DarMM said:
I've tried to work this out, but I'm not seeing it. Can you give an example with some mathematical details, even sketched not necessarily in full detail.
Okay, here's a sketch of the idea, which isn't all that profound. The minimal interpretation of quantum mechanics basically says that if you set up a system so that it is described by the state ##|\psi\rangle## and you measure some observable ##A## of the system, then you will get an eigenvalue ##a## of the corresponding operator with a probability given by: ##P_a = \langle \psi|\Pi_{A,a}|\psi\rangle##, where ##\Pi_{A,a}## is the projection operator, which has eigenvalue 1 on any state in which ##A## has definite value ##a##, and has eigenvalue 0 on any state in which ##A## has a definite value other than ##a##.
Now, the above prescription has the phrase "you measure some observable ##A##". What does that mean? Well, to measure an observable of a system means to set up an interaction between that system and a measuring device so that distinct values of that observable lead to macroscopically different states of the measuring device. In other words, for every possible value ##a## of the microscopic observable, there is a macroscopic configuration ##C_a## of the measuring device, so that the measuring device reliably ends up in configuration ##C_a## whenever the system of interest has value ##a##. For now, a macroscopic configuration is basically whatever description of a macroscopic system that one could obtain by inspection: This red light is on. There is a black dot here rather than there. This display shows such and such value. The Geiger counter is clicking.
Now, although it's an enormously difficult to completely nail down the details, I think that most physicists are fairly confident that measuring devices themselves are described by the same quantum mechanics as the systems being studied. In principle, even if intractable in practice, one could do a full-fledged quantum mechanical analysis of the system + measuring device + relevant environment, and one would find something like this:
##|\psi_a\rangle \otimes |start\rangle \Rightarrow |\psi_a\rangle \otimes |C_a\rangle##
where ##|\psi_a\rangle## is the state of the system of interest when it is an eigenstate of ##A## with eigenvalue ##a##, and ##|start\rangle## is the initial state of the measuring device + environment, and ##|C_a\rangle## is the state of the measuring device plus environment after the measurement takes place.
I say "something like this" rather than exactly this because the reality is much more complicated. There is no single state of measuring device + environment, there are enormously many microscopically distinguishable states corresponding to any macroscopic description, and a measurement process is an irreversible change, which is hard to describe using quantum mechanics. But all difficulties aside, I think most people are confident that there is nothing going on in a measurement process that isn't in principle describable by quantum mechanics.
So to the extent that my sketch can be accepted as approximately correct, with a large grain of salt, we can ask what happens if you use the same measuring setup to measure the microscopic system when it is
not in an eigenstate of ##A##. Well, since the evolution equations of quantum mechanics are linear, it follows that a superposition of initial states would lead to a superposition of final states:
##\sum_a \alpha_a |\psi_a\rangle \otimes |start\rangle ##
##\Rightarrow \sum_a \alpha_a |\psi_a\rangle \otimes |C_a\rangle##
Then the Born rule saying that there is probability ##|\alpha_a|^2## that the microscopic system will be measured to have eigenvalue ##a## is essentially the same as saying that the measuring device will later be found to be in the configuration ##C_a## with probability ##|\alpha_a|^2##. So the Born rule for the microscopic system presumably follows from the Born rule for the measuring device plus the definition of what it means to measure something.So my claim is that the empirical content of the minimal interpretation is equivalent (in principle) to the following recipe:
- Describe the whole universe (or the part that's relevant) as a quantum system.
- Let that system evolve according to the usual unitary rule.
- Then decompose the final state into a superposition of macroscopically distinguishable states, each of which has definite values for all macroscopic properties.
- Assume that the macroscopic system will be found in exactly one of those states, with a probability given by the square of the corresponding amplitude.
The Born rule for measurements would (I claim) follow from the above recipe, together with the definition of what it means to measure a microscopic quantity.
Mathematically, I think the above recipe could be formulated in terms of projection operators. Presumably any fact about the world that could be verified by observation such as "there is a black spot on the left photographic plate" corresponds to a claim of the form that some coarse-grained observable has a value in some range. Such claims can be formulated in terms of projection operators. So for every such macroscopic statement ##c##, there is presumably a corresponding projection operator ##\Pi_c##. If the observables are taken to be low-enough precision and coarse-grained enough, then all the corresponding projection operators are approximately commuting. Which means that we talk in terms of macroscopic configurations, which are just maximal collections of compatible macroscopic claims. So we can in principle come up with projection operators ##\Pi_j## on the state of the universe such that for ##j \neq k##, the projection operators ##\Pi_j## and ##\Pi_k## correspond to macroscopically distinguishable states of the universe.
Then given an initial state of the universe ##|\psi_0\rangle##, the probability that the universe will be in macroscopic configuration ##j## at a later time ##t## would be given by:
##P_j(t) = \langle \psi_0 | e^{+iHt} \Pi_j e^{-iHt} \rangle \equiv \langle \psi_0 |\Pi_j(t)| \psi_0 \rangle##
where ##\Pi_j(t)## is the time-dependent Heisenberg operator corresponding to ##\Pi_j##:
##\Pi_j(t) = e^{+iHt}\Pi_j e^{-iHt}##
If we could actually calculate ##P_j(t)##, that would give the entire empirical content of quantum mechanics.