The Idea and the Form

We can say today that $3!$ is the excelent first approximation to the theoretical model of the strong, electromagnetic and weak interactions. That $3!$ comes from

$$3! = 3 \times 2 \times 1$$ (11)

which is just an shorthand notation for

$$\lq\lq 3! '' = SU(3)_{C} \times SU(2)_{L} \times U(1)_{Y}$$ (12)

the gauge symmetry groups which give the background to the field theory of those interactions that has been developed contemporarily. This $3!$ is the so called Standard Model containing $SU(3)_{C}$ (of ``colour") for the strong interactions (QCD or quantum chromodynamics) and $SU(2)_{L}$ (of weak isospin) $\times U(1)_{Y}$ (of hypercharge) for electroweak interactions: the Glashow, Salam, Weinberg Model.

Essentially, $3!$ is a Yang Mills field theory, that is to say, a theory based on a generalization of the gauge invariance of electromagnetism mounted on the theory of Lie groups. Then, this way implies a principle of ``geometric" invariance (local phase transformations) and this principle can determine the form of the interaction.

The trascendental thing is that with symmetry as a support of the origin of the interactions a consistent theory is born. A theory which is aesthetically worthy and at the same time provides a deeper understanding of reality.