The setup for the traditional (non-quantum) problem is the following. Some experimental philosophers enlist the help of a subject, Sleeping Beauty. She will be put to sleep, and a coin is flipped. If it comes up heads, Beauty will be awoken on Monday and interviewed; then she will (voluntarily) have all her memories of being awakened wiped out, and be put to sleep again. Then she will be awakened again on Tuesday, and interviewed once again. If the coin came up tails, on the other hand, Beauty will only be awakened on Monday. Beauty herself is fully aware ahead of time of what the experimental protocol will be.

So in one possible world (heads) Beauty is awakened twice, in identical circumstances; in the other possible world (tails) she is only awakened once. Each time she is asked a question: “What is the probability you would assign that the coin came up tails?”

(Some other discussions switch the roles of heads and tails from my example.)

The Sleeping Beauty puzzle is still quite controversial. There are two answers one could imagine reasonably defending.

- “Halfer” — Before going to sleep, Beauty would have said that the probability of the coin coming up heads or tails would be one-half each. Beauty learns nothing upon waking up. She should assign a probability one-half to it having been tails.
- “Thirder” — If Beauty were told upon waking that the coin had come up heads, she would assign equal credence to it being Monday or Tuesday. But if she were told it was Monday, she would assign equal credence to the coin being heads or tails. The only consistent apportionment of credences is to assign 1/3 to each possibility, treating each possible waking-up event on an equal footing.

The Sleeping Beauty puzzle has generated considerable interest. It’s exactly the kind of wacky thought experiment that philosophers just eat up. But it has also attracted attention from cosmologists of late, because of the measure problem in cosmology. In a multiverse, there are many classical spacetimes (analogous to the coin toss) and many observers in each spacetime (analogous to being awakened on multiple occasions). Really the SB puzzle is a test-bed for cases of “mixed” uncertainties from different sources.

Chip and I argue that if we adopt Everettian quantum mechanics (EQM) and our Epistemic Separability Principle (ESP), everything becomes crystal clear. A rare case where the quantum-mechanical version of a problem is actually easier than the classical version.

In the quantum version, we naturally replace the coin toss by the observation of a spin. If the spin is initially oriented along the *x*-axis, we have a 50/50 chance of observing it to be up or down along the *z*-axis. In EQM that’s because we split into two different branches of the wave function, with equal amplitudes.

Our derivation of the Born Rule is actually based on the idea of self-locating uncertainty, so adding a bit more to it is no problem at all. We show that, if you accept the ESP, you are immediately led to the “thirder” position, as originally advocated by Elga. Roughly speaking, in the quantum wave function Beauty is awakened three times, and all of them are on a completely equal footing, and should be assigned equal credences. The same logic that says that probabilities are proportional to the amplitudes squared also says you should be a thirder.

But! We can put a minor twist on the experiment. What if, instead of waking up Beauty twice when the spin is up, we instead observe another spin. If that second spin is also up, she is awakened on Monday, while if it is down, she is awakened on Tuesday. Again we ask what probability she would assign that the first spin was down.

This new version has three branches of the wave function instead of two, as illustrated in the figure. And now the three branches don’t have equal amplitudes; the bottom one is (1/√2), while the top two are each (1/√2)^{2} = 1/2. In this case the ESP simply recovers the Born Rule: the bottom branch has probability 1/2, while each of the top two have probability 1/4. And Beauty wakes up precisely once on each branch, so she should assign probability 1/2 to the initial spin being down. This gives some justification for the “halfer” position, at least in this slightly modified setup.

All very cute, but it does have direct implications for the measure problem in cosmology. Consider a multiverse with many branches of the cosmological wave function, and potentially many identical observers on each branch. Given that you are one of those observers, how do you assign probabilities to the different alternatives?

Simple. Each observer *O _{i}* appears on a branch with amplitude

It looks easy, but note that the formula is not trivial: the weights *w _{i}* will not in general add up to one, since they might describe multiple observers on a single branch and perhaps even at different times. This analysis, we claim, defuses the “Born Rule crisis” pointed out by Don Page in the context of these cosmological spacetimes.

Sleeping Beauty, in other words, might turn out to be very useful in helping us understand the origin of the universe. Then again, plenty of people already think that the multiverse is just a fairy tale, so perhaps we shouldn’t be handing them ammunition.

]]>**Born Rule:**

The Born Rule is certainly correct, as far as all of our experimental efforts have been able to discern. But why? Born himself kind of stumbled onto his Rule. Here is an excerpt from his 1926 paper:

That’s right. Born’s paper was rejected at first, and when it was later accepted by another journal, he didn’t even get the Born Rule right. At first he said the probability was equal to the amplitude, and only in an added footnote did he correct it to being the amplitude squared. And a good thing, too, since amplitudes can be negative or even imaginary!

