Standard Model

We have to go up the following steps:

• Lagrangian Mechanics for particles. A Lagrangian function determines the dynamics of the system.
• Lagrangian field theory (classical field theory). The same, but we deal with infinite degrees of freedom, a continuum of particles. They correspond to matter fields, and the "forces" affecting the matter are encoded into the Lagrangian itself.
• Gauge theory. The matter fields are recognized as sections of a certain bundle, instead of functions. We modify the notion of differentiation (connections), so we reorganize the Lagrangians: some ugly terms of the Lagrangian of the previous step are included inside the connection. Moreover, the curvature of the connection encode the forces". They are also called fields (gauge strength tensor or gauge field or something like that). We have two Lagrangians, one for the matter field and other for the connection. See also the note quantum particle.
• Quantization. The matter fields (sections of a bundle) are promoted to operator-valued fields (fermions). Also, the gauge potentials $A_{\mu }$ are promoted to a operator (bosons).