To find out what kind of symmetry a field (as described by a Lagrangian) admits. It's called gauge symmetry: https://en.wikipedia.org/wiki/Gauge_theory https://

To find out what kind of symmetry a field (as described by a Lagrangian) admits. It's called gauge symmetry: https://en.wikipedia.org/wiki/Gauge_theory https://en.wikipedia.org/wiki/Gauge_theory It's a complex number for QED but in general, it's a unitary matrix: a rotation which preserves the magnitude of a wavefunction (a complex vector). In physics, symmetries play a fundamental role.