PeterDonis said:
I have already explained that there are multiple meanings for terms like "collapse". That means that, when you read anything that includes that word, you can't just assume it means what you would like it to mean, or that it always means the same thing. You have to read the words carefully, in context, and figure out what the words mean in that context.
Go do that.
OK. I'll try.
There are two different notions of "wave function collapse" in "standard textbook quantum theory":
1) One of these is that when a system whose quantum state is initially pure, becomes entangled with a larger environment, its state must now be described as mixed, if one wants to exclude the environment. That means instead of a single quantum vector wave function, we must use a density operator,
mathematically. In terms of conceptualization, pure states are extremal points; mixed states are in the middle.
2) The second of these is the von Neumann-Lueders "collapse postulate", which introduces a new primitive undefined term, "measurement", or "observation", into the quantum theory, in which a random replacement of one wave function with another occurs, representing a single observational outcome.
The question is what, if any, relationship there is between these two things, and what is the significance of the undefined term "measurement" or "observation". In the classical limit of quantum theory, the von Neumann-Lueders collapse looks like the "gain of information" in Bayesian probability, and moreover becomes indispensible to make sense of what we're seeing as "really being classical mechanics", and thus I think it makes sense that this interpretation should be retained in the non-classical regime as well, because the structure of the mathematical formalisms are identical; the only difference is whether ##\hat{x}## and ##\hat{p}## commute or not or, equivalently, if ##\hbar## is or isn't zero.
Here you say:
PeterDonis said:
Collapse interpretations say it means the wave function has actually, physically collapsed to a single result: i.e., the two terms in the decoherent entangled superposition of dead and alive have become one, either the dead term or the alive term.
On a collapse interpretation ... the entangled superposition goes away as soon as decoherence happens, because a collapse interpretation says that the actual, physical collapse happens when decoherence happens.
It seems to me that you're saying that
on a collapse interpretation the
physical process (whatever It means) of "collapse of the wave function" happens corresponding to decoherence: in Schroedinger's Cat, it happens long before the box is opened.
The collapse of the wave function happens whenever the quantum system initially described by the wave function becomes entangled with environment — the part of the Universe that wasn't tracked by the wave function. For example, when the particle's state evolves to the point at which it has a significant amplitude in the vicinity of the detector, the counter clicks and there is the collapse of the wavefunction.
In fact, an observation is certain physical effect that the observer has on the measured object. E.g., the Landau & Lifshitz textbook specifies that
any macroscopic object can be an observer.
The theorem of von Neumann (it can be found on "Mathematical Foundations of Quantum Mechanics") says that it doesn't make a bit of difference whether you model the cat (or anything else along the causal chain between closing the box and opening it to observe the cat) as capable of collapsing the wave function.
You'll make exactly the same testable predictions no matter where along the way you place the collapse.
Indeed, as you say:
In the basic math of QM, "collapse" means the mathematical process of updating the wave function you are going to use to predict the results of future measurements, once you know the results of the measurement that was already made. In the case of Schrodinger's cat, you don't know the result until you open the box. That is the case even though decoherence happens well before you open the box. It is pointless to say "well, the wave function collapsed when decoherence happened, even though I didn't know the result until I opened the box". In the basic math of QM, "collapse" is not something that happens in the world at some particular time; it's something that happens in your model when you get new information.
This is a "mostly subjective" viewpoint of quantum mechanics as it is really, despite looking at all the alternatives, the only one that fits the closest to the mathematics of the theory as given with no other adulterations.
On a subjective account, the wave function belongs to
you, the one outside of the box. It
models, your information or knowledge about the state of affairs in the box. The transition from "live cat" to "live or dead cat" to "dead cat" starting from the initial state is just showing how your
best knowledge,
without looking into the box, that you can predict from that initial state, changes. All we can say about the "superposition" at the in-between point is that it means the
predicted information about the answer to the question "is the cat alive?" is less than one bit.
As a subject, the cat may be assigned a wave function talking about the information it has regarding the contraption that is going to kill it. Of course, soon after that one "collapses" then there won't be wave function any more because this subject, the information-bearer, has been terminated.
Hence, from that point of view, it makes no sense to ask ourselves when physical collapse is going to occur, because the wave function models
your knowledge, not the cat's. The cat can't do anything to that.
In this sense:
PeterDonis said:
As far as the basic math of QM goes, where "collapse" is just a mathematical procedure where you update the wave function once you know the result of a measurement, obviously you don't know the result of the measurement until you open the box, so you wouldn't actually make the update to the wave function until then. But that says nothing whatever about any physical process that does or does not happen at any particular time.
In this sense:
It is pointless to say "well, the wave function collapsed when decoherence happened, even though I didn't know the result until I opened the box".
Is my explanation reasonable? Where did I go wrong?