The status of the Born Rule depends greatly on one’s preferred formulation of quantum mechanics. When we teach quantum mechanics to undergraduate physics majors, we generally give them a list of postulates that goes something like this:

- Quantum states are represented by wave functions, which are vectors in a mathematical space called Hilbert space.
- Wave functions evolve in time according to the Schrödinger equation.
- The act of measuring a quantum system returns a number, known as the eigenvalue of the quantity being measured.
- The probability of getting any particular eigenvalue is equal to the square of the amplitude for that eigenvalue.
- After the measurement is performed, the wave function “collapses” to a new state in which the wave function is localized precisely on the observed eigenvalue (as opposed to being in a superposition of many different possibilities).

It’s an ungainly mess, we all agree. You see that the Born Rule is simply postulated right there, as #4. Perhaps we can do better.

Of course we can do better, since “textbook quantum mechanics” is an embarrassment. There are other formulations, and you know that my own favorite is Everettian (“Many-Worlds”) quantum mechanics. (I’m sorry I was too busy to contribute to the active comment thread on that post. On the other hand, a vanishingly small percentage of the 200+ comments actually addressed the point of the article, which was that the potential for many worlds is automatically there in the wave function no matter what formulation you favor. Everett simply takes them seriously, while alternatives need to go to extra efforts to erase them. As Ted Bunn argues, Everett is just “quantum mechanics,” while collapse formulations should be called “disappearing-worlds interpretations.”)

Like the textbook formulation, Everettian quantum mechanics also comes with a list of postulates. Here it is:

- Quantum states are represented by wave functions, which are vectors in a mathematical space called Hilbert space.
- Wave functions evolve in time according to the Schrödinger equation.

That’s it! Quite a bit simpler — and the two postulates are exactly the same as the first two of the textbook approach. Everett, in other words, is claiming that all the weird stuff about “measurement” and “wave function collapse” in the conventional way of thinking about quantum mechanics isn’t something we need to add on; it comes out automatically from the formalism.

The trickiest thing to extract from the formalism is the Born Rule. That’s what Charles (“Chip”) Sebens and I tackled in our recent paper:

Self-Locating Uncertainty and the Origin of Probability in Everettian Quantum Mechanics

Charles T. Sebens, Sean M. CarrollA longstanding issue in attempts to understand the Everett (Many-Worlds) approach to quantum mechanics is the origin of the Born rule: why is the probability given by the square of the amplitude? Following Vaidman, we note that observers are in a position of self-locating uncertainty during the period between the branches of the wave function splitting via decoherence and the observer registering the outcome of the measurement. In this period it is tempting to regard each branch as equiprobable, but we give new reasons why that would be inadvisable. Applying lessons from this analysis, we demonstrate (using arguments similar to those in Zurek’s envariance-based derivation) that the Born rule is the uniquely rational way of apportioning credence in Everettian quantum mechanics. In particular, we rely on a single key principle: changes purely to the environment do not affect the probabilities one ought to assign to measurement outcomes in a local subsystem. We arrive at a method for assigning probabilities in cases that involve both classical and quantum self-locating uncertainty. This method provides unique answers to quantum Sleeping Beauty problems, as well as a well-defined procedure for calculating probabilities in quantum cosmological multiverses with multiple similar observers.

Chip is a graduate student in the philosophy department at Michigan, which is great because this work lies squarely at the boundary of physics and philosophy. (I guess it is possible.) The paper itself leans more toward the philosophical side of things; if you are a physicist who just wants the equations, we have a shorter conference proceeding.

Before explaining what we did, let me first say a bit about why there’s a puzzle at all. Let’s think about the wave function for a spin, a spin-measuring apparatus, and an environment (the rest of the world). It might initially take the form

(α[up] + β[down] ; apparatus says “ready” ; environment

_{0}). (1)

This might look a little cryptic if you’re not used to it, but it’s not too hard to grasp the gist. The first slot refers to the spin. It is in a superposition of “up” and “down.” The Greek letters α and β are the amplitudes that specify the wave function for those two possibilities. The second slot refers to the apparatus just sitting there in its ready state, and the third slot likewise refers to the environment. By the Born Rule, when we make a measurement the probability of seeing spin-up is |α|^{2}, while the probability for seeing spin-down is |β|^{2}.

In Everettian quantum mechanics (EQM), wave functions never collapse. The one we’ve written will smoothly evolve into something that looks like this:

α([up] ; apparatus says “up” ; environment

_{1})

+ β([down] ; apparatus says “down” ; environment_{2}). (2)

This is an extremely simplified situation, of course, but it is meant to convey the basic appearance of two separate “worlds.” The wave function has split into branches that don’t ever talk to each other, because the two environment states are different and will stay that way. A state like this simply arises from normal Schrödinger evolution from the state we started with.

