
| Gauge Bosons | Photon | W±, Z | 8 Gluons |
| Self-Interaction | No | Yes | Yes |
| Corresponding Force | Electromagnetic | Weak Force | Strong Force |
Gauge Theory is the core framework in modern physics for describing fundamental interactions. Based on the concept of symmetry, it generalizes global symmetry to local symmetry, introducing gauge fields to describe forces between particles. This theory unifies electromagnetism, weak interaction, and strong interaction, forming the basis of the Standard Model.
The Lagrangian of the system remains invariant under gauge transformations. Local gauge symmetry requires the introduction of gauge fields to compensate for changes in derivative terms.
D_μ = ∂_μ + g A_μ replaces ordinary partial derivatives, ensuring local gauge invariance and introducing coupling between particles and gauge fields.
F_{μν} = ∂_μ A_ν - ∂_ν A_μ + g[A_μ, A_ν] describes the dynamics of gauge fields, containing self-interaction terms in the non-Abelian case.
The quantized form of gauge fields mediates fundamental forces: photon (electromagnetic), W/Z bosons (weak force), gluons (strong force).
Yang-Mills theory is the foundation of non-Abelian gauge field theory, proposed by Chen-Ning Yang and Robert Mills in 1954:
This elegant formula describes the dynamics of gauge fields. In the non-Abelian case (SU(2), SU(3)), the field strength tensor contains the commutator term [A_μ, A_ν], leading to self-interactions between gauge bosons—a fundamental difference from electromagnetism (U(1))
The gauge group of the Standard Model is SU(3)_C × SU(2)_L × U(1)_Y. Through spontaneous symmetry breaking via the Higgs mechanism, SU(2)_L × U(1)_Y breaks to U(1)_{EM}, giving mass to W and Z bosons while the photon remains massless.