[epistemic status: I'm a mathematical and philosophical expert but not a QM expert; conclusions are very much tentative]

There is tension between the following two claims:

- The fundamental nature of reality consists of the wave function whose evolution follows the Schrödinger equation.
- Some discrete facts are known.

(What is discrete knowledge? It is knowledge that some nontrivial proposition X is definitely true. The sort of knowledge a Bayesian may update on, and the sort of knowledge that logic applies to.)

The issue here is that facts are facts *about* something. If quantum mechanics has any epistemic basis, then at least some things are known, e.g. the words in a book on quantum mechanics, or the outcomes of QM experiments. The question is what this knowledge is about.

If the fundamental nature of reality is the wave function, then these facts must be facts about the wave function. But, this runs into problems.

Suppose the fact in question is "A photon passed through the measurement apparatus". How does this translate to a fact about the wave function?

The wave function consists of a mapping from the configuration space (some subset of R^n) to complex numbers. Some configurations (R^n points) have a photon at a given location and some don't. So the fact of a photon passing through the apparatus or not is a fact about configurations (or configuration-histories), not about wave functions over configurations.

Yes, some wave functions assign more amplitude to configurations in which the photon passes through the apparatus than others. Still, this does not allow discrete knowledge of the wave function to follow from discrete knowledge of measurements.

The Bohm interpretation, on the other hand, has an answer to this question. When we know a fact, we know a fact about the *true* configuration-history, which is an element of the theory.

In a sense, the Bohm interpretation states that indexical information about *which world we are in* is part of fundamental reality, unlike the many-worlds interpretation which states that fundamental reality contains no indexical information. (I have discussed the trouble of indexicals with respect to physicalism previously)

Including such indexical information as "part of reality" means that discrete knowledge is possible, as the discrete knowledge is knowledge of this indexical information.

For this reason, I significantly prefer the Bohm interpretation over the many-worlds interpretation, while acknowledging that there is a great deal of uncertainty here and that there may be a much better interpretation possible. Though my reservations about the many-worlds interpretation had led me to be ambivalent about the comparison between the many-worlds interpretation and the Copenhagen interpretation, I am not similarly ambivalent about Bohm versus many-worlds; I significantly prefer the Bohm interpretation to both many-worlds and to the Copenhagen interpretation.

Say that there is some code which will run two instances of you, one where you see a blue light, and one where you see a green light. The code is run, and you see a blue light, and another you sees a green light. The you that sees a blue light gains indexical knowledge about which branch of the code they're in. But there's no need for the code to have a "reality" index parameter to allow them to gain that knowledge. You implicitly have a natural index already: the color of light you saw. I don't see why someone living in a Many-Worlds universe wouldn't be able to do the equivalent thing.

So I guess I would say that in some sense, once you've figured out the rules, measurements don't give you any knowledge about the wave function, they just give you indexical knowledge.

The wave function is a fluid in configuration space that evolves over time. You need more theory than that to talk about discrete branches of it (configurations) evolving over time.

I agree that once you have this, you can say the knowledge gained is indexical.

I think it's something like: Sometimes you find that the wavefunction |ψ⟩ is the sum of a discrete number of components |ψ⟩=|ψ1⟩+|ψ2⟩+⋯ , with the property that for any relevant observable

A, ⟨ψi|A|ψj⟩≈0 for i≠j. (Here, "≈0" also includes things like "has a value that varies quasi-randomly and super-rapidly as a function of time and space, such that it averages to 0 for all intents and purposes", and "relevant observable" likewise means "observable that might come up in practice, as opposed to artificial observables with quasi-random super-rapidly-varying spatial and time-dependence, etc.").When that situation comes up,

ifit comes up, you can start ignoring cross-terms, and calculate the time-evolution and other properties of the different |ψi⟩ as if they had nothing to do with each other, and that's where you can use the term "branch" to talk about them.There isn't a sharp line for when the cross-terms are negligible enough to properly use the word "branch", but there are exponential effects such that it's very clearly appropriate in the real-world cases of interest.

You can derive "consistent histories" by talking about things like the probability amplitude for a person right now to have memories of seeing A and B and C all happening, or for the after-effects of events A and B and C to all be simultaneously present more generally. I think...

Thanks! To the extent that discrete branches can be identified this way, that solves the problem. This is pushing the limits of my knowledge of QM at this point so I'll tag this as something to research further at a later point.