So here is the problem. After the splitting from (1) to (2), the wave function coefficients α and β just kind of go along for the ride. If you find yourself in the branch where the spin is up, your coefficient is α, but so what? How do you know what kind of coefficient is sitting outside the branch you are living on? All you know is that there was one branch and now there are two. If anything, shouldn’t we declare them to be equally likely (so-called “branch-counting”)? For that matter, in what sense are there probabilities *at all*? There was nothing stochastic or random about any of this process, the entire evolution was perfectly deterministic. It’s not right to say “Before the measurement, I didn’t know which branch I was going to end up on.” You know precisely that one copy of your future self will appear on *each* branch. Why in the world should we be talking about probabilities?

Note that the pressing question is not so much “Why is the probability given by the wave function squared, rather than the absolute value of the wave function, or the wave function to the fourth, or whatever?” as it is “Why is there a particular probability rule at all, since the theory is deterministic?” Indeed, once you accept that there should be some specific probability rule, it’s practically guaranteed to be the Born Rule. There is a result called Gleason’s Theorem, which says roughly that the Born Rule is the only consistent probability rule you can conceivably have that depends on the wave function alone. So the real question is not “Why squared?”, it’s “Whence probability?”

Of course, there are promising answers. Perhaps the most well-known is the approach developed by Deutsch and Wallace based on decision theory. There, the approach to probability is essentially operational: given the setup of Everettian quantum mechanics, how should a rational person behave, in terms of making bets and predicting experimental outcomes, etc.? They show that there is one unique answer, which is given by the Born Rule. In other words, the question “Whence probability?” is sidestepped by arguing that reasonable people in an Everettian universe will act *as if* there are probabilities that obey the Born Rule. Which may be good enough.

But it might not convince everyone, so there are alternatives. One of my favorites is Wojciech Zurek’s approach based on “envariance.” Rather than using words like “decision theory” and “rationality” that make physicists nervous, Zurek claims that the underlying symmetries of quantum mechanics pick out the Born Rule uniquely. It’s very pretty, and I encourage anyone who knows a little QM to have a look at Zurek’s paper. But it is subject to the criticism that it doesn’t really teach us anything that we didn’t already know from Gleason’s theorem. That is, Zurek gives us more reason to think that the Born Rule is uniquely preferred by quantum mechanics, but it doesn’t really help with the deeper question of why we should think of EQM as a theory of probabilities at all.

Here is where Chip and I try to contribute something. We use the idea of “self-locating uncertainty,” which has been much discussed in the philosophical literature, and has been applied to quantum mechanics by Lev Vaidman. Self-locating uncertainty occurs when you know that there multiple observers in the universe who find themselves in exactly the same conditions that you are in right now — but you don’t know which one of these observers you are. That can happen in “big universe” cosmology, where it leads to the measure problem. But it automatically happens in EQM, whether you like it or not.

Think of observing the spin of a particle, as in our example above. The steps are:

- Everything is in its starting state, before the measurement.
- The apparatus interacts with the system to be observed and becomes entangled. (“Pre-measurement.”)
- The apparatus becomes entangled with the environment, branching the wave function. (“Decoherence.”)
- The observer reads off the result of the measurement from the apparatus.

The point is that in between steps 3. and 4., the wave function of the universe has branched into two, but *the observer doesn’t yet know which branch they are on*. There are two copies of the observer that are in identical states, even though they’re part of different “worlds.” That’s the moment of self-locating uncertainty. Here it is in equations, although I don’t think it’s much help.

You might say “What if I am the apparatus myself?” That is, what if I observe the outcome directly, without any intermediating macroscopic equipment? Nice try, but no dice. That’s because decoherence happens incredibly quickly. Even if you take the extreme case where you look at the spin directly with your eyeball, the time it takes the state of your eye to decohere is about 10^{-21} seconds, whereas the timescales associated with the signal reaching your brain are measured in tens of milliseconds. Self-locating uncertainty is inevitable in Everettian quantum mechanics. In that sense, *probability* is inevitable, even though the theory is deterministic — in the phase of uncertainty, we need to assign probabilities to finding ourselves on different branches.

So what do we do about it? As I mentioned, there’s been a lot of work on how to deal with self-locating uncertainty, i.e. how to apportion credences (degrees of belief) to different possible locations for yourself in a big universe. One influential paper is by Adam Elga, and comes with the charming title of “Defeating Dr. Evil With Self-Locating Belief.” (Philosophers have more fun with their titles than physicists do.) Elga argues for a principle of *Indifference*: if there are truly multiple copies of you in the world, you should assume equal likelihood for being any one of them. Crucially, Elga doesn’t simply assert *Indifference*; he actually derives it, under a simple set of assumptions that would seem to be the kind of minimal principles of reasoning any rational person should be ready to use.

