Question

Why do (can) we impose local gauge invariance?

Answer 1

Firstly, let me say that I understand that what basically happens in gauge theories is that we keep the unphysical degrees of freedom present but in check, instead of removing them at once, which besides being generally really hard to do would cause further headaches related to Lorentz invariance.  I was trying to follow the line of thought in Ryder's Quantum Field Theory (pag. 90 - 97) to explain elementary gauge theory.  He shows that the Klein-Gordon field theory (because its action is) is invariant by the global transformation  ϕ→eiΛϕ. ϕ→eiΛϕ. However, he then argues that such a transformation would contradict the relativistic causality mantra (because it transforms the internal degrees of freedom in the whole space at the same time) and uses this fact to justify the local gauge invariance construction, which happens by letting Λ→Λ(x)Λ→Λ(x) and forcing δL=0δL=0 (since this is, initially, spoiled by the derivatives of the parameter function) by coupling a new field AμAμ to the Noether current in a smart way.

Answer 2

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[clarification needed], 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.