Mathematical formulation of the Standard Model

The Standard Model is a quantum field theory whose particle content and interactions are encoded by fields, gauge symmetries, and a Lagrangian.

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Quantum field theory: A framework that describes particles as excitations of fields and their interactions through quantum laws. The Standard Model is a specific quantum field theory.
Standard Model Lagrangian: The density function that specifies the Standard Model’s fields, kinetic terms, interactions, and symmetry-breaking terms. It is the central mathematical expression defining the theory.
Feynman rules: Rules derived from a quantum field theory’s Lagrangian for calculating scattering amplitudes with diagrams. They translate Standard Model interaction terms into calculable processes.
General relativity: A theory of gravity in which spacetime curvature is sourced by matter and energy. It describes gravity, which the Standard Model does not include as a quantum interaction.
Gauge theory: A field theory whose laws are invariant under local transformations associated with a symmetry group. Local gauge invariance determines the Standard Model’s interaction structure.
SU(3): The group of three-dimensional unitary matrices with determinant one. Its local symmetry gives the strong interaction and its eight gluon fields.
Perturbation theory: A method for approximating solutions by expanding in a small parameter. Many Standard Model predictions use expansions in interaction strengths.
Grand unified theory: A proposed framework that combines the strong, weak, and electromagnetic interactions into a larger gauge symmetry. It seeks to unify the Standard Model’s separate gauge factors at high energies.
Lie group: A group that is also a smooth manifold, allowing its symmetries to vary continuously. The Standard Model’s gauge symmetries are organized by Lie groups.
SU(2): The group of two-dimensional unitary matrices with determinant one. Its weak-isospin symmetry acts on left-handed fermion doublets.