But there is a problem! Naïvely, applying *Indifference* to quantum mechanics just leads to branch-counting — if you assign equal probability to every possible appearance of equivalent observers, and there are two branches, each branch should get equal probability. But that’s a disaster; it says we should simply ignore the amplitudes entirely, rather than using the Born Rule. This bit of tension has led to some worry among philosophers who worry about such things.

Resolving this tension is perhaps the most useful thing Chip and I do in our paper. Rather than naïvely applying *Indifference* to quantum mechanics, we go back to the “simple assumptions” and try to derive it from scratch. We were able to pinpoint one hidden assumption that seems quite innocent, but actually does all the heavy lifting when it comes to quantum mechanics. We call it the “Epistemic Separability Principle,” or *ESP* for short. Here is the informal version (see paper for pedantic careful formulations):

ESP: The credence one should assign to being any one of several observers having identical experiences is independent of features of the environment that aren’t affecting the observers.

That is, the probabilities you assign to things happening in your lab, whatever they may be, should be exactly the same if we tweak the universe just a bit by moving around some rocks on a planet orbiting a star in the Andromeda galaxy. *ESP* simply asserts that our knowledge is separable: how we talk about what happens here is independent of what is happening far away. (Our system here can still be *entangled* with some system far away; under unitary evolution, changing that far-away system doesn’t change the entanglement.)

The *ESP* is quite a mild assumption, and to me it seems like a necessary part of being able to think of the universe as consisting of separate pieces. If you can’t assign credences locally without knowing about the state of the whole universe, there’s no real sense in which the rest of the world is really separate from you. It is certainly implicitly used by Elga (he assumes that credences are unchanged by some hidden person tossing a coin).

With this assumption in hand, we are able to demonstrate that *Indifference* does not apply to branching quantum worlds in a straightforward way. Indeed, we show that you should assign equal credences to two different branches *if and only if* the amplitudes for each branch are precisely equal! That’s because the proof of *Indifference* relies on shifting around different parts of the state of the universe and demanding that the answers to local questions not be altered; it turns out that this only works in quantum mechanics if the amplitudes are equal, which is certainly consistent with the Born Rule.

See the papers for the actual argument — it’s straightforward but a little tedious. The basic idea is that you set up a situation in which more than one quantum object is measured at the same time, and you ask what happens when you consider different objects to be “the system you will look at” versus “part of the environment.” If you want there to be a consistent way of assigning credences in all cases, you are led inevitably to equal probabilities when (and only when) the amplitudes are equal.

What if the amplitudes for the two branches are not equal? Here we can borrow some math from Zurek. (Indeed, our argument can be thought of as a love child of Vaidman and Zurek, with Elga as midwife.) In his envariance paper, Zurek shows how to start with a case of unequal amplitudes and reduce it to the case of many more branches with equal amplitudes. The number of these pseudo-branches you need is proportional to — wait for it — the square of the amplitude. Thus, you get out the full Born Rule, simply by demanding that we assign credences in situations of self-locating uncertainty in a way that is consistent with *ESP*.

We like this derivation in part because it treats probabilities as epistemic (statements about our knowledge of the world), not merely operational. Quantum probabilities are really credences — statements about the best degree of belief we can assign in conditions of uncertainty — rather than statements about truly stochastic dynamics or frequencies in the limit of an infinite number of outcomes. But these degrees of belief aren’t completely subjective in the conventional sense, either; there is a uniquely rational choice for how to assign them.

Working on this project has increased my own personal credence in the correctness of the Everett approach to quantum mechanics from “pretty high” to “extremely high indeed.” There are still puzzles to be worked out, no doubt, especially around the issues of exactly how and when branching happens, and how branching structures are best defined. (I’m off to a workshop next month to think about precisely these questions.) But these seem like relatively tractable technical challenges to me, rather than looming deal-breakers. EQM is an incredibly simple theory that (I can now argue in good faith) makes sense and fits the data. Now it’s just a matter of convincing the rest of the world!

]]>Fortunately, astronomers are pushing forward to study how dark matter behaves as it’s scattered through the universe, and the results are interesting. We start with a very basic idea: that dark matter is cold and completely non-interacting, or at least has interactions (the strength with which dark matter particles scatter off of each other) that are too small to make any noticeable difference. This is a well-defined and predictive model: ΛCDM, which includes the cosmological constant (Λ) as well as the cold dark matter (CDM). We can compare astronomical observations to ΛCDM predictions to see if we’re on the right track.

At first blush, we are very much on the right track. Over and over again, new observations come in that match the predictions of ΛCDM. But there are still a few anomalies that bug us, especially on relatively small (galaxy-sized) scales.

