Predictions and Everett

Imagine this unfortunate sequence of events will certainly befall you in a classical universe:

1. You will be made to fall asleep.

2. Upon waking up, you will be shown a red square.

3. You will be made to fall asleep again.

4. While asleep, your memory will be reset to that which you had in step (1).

5. Upon waking up, you will be shown a green triangle.

6. You will be made to fall asleep for a third time.

7. While asleep, your memory will be reset again to that which you had in step (1).

8. Upon waking up, you will be shown a green circle.

9. You will then be permanently annihilated.

Questions:

10. How likely is it that you will be shown a green shape?

11. How likely is it that you will be shown a red shape?

The answers to these questions are obviously: one and one. You will be shown a green shape twice and a red shape one, and that’s certain.

Now consider a variant story where personal identity is not maintained in sleep. Perhaps each time in sleep the person who fell asleep will be annihilated and replaced by something that is in fact an exact duplicate, but that isn’t identical with the original according to the correct metaphysics of diachronic personal identity. (We can make this work on pretty much any metaphysics of diachronic personal identity. For example, we can make it work on a materialist memory theory as follows. We just suppose that before step (1), you happen to have three exact duplicates alive, who are not you. Then during the nth sleep cycle, the sleeper is annihilated, and a fresh brain is prepared and memories will be copied into it from your nth doppelganger. Since these memories don’t come from you, the resulting brain isn’t yours.)

And in the variant story, let’s ask the questions (10) and (11) again. What will the answers be? Again, it’s easy and obvious: zero and zero. You won’t be shown any shapes, because you will be annihilated in your sleep before any shapes are shown.

Now consider Everettian branching quantum mechanics. Suppose there is a quantum process that will result in your going to sleep in an equal superposition of states between having a red square, a green triangle and a green circle in front of your head, so that upon waking up an observation of the shape will be made. Now ask questions (10) and (11) again.

I contend that this is just as easy as in my classical universe story. Either the branching preserves personal identity or not. If it preserves personal identity, the answer to the questions is one and one. If it fails to preserve personal identity, the answer to the questions is zero and zero. The only relevant ontological difference between the quantum and classical stories is that in the quantum stories the wakeups might count as simultaneous while in the classical story the wakeups are sequential. And that really makes no difference.

In none of the four cases—the classical story with or without personal identity and the branching story with or without personal identity—are the answers to the questions 2/3 and 1/3. But those are in fact the right answers in the quantum case, contrary to the Everett model.

Now, one might object that we care more about decisions than predictions. Suppose that you have a choice between playing a game with one of two three-sided fair quantum dice:

• Die A is marked: red square, green triangle, green circle.

• Die B is marked: green square, red triangle, red circle.

And suppose pain will be induced if and only if the die comes up red. Which die should you prudentially choose for playing the game? Again, it depends on whether personal identity is preserved. If not, it makes no difference. If yes, clearly you should go for die A on the Everett model—and that is indeed the intuitively correct answer. But the reason for going for die A on the Everett model is different from the reason for going for it on a non-branching quantum mechanics. On the Everett model, the reason for going for die A is that it’s better to get pain once (die A) rather than twice (die B).

So far so good. But now suppose that you’ve additionally been told that if you go for die A, then before you roll A, an irrelevant twenty-sided die will be rolled. (This is a variant of an example Peter van Inwagen sent me years ago, which was due to a student of his.) Then, intuitively, if you go for die A, there will be twenty red branches and forty green branches on Everett. So on die A, you get pain twenty times if personal identity is preserved, and on die B you get pain only twice. And so you should surely go for die B, which is absurd.

One might reasonably object that there are in fact infinitely many branches no matter what. But then on the no-identity version, the choice is still irrelevant to you prudentially, while on the identity version, no matter what you do, you get pain infinitely many times no matter what you choose. And that doesn’t work, either. And if there is no fact about how many branches there will be, then the answer is just that there is no fact about which option is preferable on the identity version, and on the no-identity version, indifference still follows.

This is all basically well-known stuff. But I like the above way of making it vivid by thinking about classically sequentializing the story.