You might be interested in the work of Jess Riedel, whose research agenda is centered around finding a formal definition of wavefunction branches, e.g. https://arxiv.org/abs/1608.05377

I think in many-worlds you have to say things like "A photon passed through the measurement apparatus, in the branch of the wavefunction where we're speaking."

The more general idea is: you have a fact with a "location tag" of where that fact is true.

In a more everyday example, I can say "The temperature is 29° here". The word "here" tags a time and place.

Or in the other direction, consider "3 quarks can bind together into a proton". This

seemsto be a universal fact that therefore doesn't need any "location tag".... But is it really? No! In some plausible theories of physics, there are many universes (and this is not related to quantum many-worlds), and they all have different fundamental constants, and in some of these universes, 3 quarks cannotbind together into a proton. So you should really say "3 quarks can bind together into a proton (in the universe where I'm speaking)".So I would just go with the paradigm of "many facts need to come with a 'location tag' specifying where that fact is meant to be valid", and if you're OK with that, quantum many-worlds is fine.

Sorry if I'm misunderstanding your point. I am an expert on QM but not a mathematical and philosophical expert ;-)

Many worlds plus a location tag

isthe Bohm interpretation. You need theory for how locations evolve into other locations (in order to talk about multiple events happening in observed time), hence the nontriviality of the Bohm interpretation.Really? I don't think I agree with that. In many-worlds, you can say "The photon passed through the apparatus in the branch of the wavefunction I find myself in", and you can also say "The photon did not pass through the apparatus in other branches of the wavefunction that I do

notfind myself in". The Bohm interpretation would reject the latter.And if the measurement just happened on Earth, but you're 4 lightyears away near Alpha Centauri, space-like-separated from the measurement, you can say "The photon passed through the apparatus in some branches of the wavefunction but not others. Right now, it is not yet determined which kind of branch I will eventually find myself in. But ~4 years from now (at the soonest), there

willbe a fact of the matter about whether I am in a photon-passed-through-the-apparatus branch of the wavefunction or not, even if nobody tells me."The Bohm interpretation would reject that quote, and say there

isa fact of the matter about measurements from which you are space-like-separated.I understand you as saying "you're in some branch of the wavefunction now, and you'll be in some branch of the wavefunction tomorrow, and you need a theory relating those". I would say: That theory is the Schrodinger equation (also keeping in mind quantum decoherence theory, which is a consequence of that). Plus the postulate that you will find yourself in any given branch of the wavefunction with a probability proportional to its squared absolute amplitude. (And see also "consistent histories".) Is something missing from that?

See my reply here.

Consistent histories may actually solve the problem I'm talking about, because it discusses evolving configurations, not just an evolving wave function.

I currently disagree with your conclusion about many worlds (I want to take modus tollens where you take modus ponens), but I think this is a really nice conundrum.

There is no reason rational agents need to assign 100% probability to anything. Eliezer rejected it long ago, although he seemingly missed the implied rejection of Bayesian updates.

Bayesianism still believes in events, which are facts about the world. So the same problem comes up there, even if no fact can be known with certainty.

(in other words: the same problems that apply to 100% justification of belief apply to 99% justification of belief)

If so, it seems to require a different argument to point to the problem in that case, since your argument in the post relied on "discrete knowledge".

I don't currently see what stops a radical probabilist from interpreting evidence as unreliable information about the wave function.

(I do have an intuition that this'll be problematic; I'm just saying that I don't currently see the argument, and I think it's different from the argument in the post.)

Yes the argument has to be changed but that's mostly an issue of wording. Just replace discrete knowledge with discrete factual evidence.

If a Bayesian sees that the detector has detected a photon, how is that evidence about the wave function?

I'm a little confused: you reject physicalism, and yet you seem to be speaking from a physicalist ontology here, requiring there to be a physical fact (a true configuration) or no fact at all (no indexical information).

I believe there are physical theories and physical facts, but that not all facts are straightforwardly physical (although, perhaps these are indirectly physical in a way that requires significant philosophical and conceptual work to determine, and which has degrees of freedom).

The issue in this post is about physical facts, e.g. measurements, needing to be interpreted in terms of a physical reality. These interpretations are required to have explanatory physical theories even if there are also non-physical facts.

Hmmm. So facts aren't exclusively physical in nature, but physical theories need to do all their explanatory work on their own, without reference to any of the nonphysical facts? I'm still pretty confused. The post makes a lot more sense to me if I read it as yet another puzzle for physicalism, rather than something directly related to your actual ontology.