One such anomaly is the “too big to fail” problem. The idea here is that we can use ΛCDM to make quantitative predictions concerning how many galaxies there should be with different masses. For example, the Milky Way is quite a big galaxy, and it has smaller satellites like the Magellanic Clouds. In ΛCDM we can predict how many such satellites there should be, and how massive they should be. For a long time we’ve known that the actual number of satellites we observe is quite a bit smaller than the number predicted — that’s the “missing satellites” problem. But this has a possible solution: we only observe satellite galaxies by seeing stars and gas in them, and maybe the halos of dark matter that would ordinarily support such galaxies get stripped of their stars and gas by interacting with the host galaxy. The too big to fail problem tries to sharpen the issue, by pointing out that some of the predicted galaxies are just so massive that there’s no way they could not have visible stars. Or, put another way: the Milky Way does have *some* satellites, as do other galaxies; but when we examine these smaller galaxies, they seem to have a lot less dark matter than the simulations would predict.

Still, any time you are concentrating on galaxies that are satellites of other galaxies, you rightly worry that complicated interactions between messy atoms and photons are getting in the way of the pristine elegance of the non-interacting dark matter. So we’d like to check that this purported problem exists even out “in the field,” with lonely galaxies far away from big monsters like the Milky Way.

A new paper claims that yes, there is a too-big-to-fail problem even for galaxies in the field.

Is there a “too big to fail” problem in the field?

Emmanouil Papastergis, Riccardo Giovanelli, Martha P. Haynes, Francesco ShankarWe use the Arecibo Legacy Fast ALFA (ALFALFA) 21cm survey to measure the number density of galaxies as a function of their rotational velocity, Vrot,HI (as inferred from the width of their 21cm emission line). Based on the measured velocity function we statistically connect galaxies with their host halos, via abundance matching. In a LCDM cosmology, low-velocity galaxies are expected to be hosted by halos that are significantly more massive than indicated by the measured galactic velocity; allowing lower mass halos to host ALFALFA galaxies would result in a vast overestimate of their number counts. We then seek observational verification of this predicted trend, by analyzing the kinematics of a literature sample of field dwarf galaxies. We find that galaxies with Vrot,HI<25 km/s are kinematically incompatible with their predicted LCDM host halos, in the sense that hosts are too massive to be accommodated within the measured galactic rotation curves. This issue is analogous to the "too big to fail" problem faced by the bright satellites of the Milky Way, but here it concerns extreme dwarf galaxies in the field. Consequently, solutions based on satellite-specific processes are not applicable in this context. Our result confirms the findings of previous studies based on optical survey data, and addresses a number of observational systematics present in these works. Furthermore, we point out the assumptions and uncertainties that could strongly affect our conclusions. We show that the two most important among them, namely baryonic effects on the abundances and rotation curves of halos, do not seem capable of resolving the reported discrepancy.

Here is the money plot from the paper:

The horizontal axis is the maximum circular velocity, basically telling us the mass of the halo; the vertical axis is the observed velocity of hydrogen in the galaxy. The blue line is the prediction from ΛCDM, while the dots are observed galaxies. Now, you might think that the blue line is just a very crappy fit to the data overall. But that’s okay; the points represent upper limits in the horizontal direction, so points that lie below/to the right of the curve are fine. It’s a statistical prediction: ΛCDM is predicting how many galaxies we have at each mass, even if we don’t think we can confidently measure the mass of each individual galaxy. What we see, however, is that there are a bunch of points in the bottom left corner that are above the line. ΛCDM predicts that even the smallest galaxies in this sample should still be relatively massive (have a lot of dark matter), but that’s not what we see.

If it holds up, this result is really intriguing. ΛCDM is a nice, simple starting point for a theory of dark matter, but it’s also kind of boring. From a physicist’s point of view, it would be much more fun if dark matter particles interacted noticeably with each other. We have plenty of ideas, including some of my favorites like dark photons and dark atoms. It is very tempting to think that observed deviations from the predictions of ΛCDM are due to some interesting new physics in the dark sector.

Which is why, of course, we should be especially skeptical. Always train your doubt most strongly on those ideas that you really want to be true. Fortunately there is plenty more to be done in terms of understanding the distribution of galaxies and dark matter, so this is a very solvable problem — and a great opportunity for learning something profound about most of the matter in the universe.

]]>The movie has just been released on iTunes, so now almost everyone can watch it from the comfort of their own computer. And watch it you should — it’s a fascinating and enlightening glimpse into the world of modern particle physics, focusing on the Large Hadron Collider and the discovery of the Higgs boson. That’s not just my bias talking, either — the film is rated 95% “fresh” on RottenTomatoes.com, which represents an amazingly strong critical consensus. (Full disclosure: I’m not in it, and I had nothing to do with making it.)

