QM vs SR: The Paradox of Simultaneity in Particle States

[Fwiffo]

Very nice, I can appreciate that you're not going to construct the Fock space representation for electrons in two hydrogen atoms -- your general approach suffices :-)

So, the question remains: When do I have to use the two particle entangled state as opposed to two single particle states? The answer can't be "always," because you'd have to put every fermion in the universe into every calculation.

Again the use of language is misleading. seemingly the state |AB>-|BA> is a singlet state, and in some cases it is entangled. But in the most general case there is no "real" entanglement, this is because the particles are identical so if I discover one particle in |A> I have no idea if it was the first one or the second one. Two electrons in different parts of the universe are both in state |1>, the "proper" description if we want to give numbers to the electrons is |electron on Earth in state 1, electron on Andromeda in state 1>-|electron on Andromeda in state 1, electron on Earth in state 1> . This is not an entanglement resource in fact it is very true to say, "the electron on Earth is in state 1". It is never true to say (even after measurement) electron number 1 is the electron on earth. More to come...

 

[Fwiffo]

... sorry about braking this into two posts

so when do we have to anti-symmetrize the state? The answer is, when it becomes meaningful for our purposes. You could say the state of all fermions in the universe is at all times antisymmetric, this should be true (as far as we know). If I look at the two ground state electrons on the He atom I could say there are two electrons one in the spin up state and the other in the spin down state. This would be good in most cases, i.e it is enough to say the state is |01> (first quantized). But this is not the "real" state because the real state is |01>-|10> when we are numbering the electrons as "electron number 1" and "electron number 2" but in most cases this is meaningless. It is uninteresting except for the fact that we cannot write an antisymmetric version of |11> so that is an impossible state.

Another example: two fermions one in NY and the other in Paris, the one in NY is a proton in the Up state (1) the one in Paris is an electron in the Down state (0). It is perfectly good to write the state as |10> although a more precise way to write it would be |p,NY,1;e,P,0> and even better would be to antisymmetrise it, but that is completely unnecessary because a proton is not an electron.

So to answer the question "when dose the wave function become antisymmetric?" It always is , we just don't care about it until the "labels" become meaningful.