I always heard that three atomic force in small scale structure (Strong, weak and electromagnetic) are unified through the standard model, but I've never seen its unified equation. What is single unified equation that describes these three forces?
As the Motl answer is not really answer but kidding, I'm still wait.
If you want to print it on a T-shirt - print the Lagrangian of the Standard Model - then the T-shirt has already been printed and it looks like this:
The physicist on it is John Ellis, the 2nd most cited living particle physicist. Of course, the terms could be elaborated upon, written a bit more explicitly and accurately etc. - with all the indices and sums, the Lagrangian occupies about a page but it's still fundamentally very simple because the spirit is caught on Ellis' T-shirt.
This is the part of the electroweak Lagrangian – the strong nuclear force is omitted:
https://www.physicsforums.com/attachments/sm-lagrangian1-gif.195534/
It looks more complicated and artificial than it is. The terms related to the strong nuclear force, QCD, look similar.
One more comment. The theory we know as the Standard Model describes all the three non-gravitational forces but we don't call it "unified" because the parts of the formula responsible for the three forces may be sort of independently adjusted. A unified field theory describing these interactions is the so-called GUT (grand unified) theory which is a morally similar quantum field theory but with a simpler, more unified field contents with various fields we don't observe in practice, aside from the fields that we do, and with some extra story about "symmetry breaking" that has to be added to explain why we observe the three seemingly unrelated forces in the experiments.
To understand the meaning of the symbols in all the equations above, and how to make the calculations of the observable quantities, one needs to learn quantum field theory. It makes no sense to try to explain QFT as a part of this answer; a minimum introduction to QFT is typically a two-semester course at the university.
What I have so far understood from my limited knowledge of quantum field theory and standard model is that the unification of strong force with electroweak force is still a conjecture that needs to be verified by experiments. So far there has not been conclusive evidence of unification of two domains.
The conjecture is that that at high enough energies the quanta of interaction of three forces would have been same but as the energies were lowered (universe cooled down), new and different quanta for three forces appeared due to a process called spontaneous symmetry breaking.
Also you can refer to this - What led to the electroweak and strong forces splitting?
The so-called unified is saying that the coupling constants of three different forces are the same one when the energy scale goes to high enough. Those coupling constants are not really constant because they are changing when the energy scale goes up or down.
