This is not a homework, neither it is any exercise. It is my understanding of $U(1)$ symmetry. I would request if anybody can please correct me on any one of the following understandings:
The bottom line of particle physics is Lagrangian density which is a generalization of classical mechanics, $L= V-T$.
We come across lot of symmetries in nature which are invariant under any changes made. Like an equilateral triangle is more symmetric than other triangles...... For a symmetry, if the object is rotated or flipped, the appearance will remain unchanged.
But when we label a symmetry, we could differentiate it which is something called 'symmetry breaking'.
$U(1)$ is a Lie group. Now, typically if a circle is rotated, flipped, it appears the same, called Lie group.
There are different type of symmetries like (i) Translational --> Invariance of the laws of physics under any translation as pointed out in Noether's theorem (ii) Rotational -->Whichever direction you rotate it is the same (iii) Time symmetry --> the laws of physics are eternally unchanging (iv) Boost symmetry
The rotational and boost symmetry are grouped under Lorentz group. The translational, rotational and boost symmetry together make up the Poincare group.
Local symmetry is not a symmetry in physical spacetime.
When Einstein applied special relativity to electromagnetism, he found electromagnetic 4 potential-- the four vector (1 time like, and 3 space like, which relates the electric scalar potential and magnetic vector potential). The physicists were happy, as the electromagentic 4 potential appears in the Lagrangian density.
But the value of the electromagnetic 4 potential can be changed. The Electric field $E(t,x)$ and the Magnetic field, $B(t,x)$ can be expressed in scalar potential and vector field.The term gauge invariance refers to the property that a whole class of scalar and vector potentials, related by so-called gauge transformations, describe the same electric and magnetic fields. As a consequence, the dynamics of the electromagnetic fields and the dynamics of a charged system in a electromagnetic background do not depend on the choice of the representative $(A_0(t,x),A(t,x))$.
$U(1)$ being a circle. When the charged particle move across the $U(1)$ plane, its' mass, kinetic energy does not depend on the position of the particle. It's value depend on the rate at which it circles the plane.
This is the $U(1)$ symmetry of the standard model. If we rotate $U(1)$ at any angles, the Lagrangian density will remain unchanged.
Please do correct me where I am wrong.