I will try to argue in the spirit of Feynman's "The reason for antiparticles"
EnumaElish said:
I never quite understood this proposition. Surely, two electrons are differentiated by their positions in space?
It is formally possible to describe an antiparticle moving forward in time as its particle moving backward in time. If one chooses to do so, it is always possible in a finite system with two electrons to describe one electron exiting the system in the far future, annihilate with its antiparticle (for instance like in the diagram below)
http://upload.wikimedia.org/wikipedia/commons/b/ba/Feynman_EP_Annihilation.svg
One can construct the antiparticle as one wishes, since the annihilation takes place in the far future of the finite system. So one can claim that the antiparticle came from the distant past of the finite system. One can in the far past reverse the above diagram so that one obtains a pair creation. Thus we obtain an electron in the distant past which we can take to be emitted in the general direction of our finite system and eventually become the second electron of our system !
This can be re-interpreted as saying that the two electrons in our finite system are really one and the same electron line.
This little tale is one way to interpret the necessity that, in quantum field theory, there is only one electron field, all electrons in the Universe being excitations of this single field, and we can always arrange the electrons' configuration in a single system so as to have just one external electron line.
Please note that there is no claim that all electrons in the Universe are just the same line going back and forth in a particle and antiparticle cosmic billiard game. The claim is that quantum field theory must be able to deal with the situation if it were happening. We can never prove with a finite system that this is not what is happening. If two electrons were distinguishable, we could prove this scenario wrong.
edit
I realize that it might appear odd why this little tale is relevant to justify the depth of quantum field theory. After all, it's just a simple tale, why did people need quantum field theory to come up with it ? Because near the beginning, it claims "antiparticles moving forward in time can formally be described as particles moving backward in time". Strictly speaking, one must also reverse their handedness. This claim is an essential ingredient of the tale, and it made sense only with quantum field theory.