What I think is wrong with Everettian quantum mechanics

One can think of Everettian multiverse quantum mechanics as beginning by proposing two theses:

1. The global wavefunction evolves according to the Schroedinger equation.

2. Superpositions in the global wavefunction can be correctly interpreted as equally real branches in a multiverse.

But prima facie, these two theses don’t fit with observation. If one prepares a quantum system in a (3/5)|↑⟩+(4/5)|↓⟩ spin state, and then observes the spin, one will will observe spin up in |3/5|^2=9/25 cases and spin down in |4/5|^2=16/25 cases. But (roughly speaking) there will be two equally real branches corresponding to this result, and so prima facie one would expect equally likely observations, which doesn't fit observation. But the Everettian adds a third thesis:

3. One ought to make predictions as to which branch one will observe proportionately to the square of the modulus of the coefficients that the branch has in the global wavefunction.

Since Aristotelian science has been abandoned, there has been a fruitful division of labor between natural science and philosophy, where investigation of normative phenomena has been relegated to philosophy while science concerned itself with the non-normative. From that point of view, while (1) and (less clearly but arguably) (2) belong to the domain of science, (3) does not. Instead, (3) belongs to epistemology, which is study of the norms of thought.

This point is not a criticism. Just as a doctor who has spent much time dealing sensitively with complex cases will have unique insights into bioethics, a scientist who has spent much time dealing sensitively with evidence will have unique insights into scientific epistemology. But it is useful, because the division of intellectual labor is useful, to remember that (3) is not a scientific claim in the modern sense. And there is nothing wrong with that as such, since many non-scientific claims, such as that one shouldn’t lie and that one should update by conditionalization, are true and important to the practice of the scientific enterprise.

But (3) is a non-scientific claim that is absurd. Imagine that a biologist came up with a theory that predicted, on the basis of their genetics and environment, that:

4. There are equal numbers of male and female infant spider monkeys.

You might have thought that this theory is empirically disproved by observations of a lot more female than male infant spider monkeys. But our biologist is clever, and comes up with this epistemological theory:

5. One ought to make predictions as to the sex of an infant spider monkey one will observe in inverse proportion to the ninth power of the average weight of that sex of spider monkeys.

And now, because male spider monkeys are slightly larger than females, we will make predictions that roughly fit our observations.

Here’s what went wrong in our silly biological example. The biologist’s epistemological claim (5) was not fitted to the actual ontology of the biologist’s theory. Instead, basically, the biologist said: when making predictions of future observations, make them in the way that you should if you thought the sex ratios were inversely proportional to the ninth power of the average weights, even though they aren’t.

This is silly. But exactly the same thing is going on in the Everett case. We are being told to make predictions in the way you should if the modulus squares of the weights in the superposition were chances of collapse. But they are not.

It is notorious that any scientific theory can be saved from empirical disconfirmation by adding enough auxiliary scientific hypotheses. But one can also save any scientific theory from empirical disconfirmation by adding an auxiliary philosophical hypothesis as to how confirmation or disconfirmation ought to proceed. And doing that may be worse than obstinately adding auxiliary scientific hypotheses. For auxiliary scientific hypotheses can often be tested and disproved. But an auxiliary epistemological hypothesis may simply close the door to refutation.

To put it positively, we want a certain degree of independence between epistemological principles and the ontology of a theory so that the ontology of the theory can be judged by the principles.

A modified consciousness-causes-collapse interpretation of quantum mechanics

Here are two technical problems with consciousness causes collapse (ccc) interpretations of quantum mechanics. In both, suppose a quantum experiment with two possible outcomes, A and B, of equal probability 1/2.

1. The sleeping experimenter: The experimenter is dreamlessly asleep in the lab and the experiment is rigged to wake her up on measuring A by ringing a bell. If conscious observation causes collapse, then when A is measured, the experimenter is woken up, and collapse occurs. Presumably, this happens half the time. But what happens the other half the time? No conscious observation occurs, so no collapse occurs, so the system remains in a superposition of A and B states. But that means that when the experimenter naturally wakes up several hours later, and then collapse will happen. However, when collapse happens then, it has both A and B outcome options at equal chances. But that means that overall, there is a 75% chance of an A outcome, which is wrong.

2. Order of explanation: The experimenter is awake. On outcome A, a bell rings. On B, a red light goes on. In fact, A is observed. What caused the collapse? It wasn’t the observer’s hearing the bell, because the bell’s occurrence is explanatorily posterior to the collapse. But we said that it is conscious observation that causes the collapse. Which conscious observation was that, if it wasn’t the hearing of the bell? Note that the observer need not have been conscious prior to hearing the bell or seeing the light—the experiment can be rigged so that either the bell or the light wakes up the observer. Perhaps the cause of the collapse was the state of being about to hear a bell or see a red light, or maybe it was the disjunctive state of hearing a bell or seeing a red light. But the former is a strange kind of cause, and the second would be a weird case where the disjunction is prior to its true disjunct.

The first problem strikes me as more serious than the second—the second is a matter of strangeness, while the first yields incorrect predictions.

I’ve been thinking about a curious ccc interpretation that escapes both problems. On this interpretation, the universe branches like in Everett-style multiverse explanations, but a conscious observation in any branch causes collapse. Collapse is the termination of a bunch of branches, including perhaps the termination of the branch in which the collapse-causing observation occurred. The latter isn’t some sort of weird retroactive thing—it’s just that the branch terminates right after the observation.

In case 2, the universe branches into an A-universe and a B-universe (or into pluralities of universes of both sorts). In the A-universe a bell is heard by the observer. In the B-universe a red light is seen by her. When this happens, collapse occurs, and there is no future to the observer after the observation of the red light, because in fact (or so case 2 was set up) it is the observation of A that won out. Or at least this is how it is when the two observations would be simultaneous. Suppose next that the bell observation would be made slightly earlier. Then as soon as the bell observation is made, the B-branch is terminated, and the red light observation is never made. On the other hand, if the light observation is timed to come first, then as soon as the light observation is made in the B-branch, this observation terminates the B-branch, and shortly afterwards the bell is heard in the remaining branch, the A-branch.

Case 1, then, works as follows. The universe branches into an A-universe, with a bell, and a silent B-universe. As soon as the bell is heard in the A-universe, the observation causes collapse, and one of the branches is terminated. If it’s the A-branch that’s terminated, then the observer heard the bell, but the future of that observation is annihilated. Instead, a couple of hours later the observer wakes up in the B-branch, and deduces that B must have been measured. If it’s the B-branch that’s terminated, on the otehr hand, then the observer’s observing of the bell has a future.

Prior to collapse, on this interpretation, we are located in multiple branches. And then our multilocation is wholly or partly resolved by collapse in favor of location in a proper subset of the branches where we were previously located. What happened to us in the other branches really did happen to us, but we never remember it, because it’s not recorded to memory.

On this interpretation, various things are observed by us which we never remember, because they have no future. This is a bit disquieting. Suppose that instead of the red light in case 2, the experimenter is poked with a red hot poker. Then if she hears the bell ring, she is relieved to have escaped the pain. But she didn’t: for if the poking is timed at or before the ringing, then the poking really did happen to her, albeit in another branch and not recorded to memory.

Fortunately for us, the futureless unremembered bad things were very brief: they only lasted for as short a period of time as was needed to establish them as phenomenologically different from the other possible outcome. So in the poked-with-a-poker branch, one only feels the pain for the briefest moment. And that’s not a big deal.

I worry a bit about quantum Zeno issues with this interpretation.

No-collapse interpretations without a dynamically evolving wavefunction in reality

Bohm’s interpretation of quantum mechanics has two ontological components: It has the guiding wave—the wavefunction—which dynamically evolves according to the Schrödinger equation, and it has the corpuscles whose movements are guided by that wavefunction. Brown and Wallace criticize Bohm for this duality, on the grounds that there is no reason to take our macroscopic reality to be connected with the corpuscles rather than the wavefunction.

I want to explore a variant of Bohm on which there is no evolving wavefunction, and then generalize the point to a number of other no-collapse interpretations.

So, on Bohm’s quantum mechanics, reality at a time t is represented by two things: (a) a wavefunction vector |ψ(t)⟩ in the Hilbert space, and (b) an assignment of values to hidden variables (e.g., corpuscle positions). The first item evolves according to the Schrödinger equation. Given an initial vector |ψ(0)⟩, the vector at time t can be mathematically given as |ψ(t)⟩ = U_t|ψ(0)⟩ where U_t is a mathematical time-evolution operator (dependent on the Hamiltonian). And then by a law of nature, the hidden variables evolve according to a differential equation—the guiding equation—that involves |ψ(t)⟩.

