Say there are two particles in the x axis. Initially, with high probability one staying at $x=-1$ and going to the right, and the other $x=1$ going to the left. Will they collapse when both arriving at $x=0$?

If one does not use the Born rule, one has only a combined $\psi(x_1,x_2,p_1,p_2,t):\mathbb{R}^4\times\mathbb{R} \rightarrow \mathbb{C}$ wave function, where $|\psi(x_1,x_2,p_1,p_2,t)|^2$ is the probability of particles being respectively at position $x_1$ and $x_2$ at time $t$ with momentum $p_1, p_2$. In this case, will the particles eventually collapse? If not, what does it mean to "measure"? Is "measuring" a primitive concept like "particle", not defined in terms of other concepts? So that it is implicitly understood what does it mean when physicists say "measuring".

Born rule says if a measurement is performed on the $x$ axis, then $\psi$ will become $\phi$, with $|\phi(x_1,x_2,p_1,p_2)|^2=1$ for some $x_1,x_2,p_1,p_2$. And this happens with probability $|\psi(x_1,x_2,p_1,p_2)|^2$.

Is it right to say that "if I am not there" no collapse occur?

So, if one considers the whole universe, who is making the universe collapse?

Is this question solved by defining another concept: a particle is either classical either quantum? And saying that "a collapse occurs when a classical system interacts with a quantum system". In this case, how "interaction" is defined? In the example, above, say the particle on the right is classical: its probability of being is some $x(t)$ with momentum $p(t)$ is 1 at all times. Will the two particles interact as soon the probability of the two of being near each other is non zero? $\exists \epsilon,x: \; |\psi(x-\epsilon,x+\epsilon)|^2 > 0$. And in this case the Born rule will roll the dice and take a definite position for the left particle?


"Measurement" is only fuzzily defined.

There are situations in which it's certainly safe to apply the Born rule and collapse the wave function. Those are situations in which the observable you care about has become so thoroughly entangled with the environment that there's no hope of undoing the entanglement precisely enough to do an interference experiment that could determine whether the wave function has collapsed or not. This is the case if, for instance, a Geiger counter has emitted a click, which is a sound wave that alters the positions of a vast number of atmospheric molecules. When people talk about a "classical system" in this context, they essentially mean a system so complicated that you'll never successfully disentangle your observable from it.

There are situations in which you definitely can't collapse the wave function. Those are situations that are similar to experiments we've done that found quantum interference that wouldn't have been there if the wave function had collapsed. Your example of two quantum particles interacting is one of those. If a mere interaction of two particles caused a collapse then we would never have formulated quantum mechanics because we would have no evidence for it.

In between those situations, there's a grey area. There have been attempts to narrow it by, for example, doing double-slit interference experiments with viruses, but I don't know the current status of those efforts. It's difficult because if you don't find interference, it's hard to know whether that's because a physical collapse happened or because you failed to erase all of the which-path information.

Incidentally, when you say that $ψ$ is a function of $x_1,x_2,p_1,p_2$, that's incorrect. It can be written as a function of $x_1$ and $x_2$ or as a function of $p_1$ and $p_2$ but not of all four.

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