U (1) symmetry

[taylordnz]

could someone explain this

[selfAdjoint]

It's just the symmetry of a rotating circle.

A little deeper, take the complex numbers, they form a plane; take the unit circle about the origin in that plane, its equation is z = e^{i\theta}. Imagine unitary transformations on the plane, they take every complex number into another one and they preserve that circle, but map one angle on it into another one.

Now suppose that \lambda e^{i\theta} decribes a wave of phase \theta; the unitary transformation will not affect the wave length but will change the phase, \theta.

[taylordnz]

then could you explain

U (2) symmtery
U (3) symmtery