Huge kudos to Mark (who went to grad school in physics before becoming a filmmaker) and David (who did a brief stint as an undergraduate film major before switching to physics) for pulling this off. It’s great for the public appreciation of science, but it’s also just an extremely enjoyable movie, no matter what your background is. Watch it with a friend!

]]>Any discussion of Everettian quantum mechanics (“EQM”) comes with the baggage of pre-conceived notions. People have heard of it before, and have instinctive reactions to it, in a way that they don’t have to (for example) effective field theory. Hell, there is even an app, universe splitter, that lets you create new universes from your iPhone. (Seriously.) So we need to start by separating the silly objections to EQM from the serious worries.

The basic silly objection is that EQM postulates too many universes. In quantum mechanics, we can’t deterministically predict the outcomes of measurements. In EQM, that is dealt with by saying that *every measurement outcome “happens,”* but each in a different “universe” or “world.” Say we think of Schrödinger’s Cat: a sealed box inside of which we have a cat in a quantum superposition of “awake” and “asleep.” (No reason to kill the cat unnecessarily.) Textbook quantum mechanics says that opening the box and observing the cat “collapses the wave function” into one of two possible measurement outcomes, awake or asleep. Everett, by contrast, says that the universe splits in two: in one the cat is awake, and in the other the cat is asleep. Once split, the universes go their own ways, never to interact with each other again.

And to many people, that just seems like too much. Why, this objection goes, would you ever think of inventing a huge — perhaps infinite! — number of different universes, just to describe the simple act of quantum measurement? It might be puzzling, but it’s no reason to lose all anchor to reality.

To see why objections along these lines are wrong-headed, let’s first think about classical mechanics rather than quantum mechanics. And let’s start with one universe: some collection of particles and fields and what have you, in some particular arrangement in space. Classical mechanics describes such a universe as a point in phase space — the collection of all positions and velocities of each particle or field.

What if, for some perverse reason, we wanted to describe two copies of such a universe (perhaps with some tiny difference between them, like an awake cat rather than a sleeping one)? We would have to double the size of phase space — create a mathematical structure that is large enough to describe both universes at once. In classical mechanics, then, it’s quite a bit of work to accommodate extra universes, and you better have a good reason to justify putting in that work. (Inflationary cosmology seems to do it, by implicitly assuming that phase space is already infinitely big.)

That is *not what happens in quantum mechanics*. The capacity for describing multiple universes is *automatically there*. We don’t have to add anything.

The reason why we can state this with such confidence is because of the fundamental reality of quantum mechanics: the existence of superpositions of different possible measurement outcomes. In classical mechanics, we have certain definite possible states, all of which are directly observable. It will be important for what comes later that the system we consider is microscopic, so let’s consider a spinning particle that can have spin-up or spin-down. (It is directly analogous to Schrödinger’s cat: cat=particle, awake=spin-up, asleep=spin-down.) Classically, the possible states are

“spin is up”

or

“spin is down”.

Quantum mechanics says that the state of the particle can be a superposition of both possible measurement outcomes. It’s not that we don’t know whether the spin is up or down; it’s that it’s really in a superposition of both possibilities, at least until we observe it. We can denote such a state like this:

(“spin is up” + “spin is down”).

While classical states are points in phase space, quantum states are “wave functions” that live in something called Hilbert space. Hilbert space is very big — as we will see, it has room for lots of stuff.

To describe measurements, we need to add an observer. It doesn’t need to be a “conscious” observer or anything else that might get Deepak Chopra excited; we just mean a macroscopic measuring apparatus. It could be a living person, but it could just as well be a video camera or even the air in a room. To avoid confusion we’ll just call it the “apparatus.”

In any formulation of quantum mechanics, the apparatus starts in a “ready” state, which is a way of saying “it hasn’t yet looked at the thing it’s going to observe” (*i.e.*, the particle). More specifically, the apparatus is not entangled with the particle; their two states are independent of each other. So the quantum state of the particle+apparatus system starts out like this:

(“spin is up” + “spin is down” ; apparatus says “ready”) (1)

The particle is in a superposition, but the apparatus is not. According to the textbook view, when the apparatus observes the particle, the quantum state collapses onto one of two possibilities:

(“spin is up”; apparatus says “up”)

or

(“spin is down”; apparatus says “down”).

When and how such collapse actually occurs is a bit vague — a huge problem with the textbook approach — but let’s not dig into that right now.

But there is clearly another possibility. If the particle can be in a superposition of two states, then so can the apparatus. So nothing stops us from writing down a state of the form

(spin is up ; apparatus says “up”)

+ (spin is down ; apparatus says “down”). (2)

The plus sign here is crucial. This is not a state representing one alternative or the other, as in the textbook view; it’s a superposition of both possibilities. In this kind of state, the spin of the particle is entangled with the readout of the apparatus.

