Gauge theory & Yang-Mills theory
Group theory and gauge theory
"Gauge theory " means the Lagrangian for field theory is invariant under a continuous group of local transformation, and the corresponding symmetry called "gauge symmetry".
In gauge theory, all theory of QFT can be described by a group:
- Electromagnetic interaction: U(1)
- weak interaction: SU(2)
- Strong interaction: SU(3)
Similarly, the relevant unified theory have corresponding groups as below:
- Electroweak theory: SU(2)× U(1)
- Grand unified theory: SU(3)× SU(2)× U(1)
where "grand unified theory" (GUT) is a model that unifying strong, weak, and electromagnetic interaction. The classical version of GUT is SU(5), which subgroup is SU(3)× SU(2)× U(1).
In gauge theory, all theory of QFT can be described by a group:
- Electromagnetic interaction: U(1)
- weak interaction: SU(2)
- Strong interaction: SU(3)
Similarly, the relevant unified theory have corresponding groups as below:
- Electroweak theory: SU(2)× U(1)
- Grand unified theory: SU(3)× SU(2)× U(1)
where "grand unified theory" (GUT) is a model that unifying strong, weak, and electromagnetic interaction. The classical version of GUT is SU(5), which subgroup is SU(3)× SU(2)× U(1).
Yang-Mills theory
Classical gauge theory is based on Abelian Lie group. Yang-Mills theory using non-Abelian Lie group to unify electroweak interaction and strong interaction.
Lagrangian for YM theory as below:
Lagrangian for YM theory as below:
Lagrangian for field theory
Lagrangian for field theory as below:
- QED: (g is coupled constant, F is electromagnetic field tensor)
- QED: (g is coupled constant, F is electromagnetic field tensor)
- QCD: (G is gluon field tensor)
- Electroweak theory:
where, g means the term of W and B particles, f means kinetic term of fermion, h means Higgs field, y means Yukawa interaction.
As a result, the Lagrangian for standard model can be written as below:
Interestingly, the detailed Lagrangian for all the interactions of SM would be written as below: