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a try using Engnish to record the study.

Introduction

Gauge theory has the long history in developing Physics theories. It firstly is used for depicting the local symmetry of the construction of theory. The term gauge refers to any specific mathematical formalism to regulate redundant degrees of freedom in the Lagrangian of a physical system. The transformations between possible gauges, called gauge transformations, form a Lie group—referred to as the symmetry group or the gauge group of the theory. Associated with any Lie group is the Lie algebra of group generators. For each group generator there necessarily arises a corresponding field (usually a vector field) called the gauge field. Gauge fields are included in the Lagrangian to ensure its invariance under the local group transformations (called gauge invariance). When such a theory is quantized, the quanta of the gauge fields are called gauge bosons. If the symmetry group is non-commutative, then the gauge theory is referred to as non-abelian gauge theory, the usual example being the Yang–Mills theory¹.

Likewise, some methods from gauge field theory has been applied in condensed state physics resently. With the help of the topological views, it reveals the significance of phase position about lattice electrodynamics. Therefore, there is a need to visit this scenery taking symmetry as the starting point. This note just is for reference only.

Classical gauge: The Choice of Magnetical Vector

We often encounter the concept about gauge when electrodynamics

Theoretical mechanic frame: From the principle of least action to Noether theorem

The beginning of latice gauge field theory: Bloch-Floquet theory

Spinor and Chiral

Phase position and Berry gauge

Reference

1. https://en.wikipedia.org/wiki/Gauge_theory