Hello Wei Dai. Your paradigm is a bit opaque to me. There’s a cosmology here which involves programs, program outputs, and probability distributions over each, but I can’t tell what’s supposed to exist. Just the program outputs? The program outputs and the programs? Does the program correspond to “basic physical law”, and program output to “the physical world”?
If I try to abstract away from the metaphysical idiosyncrasies, the idea seems to be that Born’s rule is true because the worlds which function according to Born’s rule are the majority of the worlds in which sentient beings show up. Well, it could be true. But here’s an interesting Bohmian fact: if you start out with an ensemble of Bohmian worlds deviating from the Born distribution, they will actually converge on it, solely due to Bohmian dynamics. (See quant-ph/0403034.) So something like the Bohmian equation of motion may actually be the more fundamental fact.
In general, I think what exists are mathematical structures, which include computations as a subclass.
But here’s an interesting Bohmian fact: if you start out with an ensemble of Bohmian worlds deviating from the Born distribution, they will actually converge on it, solely due to Bohmian dynamics.
Thanks for the link. That looks interesting, and I have a couple of questions that maybe you help me with.
Why do they converge to the Born distribution? The authors make an analogy with thermal relaxation, but there is a standard explanation of the second law of thermodynamics in terms of sizes of macrostates in configuration space, and I don’t see what the equivalent explanation is for Bohmian relaxation.
What about decoherence? Suppose you have a wavefunction that has decohered into two approximately non-interacting branches occupying different parts of configuration space. If you start with a Bohmian world that belongs to one branch, then in all likelihood its future evolution will stay within that branch, right? Now if you take an ensemble of Bohmian worlds that all belong to that branch, how will it converge to the Born distribution, which occupies both branches?
This is more of an objection to the Bohmian ontology than a question. If you look at Bohmian Mechanics as a computation, it consists of two parts: (1) evolution of the wavefunction, and (2) evolution of a point in configuration space, guided by the wavefunction. But it seems like all of the real work is being done in part 1. If you wanted to simulate a quantum system, for example, it seems sufficient to just do part 1, and then sample the resulting wavefunction according to Born’s rule, and part 2 adds more complexity and computational burden without any apparent benefit.
Let’s distinguish two versions of this question. First version: why does a generic non-Born ensemble of Bohmian worlds tend to become Born-like? I think the technical answer is to be found in footnote 9 and the discussion around equation 20. But ultimately I think it will come back to a Liouville theorem in the space of distributions. There is some natural metric under which the Born-like distributions are the majority. (Or perhaps it is that non-Born regions are traversed relatively quickly.)
Second version: why does an individual Bohmian world contain a Born distribution of outcomes? This follows from the first part. An individual Bohmian world consists of a universal wavefunction and a quasiclassical trajectory. If you pick just a few of the classical variables, you can construct a corresponding reduced density matrix in the usual fashion, and a reduced Bohmian equation of motion in which the evolution of those variables depends on that density matrix and on influences coming from all the degrees of freedom that were traced over. So when you look at all the instances, within a single Bohmian history, of a particular physical process, you are looking at an ensemble of noisy Bohmian microhistories. The argument above suggests that even if this starts as a non-Born ensemble, it will evolve into a Born-like ensemble. The only complication is the noise factor. But it is at least plausible that in the majority of Bohmian worlds, this nonlocal noise is just noise and does not introduce an anti-Born tendency.
From an all-worlds-exist perspective, which we both favor, I would summarize as follows: (1) the Born distribution is the natural measure on the subset of worlds consisting of the Bohmian worlds (2) most Bohmian worlds will exhibit an internal Born distribution of physical outcomes. At present these are conjectures rather than theorems, but I would consider them plausible conjectures in the light of Valentini’s work.
“What about decoherence?”
As we’ve just discussed, Bohmian dynamics both preserves exact Born distributions and evolves non-Born distributions towards Born-like distributions (and this is true for subsystems of a Bohmian world as well as for the whole). So the sub-ensembles in the decohered branches will preserve or evolve towards Born.
“part 2 adds more complexity and computational burden without any apparent benefit”
This is a complicated matter to discuss, not least because there is an interpretation of Bohmian mechanics, the nomological interpretation, according to which the “wavefunction” is a law of motion and not a thing. In nomological Bohmian mechanics, the configuration is all that exists, evolving according to a nonlocal potential.
