There is a simple question about quantum theory that has been increasingly bothering me. I keep hoping that my reading about interpretational issues will turn up a discussion of this point, but that hasn’t happened. I’m hoping someone expert in such issues can provide an answer and/or pointers to places where this question is discussed.

In the last posting I commented that I’m not sympathetic to recent attempts to “reconstruct” the foundations of quantum theory along some sort of probabilistic principles. To explain why, note that I wrote a long book about quantum mechanics, one that delved deeply into a range of topics at the fundamentals of the subject. Probability made no appearance at all, other than in comments at the beginning that it appeared when you had to come up with a “measurement theory” and relate elements of the quantum theory to expected measurement results. What happens when you make a “measurement” is clearly an extremely complex topic, involving large numbers of degrees of freedom, the phenomenon of decoherence and interaction with a very complicated environment, as well as the emergence of classical behavior in some particular limits of quantum mechanics. It has always seemed to me that the hard thing to understand is not quantum mechanics, but where classical mechanics comes from (in the sense of how it emerges from a “measurement”).

A central question of the interpretation of quantum mechanics is that of “where exactly does probability enter the theory?”. The simple question that has been bothering me is that of why one can’t just take as answer the same place as in the classical theory: in one’s lack of precise knowledge about the initial state. If you do a measurement by bringing in a “measuring apparatus”, and taking into account the environment, you don’t know exactly what your initial state is, so have to proceed probabilistically.

One event that made me think more seriously about this was watching Weinberg’s talk about QM at the SM at 50 conference. At the end of this talk Weinberg gets into a long discussion with ‘t Hooft about this issue, although I think ‘t Hooft is starting from some unconventional point of view about something underlying QM. Weinberg ends by saying that Tom Banks has made this argument to him, but that he thinks the problem is you need to independently assume the Born rule.

One difficulty here is that you need to precisely define what a “measurement” is, before you can think about “deriving” the Born rule for results of measurements, and I seem to have difficulty finding such a precise definition. What I wonder about is whether it is possible to argue that, given that your result is going to be probabilistic, and given some list of properties a “measurement” should satisfy, can you show that the Born rule is the only possibility?

So, my question for experts is whether they can point to good discussions of this topic. If this is a well-known possibility for “interpreting” QM, what is the name of this interpretation?

**Update**: I noticed that in 2011 Tom Banks wrote a detailed account of his views on the interpretation of quantum mechanics, posted at Sean Carroll’s blog, with an interesting discussion in the comment section. This makes somewhat clearer the views Weinberg was referring to. To clarify the question I’m asking, a better version might be: “is the source of probability in quantum mechanics the same as in classical mechanics: uncertainty in the initial state of the measurement apparatus + environment?”. I need to read Banks more carefully, together with his discussion with others, to understand if his answer to this would be “yes”, which I think is what Weinberg was saying.

**Update**: My naive questions here have attracted comments pointing to very interesting work I wasn’t aware of that is along the lines of what I’ve been looking for (a quantum model of what actually happens in a measurement that leads to the sort of classical outcomes expected, such that one could trace the role of probability to the characterization of the initial state and its decomposition into a system + apparatus). What I learned about was

In these last references the implications for the measurement problem are discussed in great detail, but I’m still trying to absorb the subtleties of this story.

I’d be curious to hear what experts think of Landsman’s claim that there’s a possible distinct “instability” approach to the measurement problem that may be promising.

**Update**: From the comments, an explanation of the current state of my confusion about this.

The state of the world is described at a fixed time by a state vector, which evolves unitarily by the Schrodinger equation. No probability here.

If I pick a suitable operator, e.g. the momentum operator, then if the state is an eigenstate, the world has a well-defined momentum, the eigenvalue. If I couple the state to an experimental apparatus designed to measure momenta, it produces a macroscopic, classically describable, readout of this number. No probability here.

If I decide I want to know the position of my state, one thing the basic formalism of QM says is “a momentum eigenstate just doesn’t have a well-defined position, that’s a meaningless question. If you look carefully at how position and momentum work, if you know the momentum, you can’t know the position”. No probability here.

If I decide that, even though my state has no position, I want to couple it to an experimental apparatus designed to measure the position (i.e. one that gives the right answer for position eigenstates), then the Born rule tells me what will happen. In this case the “position” pointer is equally likely to give any value. Probability has appeared.

So, probability appeared when I introduced a macroscopic apparatus of a special sort: one with emergent classical behavior (the pointer) specially designed to behave in a certain way when presented with position eigenstates. This makes me tempted to say that probability has no fundamental role in quantum theory, it’s a subtle feature of the emergence of classical behavior from the more fundamental quantum behavior, that will appear in certain circumstances, governed by the Born rule. Everyone tells me the Born rule itself is easily explicable (it’s the only possibility) once you assume you will only get a probabilistic answer to your question (e.g. what is the position?)

A macroscopic experimental apparatus never has a known pure state. If I want to carefully analyze such a setup, I need to describe it by quantum statistical mechanics, using a mixed state. Balian and collaborators claim that if they do this for a specific realistic model of an experimental apparatus, they get as output not the problematic superposition of states of the measurement problem, but definite outcomes, with probabilities given by the Born rule. When I try and follow their argument, I get confused, realize I am confused by the whole concept: tracking a mixed quantum state as it evolves through the apparatus, until at some point one wants to talk about what is going on in classical terms. How do you match your classical language to the mixed quantum state? The whole thing makes me appreciate Bohr and the Copenhagen interpretation (in the form “better not to try and think about this”) a lot more…