Gauge group

In physics, a gauge theory is a type of field theory in which the Lagrangian is invariant under a continuous group of local transformations. An invariant is a model that holds true no matter the mathematical procedure applied to it. This is the concept behind gauge invariance. The idea of fields as described by Michael Faraday in his study of electromagnetism led to the postulate that fields could be described mathematically as scalars and vectors. When a field is transformed, but the result is not, this is called gauge invariance or gauge symmetry. Applying gauge theory creates a unification which describes mathematical formulas or models that hold good for all fields of the same class.

The term gauge refers to any specific mathematical formalism to regulate redundant degrees of freedom in the Lagrangian. 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 theory.

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