stevendaryl said:
That's the practical approach to resolving paradoxes in QM: Assume that there is no such thing as a pure state consisting of a superposition of macroscopically different states. There can only be mixed states.
However, I find that assumption to be a "soft contradiction". QM does not in any way limit the size or complexity of the systems that can be described by it.
Let me state, more precisely, what I think the contradiction is. But first, let me formulate an alternative to the Born rule: Instead of saying "A measurement of a system always produces an eigenvalue of the operator that is measured, with a probability given by blah blah blah", you say: "At any given time, the probability that the universe is in some particular macroscopic configuration is given by blah blah blah". This is a specific case of the Born rule, in which the system is the entire universe, and the observable is the universe's macroscopic configuration. You don't need the general Born rule because by definition, a measurement of a microscopic quantity means setting up a measurement so that the microscopic quantity is amplified to produce a macroscopic effect. So if you know the probabilities for macroscopic configurations, then that tells you the probabilities for various measurement results (since they are macroscopically different).
This form of the Born rule can be mathematically described this way: If we let ##\Pi_j## be the projection of the state of the universe onto the macroscopic state number ##j## (you can't have continuum many macroscopically distinguishable configurations, so it's enough to consider a countable collection of projection operators), and you let ##|\psi(0)\rangle## be the initial state, at time ##t=0##, then the probability of being in state ##j## at a later time ##t > 0## is given by:
##P(j) = \langle \psi| e^{iHt/\hbar} \Pi_j e^{-iHt/\hbar} |\psi\rangle##
Then the issue of collapse can be stated this way: What if, we ask what the probability is of being in state ##j## at time ##t_1## and then later being in state ##k## at time ##t_2##? Here are two possible answers:
- ##P_{collapse}(j,k) = \langle \psi| e^{iH t_1/\hbar} \Pi_j e^{iH(t_2 - t_1)/\hbar} \Pi_k e^{-iH(t_2 - t_1)/\hbar} \Pi_j e^{-iHt_1/\hbar} |\psi\rangle##
- ##P_{no-collapse}(j,k) = \langle \psi| e^{iH t_2/\hbar} \Pi_k e^{-iH t_2/hbar} |\psi\rangle##
If the state ##|\psi\rangle## is itself a pure macroscopic state (##\Pi_i |\psi\rangle = |\psi\rangle## for some macroscopic configuration ##i##), then there will be negligible difference between these two numbers. That's because the macroscopic configuration ##k## includes the record of having previously gotten some particular result for some past measurement. So for each ##k##, the probability ##P_{collapse}(j,k)## will be approximately zero for all except one value of ##j##. For that value of ##j##, there will be a negligible difference between ##P_{collapse}## and ##P_{no-collapse}##.
So under the assumption that the world currently is in a macroscopically pure state, the collapse assumption is harmless. Assume it or not, it makes no difference.
Eventually, though, the state of the universe will drift away from being macroscopically pure, and the distinction between ##P_{collapse}## and ##P_{no-collapse}## will grow larger and larger.
