I am an undergraduate familiarizing myself with EPR for my upcoming bachelor's thesis, but this is probably pretty basic stuff for you out there.
I am referring to the famous Einstein-Podolsky-Rosen thought experiment / paper. Especially how it's outlined in Leslie Ballentine's 'Quantum Mechanics - A Modern Development'
Def. Δx := x_2 - x_1
P := p_1 + p_2
I can see how [Δx;P] = 0, therefore Δx & P share (a complete set of) eigenstate(s). I.e. at a time t = t_0 both quantities represented by the above-mentioned can be sharply measured without mutual disturbance.
Since P~H is a constant of the system (given no external interference, V=0), I can see how measuring p_1 at a later time would determine p_2 exactly, since P is known from the beginning.
However the point of confusion for me that most books seem to skip over is the following.
Δx is not a constant of the system. If I were to measure x_1 at some later time, how does that get me x_2?
I had two quick ideas that rely on exact knowledge of the later time, or masses of the particles involved respectively.
A) I think it would be legal (without breaking the |Δx> eigenstate)* to define x_1(0) as 0. I.e. the center of mass at t=0 is x_2(0). And since the center of mass is a constant of an unperturbed system, it would still be x_2(0) at all later stages as well. And therefore I am able to determine x_2(t) by inference from x_1(t). But I suspect my premise * is wrong.
B) Δx(t) = Δx(0) + P * t, and determine x_2 through that.
But nobody seems to mention this and x_2(t) seems to automatically be implied from x_1(t).
Can you help me out?