But now suppose we change the ontology. We keep the assignment of values to hidden variables at times. But instead of supposing that reality has something corresponding to the wavefunction vector at every time, we merely suppose that reality has something corresponding to an initial wavefunction vector |ψ₀⟩. There is nothing in physical reality corresponding to the wavefunction at t if t > 0. But nonetheless it makes mathematical sense to talk of the vector U_t|ψ₀⟩, and then the guiding equation governing the evolution of the hidden variables can be formulated in terms of U_t|ψ₀⟩ instead of |ψ(t)⟩.

If we want an ontology to go with this, we could say that the reality corresponding to the initial vector |ψ₀⟩ affects the evolution of the hidden variables for all subsequent times. We now have only one aspect of reality—the hidden variables of the corpuscles—evolving dynamically instead of two. We don’t have Schrödinger’s equation in the laws of nature except as a useful mathematical property of the U_t operator described by the initial vector). We can talk of the wavefunction U_t|ψ₀⟩ at a time t, but that’s just a mathematical artifact, just as m₁m₂ is a part of the equation expressing Newton’s law of gravitation rather than a direct representation of physical reality. Of course, just as m₁m₂ is determined by physical things—the two masses—so too the wavefunction U_t|ψ₀⟩ is determined by physical reality (the initial vector, the time, and the Hamiltonian). This seems to me to weaken the force of the Brown and Wallace point, since there no longer is a reality corresponding to the wavefunction at non-initial times, except highly derivatively.

Interestingly, the exact same move can be made for a number of other no-collapse interpretations, such as Bell’s indeterministic variant of Bohm, other modal interpretations, the many-minds interpretation, the traveling minds interpretation and the Aristotelian traveling forms interpretation. There need be no time-evolving wavefunction in reality, but just an initial vector which transtemporally affects the evolution of the other aspects of reality (such as where the minds go).

Or one could suppose a static background vector.

It’s interesting to ask what happens if one plugs this into the Everett interpretation. There I think we get something rather implausible: for then all time-evolution will disappear, since all reality will be reduced to the physical correlate of the initial vector. So my move above is only plausible for those no-collapse interpretations on which there is something more beyond the wavefunction.

There is also a connection between this approach and the Heisenberg picture. How close the connection is is not yet clear to me.

Multi-Bohm

I am exploring what seem to me to be under-explored parts of the logical space of interpretations of Quantum Mechanics. I may be wasting my time: there may be good reasons why those parts of logical space are not explored much. But I am also hoping that such exploration will broaden my mind.

So, here’s a curious interpretation: multi-Bohm. Assume no collapse as in Everett. At any given time t, there is the set S_t of all particle position assignments compatible with the value of the wavefunction ψ(t) (we can extend to spin and other things in the same way that Bohm gets extended to spin and other things). Typically, this set will include every possible position assignment, and will have continuum cardinality.

Now on Bohm’s interpretation, one member of S_t is privileged: it is the actual positions of the particles. But drop that privileging. Suppose instead that all the assignments of S_t are on par. Then S_t gives us a synchronic decomposition of the Everettian multiverse into "branches". Now stitch the synchronic decomposition into trajectories using the guiding equation: a position assignment s_t ∈ S_t is part of the same trajectory as a position assignment s_t′ ∈ S_t′ if and only if the guiding equation evolves s_t into s_t′ over the time span from t to t′ given the actual wavefunction ψ.

We can think of the above as a story with infinitely many (continuum many) parallel Bohmian universes. But that bloats the ontology by including infinitely many ensembles of particles. Since the wavefunction fully determines the sets S_t of position assignments (or so I assume—there are some worries about null-measure stuff that I am not perfectly sure of), we can stop thinking about real particles and just as a way of speaking superimposed on top of the many-worlds interpretation.

This means that we can interpret the many-worlds interpretation not as a branching-worlds story, but as a deterministic parallel worlds reading. For given the two-way (I assume) determinism in the guiding equation, the trajectories never meet: the branches always stay separate and parallel. Moreover, the probability problem of the many-worlds interpretation is unsolved, and so we cannot say that the story fits better with one set of experimental results rather than another.

This isn’t very attractive…