bluecap said:
Oh actually first time to hear about this. Ill read Ballentine tomorrow curious to see what's all the fuss about it. Thanks for the tips. Btw do you consider the quantum state as objective or concern only the bayerian and frequentist aspects or side of it? Then you are a genuine Ensemble Interpretation proponent while Vanees71 is more a hybrid Ensembler/Copenhagen right? He believes the quantum state is objective while you are agnostic. We mustn't use categorication from book only or author but from technical consideration. Many thanks.
Well, Vanhees says that the quantum state is objective because it is an equivalence class of preparation procedures. That's what I would call
subjectve. It seems the same to me as the idea that the quantum state represents our information about the system, which is a subjective notion of the state.
What's weird about QM is that there are two interpretations that people freely switch back and forth between, even though they seem completely different. No, I don't mean Copenhagen versus Many Worlds versus Bohmian.The two interpretations are:
- QM is a deterministic theory about microscopic systems. (This initial state will deterministically evolve into that state, according to Schrodinger's equation)
- QM is a stochastic (nondeterministic) theory about macroscopic systems. (If you perform this experiment, you will get one of these results, with such and such probabilities)
Here's my feeble attempt to bridge the gap between these two interpretations, which I think is compatible with Copenhagen.
- Let's suppose that we have a Hamiltonian H for the entire universe, and a corresponding Hilbert space of possible pure states.
- Assume for simplicity that our universe is finite.
- Pick a complete basis |\psi_\lambda\rangle.
- Assume a finite degree of precision for any measurement of a quantity.
- This implies a countable (or maybe even finite) set of possible distinguishable "classical states" for the universe. Call them S_i
- Then presumably the "classical state" of the universe can in principle be defined via a countable (or even finite) indexed collection of projection operators \Pi_j. The meaning of this is that if the universe is in state |\Psi\rangle, then it's in the classical state j provided that \Pi_j |\Psi\rangle = |\Psi\rangle.
- At this point, I have a bit of a problem. To describe the dynamics of classical states, it's not enough to know the projection operator. You also need a density matrix. The operators \Pi_j are massively degenerate; there are many, many microscopic states corresponding to the same macroscopic state. So if all you know is the macroscopic state (which is all we ever can know), then the best we can do is to have a probability distribution on microscopic states. This can be described by the numbers p_{i,\lambda}, the probability that the microstate is |\psi_\lambda\rangle given that the macrostates is i. Or equivalently, it can be described by the density matrix \rho_i = \sum_\lambda p_{i,\lambda} |\psi_\lambda\rangle \langle \psi_\lambda|
Now, we can give the classical dynamics. If the universe starts in the classical state S_i at time t_1, then the probability that it will be in classical state S_j at time t_2 will be given by:
P(i,t_1, j, t_2) = \sum_\lambda p_{i,\lambda} \langle \psi_\lambda|\Pi_j(t_2 - t_1) |\psi_\lambda\rangle
where \Pi_j(t_2 - t_1) is the operator \Pi_j in the Heisenberg picture: \Pi_j(t_2 - t_1) = e^{+i H (t_2 - t_1)} \Pi_j e^{-i H (t_2 - t_1)}
This transition function P(i,t_1, j, t_2) in a sense tells us everything we need to know, and everything that we can test experimentally. The details of complex-valued wave functions that evolve unitarily can be seen as just calculational tools for deriving this macroscopic dynamics.
But there are many strange aspects to this macroscopic dynamics, but perhaps that would consume another thread.
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