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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.

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Lubos' answer is the correct answer to your question. –  user1504 Apr 25 '13 at 12:03
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Read Lubos' answer carefully. The standard model doesn't unify the electroweak and strong force. Minor extensions (GUT) to the standard model appear to unify the three forces. The trouble is that nothing works out beautifully and so far no equation has been found which produces everything else without a lot of manual adjusting (fine tuning). The answer to your question (the equation) hasn't been found yet and may not exist. –  Brandon Enright Apr 29 '13 at 16:11
    
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2 Answers

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:

http://www.flickr.com/photos/joshmt/7503048090/

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:

http://blogs.discovermagazine.com/cosmicvariance/files/uploads/sm-lagrangian1.gif

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.

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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.

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It's my understanding that they aren't the same but are very close. If you add in SUSY they get even closer (or are the same). Either way this doesn't answer the question without a lot more elucidation. (-1). –  Brandon Enright Apr 29 '13 at 16:14
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