Naively (ie in my naive understanding) it seems like an agent-centric perspective (ie the opposite of a view from nowhere) is more or less like Solomonoff induction (so e.g. solves anthropic reasoning via UDASSA). The world is built outward from the agent, rather than the other way around, but we still get something like indexical facts. So many-worlds seems ok.

It's rather nonstandard to consider things like photon measurements to be nonphysical facts. Presumably, these come within the domain of physical theories.

Suppose we go with Solomonoff induction. Then we only adopt physical theories that explain observations happening over subjective time. These observations include discrete physical measurements.

It's not hard to see how Bohm explains these measurements: they are facts about the true configuration history.

It is hard to see how many worlds explains these measurements. Some sort of bridge law is required. The straightworward way of specifying the bridge law

isthe Bohm interpretation.Preferences do not make science. Philosophy, for sure.

Odds are, once mesoscopic quantum effects become accessible to experiment, we will find that none of the interpretational models reflect the observations well. I would put 10:1 odds that the energy difference of entangled states cannot exceed about one Planck mass, a few micrograms. Whether there is a collapse of some sort, hidden variables, superdeterminism, who knows.

Anyway, in general I find this approach peculiar, picking a model based on emotional reasoning like "I like indexicality", or "String theory is pretty". It certainly can serve as a guide of what to put one's efforts in as a promising research area, but it's not a matter of preference, the observations will be the real arbiter.

We model some discrete facts as being known. But Bohm interpretation or Everett, there's still just a bunch of microphysical stuff moving around. To expect the microphysical stuff to be different so that our model of how things are known works seems backwards to me.

I'm saying that our microphysical theories should explain our macrophysical observations. If they don't then we toss out the theory (Occam's razor).

Macrophysical observations are discrete.

I model macrophysical observations as discrete too. But I also model tables and chairs as discrete, without needing to impose any requirements that they not be made of non-discrete stuff. A microphysical explanation of discrete observations doesn't need to be made up of discrete parts.

Then you need a theory of how the continuous microstate determines the discrete macrostate. E.g. as a function from reals to booleans. What is that theory in the case of the wave function determining photon measurements?

If the microphysical theory is like quantum mechanics (Bohm-ish mechanics very much included), this is basically Schrödinger's cat argument. It would be

absurdif there was not some function from the microphysical state of the world to the truth of the macrophysical fact of whether the cat in the box is alive or dead. Therefore, there is some such function, and if quantum mechanics doesn't support it then quantum mechanics is incomplete.Schrödinger was wrong about the cat thing, as far as we can tell. His knowledge of discrete macrophysical states of cats had an

explanation, but didn't directly reflect reality.There are absurd quantum states that don't allow for a function from the microphysical state of the world to whether I observe a photon as having spin left or spin right. If I believe otherwise, my beliefs deserve an explanation, but that doesn't mean they directly reflect reality.

I'm not asking for there to be a function to the entire world state, just a function to observations. Otherwise the theory does not explain observations!

(aside: I think Bohm does say there is a definite answer in the cat case, as there is a definite configuration that is the true one; it's Copenhagen that fails to say it is one way or the other)

Bohmian mechanics is not relativistic and has not been coherently formulated for spin-1/2 or spin 1 fields.

The Copenhagen interpretation is the best (most accurate) interpretation of quantum mechanics, so long as it is understood as a purely "epistemic" interpretation. That is: unlike pre-quantum theories, quantum mechanics does not provide a complete ontology of the world. There are physical properties (the observables) that can take various values, and the theory gives conditional probabilities for these possibilities, but no picture of what happens in between.

Consistent histories, applied to cosmology, is a slight adaptation of Copenhagen, in which one can obtain a probability for an entire history of the universe (specified in terms of observables-taking-values), given a "wavefunction of the universe" and a set of "mutually decoherent histories". However, it is still not ontologically complete, as it is still up to the user to decide when and where in each history, observables shall take values. The only constraint is mutual decoherence, i.e. not violating the uncertainty principle. One might look for a set of maximally specified decoherent histories, as a determining ontological principle, but there's still a very large number of ways to do this.

If simplest physics contradicts epistemology, you should change epistemology - it would be nice to develop some weird quantum knowledge theory without fundamental discrete facts.

Let me know if anyone succeeds at that. I've thought in this direction and found it very difficult.