gaugeinvariant

Gauge invariance, or gauge symmetry, is a principle in which the fundamental equations of a field theory remain unchanged under local transformations of the fields according to a gauge group. In a local (position-dependent) gauge transformation, the fields are altered in a way that leaves observable predictions intact. This redundancy means that many mathematically distinct field configurations describe the same physical situation, and only gauge-invariant quantities have direct physical meaning.

In electromagnetism, for example, the four-potential Aμ can be transformed as Aμ → Aμ + ∂μΛ without changing the electric and magnetic fields. The gauge-invariant quantities are the field strength tensor Fμν and combinations built from it. In non-abelian gauge theories, such as those underlying the Standard Model, gauge fields transform more complexly under a group like SU(N). Physical observables must be gauge-invariant; common examples include traces of field-strength contractions and Wilson loops, which are path-ordered exponentials of the gauge field around a closed loop.

Gauge invariance has several important roles. It dictates the interactions of gauge bosons with matter fields and constrains the form of the Lagrangian, providing a foundation for the Standard Model’s interactions. When quantizing gauge theories, one typically fixes a gauge to remove redundant degrees of freedom, which introduces artificial gauge-dependent terms and ghost fields, while physical observables remain gauge-invariant. Gauge invariance also guides nonperturbative methods, such as lattice gauge theory, where gauge-invariant operators like Wilson loops and plaquettes are central to extracting physical information.