Part 4/8:
Introducing Symmetry: U1 Symmetry
To harness this mathematical structure into a useful theory, we impose a symmetry onto our equation. The concept of symmetry dictates that if we rotate our field (analogue to rotating a circle), the fundamental properties should remain unchanged. In technical terms, we refer to this as U1 symmetry.
This symmetry assumption becomes crucial as we explore local transformations — assessing how our equation behaves under variations at specific locations in space-time rather than the entire universe. However, this local consideration introduces complications; in its essence, the symmetry insists that the original equation does not change under these variations.