Nature chooses symmetries. Making those symmetries local creates connections (fields). Different representations of those symmetries become particles.

Empirically, the universe respects internal rotation symmetries:$$SU(3)\times SU(2)\times U(1)$$They are the smallest groups that match:

• color triplets
• weak doublets
• electric phases

When symmetry is global → nothing happens.
When symmetry is local → derivatives break symmetry → so nature must introduce compensating fields. This creates lie-algebra-valued-1-form Aμ​=Aμa​Ta

But A field is not “a particle”. A field is a section of a bundle. Different particles correspond to different ways of transforming under the symmetry group. Mathematically: particles = representations of SU(3) x SU(2) x U(1).

Now I’d like to connect these understanding to spinors in Dirac Equation. When we talk about spinors in the Dirac equation, we mean the 4-component wavefunction $\psi$ itself.

That’s how nature works!