What would it be like to live in a world with the kind of quantum state we have written in (2)? It might seem a bit unrealistic at first glance; after all, when we observe real-world quantum systems it always *feels like* we see one outcome or the other. We never think that we ourselves are in a superposition of having achieved different measurement outcomes.

This is where the magic of decoherence comes in. (Everett himself actually had a clever argument that didn’t use decoherence explicitly, but we’ll take a more modern view.) I won’t go into the details here, but the basic idea isn’t too difficult. There are more things in the universe than our particle and the measuring apparatus; there is the rest of the Earth, and for that matter everything in outer space. That stuff — group it all together and call it the “environment” — has a quantum state also. We expect the apparatus to quickly become entangled with the environment, if only because photons and air molecules in the environment will keep bumping into the apparatus. As a result, even though a state of this form is in a superposition, the two different pieces (one with the particle spin-up, one with the particle spin-down) will never be able to interfere with each other. Interference (different parts of the wave function canceling each other out) demands a precise alignment of the quantum states, and once we lose information into the environment that becomes impossible. That’s decoherence.

Once our quantum superposition involves macroscopic systems with many degrees of freedom that become entangled with an even-larger environment, the different terms in that superposition proceed to evolve completely independently of each other. *It is as if they have become distinct worlds* — because they have. We wouldn’t think of our pre-measurement state (1) as describing two different worlds; it’s just one world, in which the particle is in a superposition. But (2) has two worlds in it. The difference is that we can imagine undoing the superposition in (1) by carefully manipulating the particle, but in (2) the difference between the two branches has diffused into the environment and is lost there forever.

All of this exposition is building up to the following point: in order to describe a quantum state that includes two non-interacting “worlds” as in (2), *we didn’t have to add anything at all* to our description of the universe, unlike the classical case. All of the ingredients were already there!

Our only assumption was that the apparatus obeys the rules of quantum mechanics just as much as the particle does, which seems to be an extremely mild assumption if we think quantum mechanics is the correct theory of reality. Given that, we know that the particle can be in “spin-up” or “spin-down” states, and we also know that the apparatus can be in “ready” or “measured spin-up” or “measured spin-down” states. And if that’s true, the quantum state has the built-in ability to describe superpositions of non-interacting worlds. Not only did we not need to add anything to make it possible, we had no choice in the matter. **The potential for multiple worlds is always there in the quantum state, whether you like it or not.**

The next question would be, do multiple-world superpositions of the form written in (2) ever actually come into being? And the answer again is: yes, automatically, without any additional assumptions. It’s just the ordinary evolution of a quantum system according to Schrödinger’s equation. Indeed, the fact that a state that looks like (1) evolves into a state that looks like (2) under Schrödinger’s equation is what we mean when we say “this apparatus measures whether the spin is up or down.”

The conclusion, therefore, is that multiple worlds automatically occur in quantum mechanics. They are an inevitable part of the formalism. The only remaining question is: what are you going to do about it? There are three popular strategies on the market: anger, denial, and acceptance.

The “anger” strategy says “I hate the idea of multiple worlds with such a white-hot passion that I will *change the rules of quantum mechanics* in order to avoid them.” And people do this! In the four options listed here, both dynamical-collapse theories and hidden-variable theories are straightforward alterations of the conventional picture of quantum mechanics. In dynamical collapse, we change the evolution equation, by adding some explicitly stochastic probability of collapse. In hidden variables, we keep the Schrödinger equation intact, but add new variables — hidden ones, which we know must be explicitly non-local. Of course there is currently zero empirical evidence for these rather *ad hoc* modifications of the formalism, but hey, you never know.

The “denial” strategy says “The idea of multiple worlds is so profoundly upsetting to me that I will *deny the existence of reality* in order to escape having to think about it.” Advocates of this approach don’t actually put it that way, but I’m being polemical rather than conciliatory in this particular post. And I don’t think it’s an unfair characterization. This is the quantum Bayesianism approach, or more generally “psi-epistemic” approaches. The idea is to simply deny that the quantum state represents anything about reality; it is merely a way of keeping track of the probability of future measurement outcomes. Is the particle spin-up, or spin-down, or both? Neither! There is no particle, there is no spoon, nor is there the state of the particle’s spin; there is only the probability of seeing the spin in different conditions once one performs a measurement. I advocate listening to David Albert’s take at our WSF panel.

The final strategy is acceptance. That is the Everettian approach. The formalism of quantum mechanics, in this view, consists of quantum states as described above and nothing more, which evolve according to the usual Schrödinger equation and nothing more. The formalism predicts that there are many worlds, so we choose to accept that. This means that the part of reality we experience is an indescribably thin slice of the entire picture, but so be it. Our job as scientists is to formulate the best possible description of the world as it is, not to force the world to bend to our pre-conceptions.

