A dramatization of what this means… Suppose there was a process which put me into a superposition of happy, sad, and dead. I suppose that positionists would like to think that this quantum state corresponds to the existence of three worlds, each of them a distinct spatial configuration. Schematically:
But non-position-basis states, when viewed in terms of configurations, are themselves superpositions. Even if they are peaked at a certain region of configuration space, they will contain a residual nonzero amplitude for everything else as well. And so the happy/sad/dead superposition will resolve into something like this:
… where the epsilons indicate the presence of a small but nonzero amplitude for the “wrong” configurations, even though we are now supposedly talking about a “world” or “branch” of the wavefunction which can be identified with the empirical reality of an entity being in one particular state.
The point of the many worlds interpretation is that we can identify components of the wavefunction with the diverse, mutually exclusive outcomes we see empirically. Thus, the cat which is in the superposition “dead plus alive” turns out to be duplicated; in one world it is alive, in the other world it is dead. That’s the idea. But when we try to implement the idea in this way, it turns out that the cat in one world is alive plus epsilon dead, and in the other world it is dead plus epsilon alive. This suggests to me that there is a problem.
Wouldn’t people who support a preferred basis agree that you can write a given state as a linear combination of one of the non-preferred bases? Wouldn’t they just say that the linear-algebraic Hilbert-space formalism, which allows this, fails to capture some fundamental physical distinction among the bases?
I think that what I’m missing is how this comment bears on its parent (which I didn’t understand, because I haven’t read Hanson’s paper).
ETA: So, I’ve looked at Hanson’s paper. It looks like his projection operators are state-dependent, so that, as you say, they are time-dependent as the state evolves. And, associated to the evolving projection operator, there is an evolving basis. This evolving basis is important for the mangled-worlds approach, because it keeps track of which worlds are resistant to mangling.
But a supporter of the position basis might still maintain that, nonetheless, and all the while, the position basis retains some fundamental ontological significance. Personally, I don’t find the arguments for preferring a basis to be all that convincing. But “positionists” already prefer the position basis over the energy basis, despite the fact that (AFAIK) the cleanest presentation of the Schrödinger equation is in terms of the Hamiltonian. So what’s to stop them from disregarding the role of Hanson’s evolving projection operator’s basis?
The evolving basis of mangled worlds is meant to explain the Born probabilities, by producing worlds in the correct multiplicities to reproduce observed frequencies. If you ignore this, you’re discarding the very rationale of mangled worlds.
I didn’t express myself clearly when I wrote, “So what’s to stop them from disregarding the role of Hanson’s evolving projection operator’s basis?”. I didn’t mean that “positionists” would disregard the role that Hanson’s bases might have in explaining the Born probabilities. I just meant that positionists would deny that this role confers the “ontological fundamentalness” that they reserve for the position basis.
Normally, if a basis is regarded as ontologically fundamental, it is because all one’s worlds are basis vectors in that basis. “Alive plus epsilon dead” and “dead plus epsilon alive” are definitely not basis vectors from the position basis.
Anyway, the important fact is that for Robin’s scheme to work, each individual world must have small amplitudes of other configurations shadowing the dominant configuration. It’s a sign that it’s a contrivance, that it doesn’t work. Do the small-amplitude copies of me in other states that shadow me in this individual world also have experiences? If so, doesn’t that screw up the reproduction of the Born probabilities? Because that is all about just counting the dominant configuration.
A dramatization of what this means… Suppose there was a process which put me into a superposition of happy, sad, and dead. I suppose that positionists would like to think that this quantum state corresponds to the existence of three worlds, each of them a distinct spatial configuration. Schematically:
But non-position-basis states, when viewed in terms of configurations, are themselves superpositions. Even if they are peaked at a certain region of configuration space, they will contain a residual nonzero amplitude for everything else as well. And so the happy/sad/dead superposition will resolve into something like this:
… where the epsilons indicate the presence of a small but nonzero amplitude for the “wrong” configurations, even though we are now supposedly talking about a “world” or “branch” of the wavefunction which can be identified with the empirical reality of an entity being in one particular state.
The point of the many worlds interpretation is that we can identify components of the wavefunction with the diverse, mutually exclusive outcomes we see empirically. Thus, the cat which is in the superposition “dead plus alive” turns out to be duplicated; in one world it is alive, in the other world it is dead. That’s the idea. But when we try to implement the idea in this way, it turns out that the cat in one world is alive plus epsilon dead, and in the other world it is dead plus epsilon alive. This suggests to me that there is a problem.
Wouldn’t people who support a preferred basis agree that you can write a given state as a linear combination of one of the non-preferred bases? Wouldn’t they just say that the linear-algebraic Hilbert-space formalism, which allows this, fails to capture some fundamental physical distinction among the bases?
I think that what I’m missing is how this comment bears on its parent (which I didn’t understand, because I haven’t read Hanson’s paper).
ETA: So, I’ve looked at Hanson’s paper. It looks like his projection operators are state-dependent, so that, as you say, they are time-dependent as the state evolves. And, associated to the evolving projection operator, there is an evolving basis. This evolving basis is important for the mangled-worlds approach, because it keeps track of which worlds are resistant to mangling.
But a supporter of the position basis might still maintain that, nonetheless, and all the while, the position basis retains some fundamental ontological significance. Personally, I don’t find the arguments for preferring a basis to be all that convincing. But “positionists” already prefer the position basis over the energy basis, despite the fact that (AFAIK) the cleanest presentation of the Schrödinger equation is in terms of the Hamiltonian. So what’s to stop them from disregarding the role of Hanson’s evolving projection operator’s basis?
The evolving basis of mangled worlds is meant to explain the Born probabilities, by producing worlds in the correct multiplicities to reproduce observed frequencies. If you ignore this, you’re discarding the very rationale of mangled worlds.
I didn’t express myself clearly when I wrote, “So what’s to stop them from disregarding the role of Hanson’s evolving projection operator’s basis?”. I didn’t mean that “positionists” would disregard the role that Hanson’s bases might have in explaining the Born probabilities. I just meant that positionists would deny that this role confers the “ontological fundamentalness” that they reserve for the position basis.
Normally, if a basis is regarded as ontologically fundamental, it is because all one’s worlds are basis vectors in that basis. “Alive plus epsilon dead” and “dead plus epsilon alive” are definitely not basis vectors from the position basis.
Anyway, the important fact is that for Robin’s scheme to work, each individual world must have small amplitudes of other configurations shadowing the dominant configuration. It’s a sign that it’s a contrivance, that it doesn’t work. Do the small-amplitude copies of me in other states that shadow me in this individual world also have experiences? If so, doesn’t that screw up the reproduction of the Born probabilities? Because that is all about just counting the dominant configuration.