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I am reading that it is proton magnetism that keeps us from falling through a bed if we land on it- That is to say that the proton magnetism in the bed repulses the protons in us, our clothes etc. If proton magnetism rejects nearby protons due to their magnetism why don't all of our proton rich cells simply fly apart?

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In everyday matter around us, individual protons are fixed inside the nuclei of the constituent atoms. They cannot do free streaming inside everyday materials. There are two fundamental interactions that keep our nuclei stable: electromagnetic interaction (keeping electrons inside their orbits around nucleus) and strong interaction (keeping nucleons from disintegration). But the first one is many orders of magnitude smaller than the second one. In small scale, We need both of them to survive.

Again, the use of everyday is crucial here as you want to break the matter apart; first you can remove electrons by low-energy ionization and then separate individual protons by high-energy collisions. But, no matter what is the amount of energy, you will never have energy enough to break protons into their constituent particles (gluons and quarks) for a considerable time duration due to confinement. However, we believe that this was easily achieved in early Universe where it is thought infinite amount of energy was somehow reached for a very very very small fraction of time.

Benjamin
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    You can't break a free isolated proton into quarks because if you supply the energy to separate the quarks then you end up providing enough energy to make new quarks and if they are isolated they can form multiple nucleons instead of free quarks. The early universe had a quark gluon plasma because if you have enough energy then you can have all the quarks and gluons interacting with each other instead of clumping into nucleons. – Timaeus Mar 16 '16 at 00:17
  • @Timaeus. Your thought logics is appreciated. – Benjamin Mar 16 '16 at 01:28