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Basically, are the atoms that make up my body right now something that has existed since the big bang?

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Not exactly. Fusion of atoms in Supernova nucleosynthesis is thought to be responsible for the various atoms that make up the periodic table. While there hasn't been one in our part of the galaxy for quite some time, plenty of Supernova are occurring through out the universe right now. So, while you are made of old stuff, in terms of atoms, most of it probably isn't as old as the big bang itself.

“The nitrogen in our DNA, the calcium in our teeth, the iron in our blood, the carbon in our apple pies were made in the interiors of collapsing stars. We are made of star stuff.” -Carl Sagan

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Watch out, though, because particles can be indistinguishable. Roughly, you can't write a label on an atom, even in principle, so to say that your hydrogen atoms are this old and my hydrogen atoms are that old is not very well defined. What is closer to reality is that there are this many atoms in this state, and that many atoms in that state, and which is which is none of your business :)

This is easier to understand if you think of elementary particles (quarks, electrons, etc.) as being excited states of a single "thing" called a field. There is only one electron field, and, depending on what quantum state that field is in, there exist a number of electrons, all of which are identical. You can't claim ownership of a particular electron any more than you can claim ownership of the letters in your name.

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