Hello Wei Dai. Your paradigm is a bit opaque to me. There’s a cosmology here which involves programs, program outputs, and probability distributions over each, but I can’t tell what’s supposed to exist. Just the program outputs? The program outputs and the programs? Does the program correspond to “basic physical law”, and program output to “the physical world”?
If I try to abstract away from the metaphysical idiosyncrasies, the idea seems to be that Born’s rule is true because the worlds which function according to Born’s rule are the majority of the worlds in which sentient beings show up. Well, it could be true. But here’s an interesting Bohmian fact: if you start out with an ensemble of Bohmian worlds deviating from the Born distribution, they will actually converge on it, solely due to Bohmian dynamics. (See quant-ph/0403034.) So something like the Bohmian equation of motion may actually be the more fundamental fact.
In general, I think what exists are mathematical structures, which include computations as a subclass.
Thanks for the link. That looks interesting, and I have a couple of questions that maybe you help me with.
Why do they converge to the Born distribution? The authors make an analogy with thermal relaxation, but there is a standard explanation of the second law of thermodynamics in terms of sizes of macrostates in configuration space, and I don’t see what the equivalent explanation is for Bohmian relaxation.
What about decoherence? Suppose you have a wavefunction that has decohered into two approximately non-interacting branches occupying different parts of configuration space. If you start with a Bohmian world that belongs to one branch, then in all likelihood its future evolution will stay within that branch, right? Now if you take an ensemble of Bohmian worlds that all belong to that branch, how will it converge to the Born distribution, which occupies both branches?
This is more of an objection to the Bohmian ontology than a question. If you look at Bohmian Mechanics as a computation, it consists of two parts: (1) evolution of the wavefunction, and (2) evolution of a point in configuration space, guided by the wavefunction. But it seems like all of the real work is being done in part 1. If you wanted to simulate a quantum system, for example, it seems sufficient to just do part 1, and then sample the resulting wavefunction according to Born’s rule, and part 2 adds more complexity and computational burden without any apparent benefit.
“Why do they converge to the Born distribution?”
Let’s distinguish two versions of this question. First version: why does a generic non-Born ensemble of Bohmian worlds tend to become Born-like? I think the technical answer is to be found in footnote 9 and the discussion around equation 20. But ultimately I think it will come back to a Liouville theorem in the space of distributions. There is some natural metric under which the Born-like distributions are the majority. (Or perhaps it is that non-Born regions are traversed relatively quickly.)
Second version: why does an individual Bohmian world contain a Born distribution of outcomes? This follows from the first part. An individual Bohmian world consists of a universal wavefunction and a quasiclassical trajectory. If you pick just a few of the classical variables, you can construct a corresponding reduced density matrix in the usual fashion, and a reduced Bohmian equation of motion in which the evolution of those variables depends on that density matrix and on influences coming from all the degrees of freedom that were traced over. So when you look at all the instances, within a single Bohmian history, of a particular physical process, you are looking at an ensemble of noisy Bohmian microhistories. The argument above suggests that even if this starts as a non-Born ensemble, it will evolve into a Born-like ensemble. The only complication is the noise factor. But it is at least plausible that in the majority of Bohmian worlds, this nonlocal noise is just noise and does not introduce an anti-Born tendency.
From an all-worlds-exist perspective, which we both favor, I would summarize as follows: (1) the Born distribution is the natural measure on the subset of worlds consisting of the Bohmian worlds (2) most Bohmian worlds will exhibit an internal Born distribution of physical outcomes. At present these are conjectures rather than theorems, but I would consider them plausible conjectures in the light of Valentini’s work.
“What about decoherence?”
As we’ve just discussed, Bohmian dynamics both preserves exact Born distributions and evolves non-Born distributions towards Born-like distributions (and this is true for subsystems of a Bohmian world as well as for the whole). So the sub-ensembles in the decohered branches will preserve or evolve towards Born.
“part 2 adds more complexity and computational burden without any apparent benefit”
This is a complicated matter to discuss, not least because there is an interpretation of Bohmian mechanics, the nomological interpretation, according to which the “wavefunction” is a law of motion and not a thing. In nomological Bohmian mechanics, the configuration is all that exists, evolving according to a nonlocal potential.