Such brave declarations aren’t enough on their own, of course. The fierce austerity of EQM is attractive, but we still need to verify that its predictions map on to our empirical data. This raises questions that live squarely at the physics/philosophy boundary. Why does the quantum state branch into certain kinds of worlds (*e.g.*, ones where cats are awake or ones where cats are asleep) and not others (where cats are in superpositions of both)? Why are the probabilities that we actually observe given by the Born Rule, which states that the probability equals the wave function squared? In what sense are there probabilities *at all*, if the theory is completely deterministic?

These are the *serious* issues for EQM, as opposed to the silly one that “there are just too many universes!” The “why those states?” problem has essentially been solved by the notion of pointer states — quantum states split along lines that are macroscopically robust, which are ultimately delineated by the actual laws of physics (the particles/fields/interactions of the real world). The probability question is trickier, but also (I think) solvable. Decision theory is one attractive approach, and Chip Sebens and I are advocating self-locating uncertainty as a friendly alternative. That’s the subject of a paper we just wrote, which I plan to talk about in a separate post.

There are other silly objections to EQM, of course. The most popular is probably the complaint that it’s not falsifiable. That truly makes no sense. It’s trivial to falsify EQM — just do an experiment that violates the Schrödinger equation or the principle of superposition, which are the only things the theory assumes. Witness a dynamical collapse, or find a hidden variable. Of course we don’t see the other worlds directly, but — in case we haven’t yet driven home the point loudly enough — those other worlds are not added on to the theory. They come out automatically if you believe in quantum mechanics. If you have a physically distinguishable alternative, by all means suggest it — the experimenters would love to hear about it. (And true alternatives, like GRW and Bohmian mechanics, are indeed experimentally distinguishable.)

Sadly, most people who object to EQM do so for the silly reasons, not for the serious ones. But even given the real challenges of the preferred-basis issue and the probability issue, I think EQM is way ahead of any proposed alternative. It takes at face value the minimal conceptual apparatus necessary to account for the world we see, and by doing so it fits all the data we have ever collected. What more do you want from a theory than that?

]]>These days, all you have to do is fire up the YouTube and watch lectures on your own time. MIT has just released an entire undergraduate quantum course, lovingly titled “8.04″ because that’s how MIT rolls. The prof is Allan Adams, who is generally a fantastic lecturer — so I’m suspecting these are really good even though I haven’t actually watched them all myself. Here’s the first lecture, “Introduction to Superposition.”

Allan’s approach in this video is actually based on the first two chapters of *Quantum Mechanics and Experience* by philosopher David Albert. I’m sure this will be very disconcerting to the philosophy-skeptics haunting the comment section of the previous post.

This is just one of many great physics courses online; I’ve previously noted Lenny Susskind’s GR course. But, being largely beyond my course-taking days myself, I haven’t really kept track. Feel free to suggest your favorites in the comments.

]]>Roughly speaking, physicists tend to have three different kinds of lazy critiques of philosophy: one that is totally dopey, one that is frustratingly annoying, and one that is deeply depressing.

**“Philosophy tries to understand the universe by pure thought, without collecting experimental data.”**

This is the totally dopey criticism. Yes, most philosophers do not actually go out and collect data (although there are exceptions). But it makes no sense to jump right from there to the accusation that philosophy completely *ignores* the empirical information we have collected about the world. When science (or common-sense observation) reveals something interesting and important about the world, philosophers obviously take it into account. (Aside: of course there are *bad* philosophers, who do all sorts of stupid things, just as there are bad practitioners of every field. Let’s concentrate on the good ones, of whom there are plenty.)

Philosophers do, indeed, tend to think a lot. This is not a bad thing. All of scientific practice involves some degree of “pure thought.” Philosophers are, by their nature, more interested in foundational questions where the latest wrinkle in the data is of less importance than it would be to a model-building phenomenologist. But at its best, the practice of philosophy of physics is continuous with the practice of physics itself. Many of the best philosophers of physics were trained as physicists, and eventually realized that the problems they cared most about weren’t valued in physics departments, so they switched to philosophy. But those problems — the basic nature of the ultimate architecture of reality at its deepest levels — are just physics problems, really. And some amount of rigorous thought is necessary to make any progress on them. Shutting up and calculating isn’t good enough.

**“Philosophy is completely useless to the everyday job of a working physicist.”**

Now we have the frustratingly annoying critique. Because: duh. If your criterion for “being interesting or important” comes down to “is useful to me in my work,” you’re going to be leading a fairly intellectually impoverished existence. Nobody denies that the vast majority of physics gets by perfectly well without any input from philosophy at all. (“We need to calculate this loop integral! Quick, get me a philosopher!”) But it also gets by without input from biology, and history, and literature. Philosophy is interesting because of its intrinsic interest, not because it’s a handmaiden to physics. I think that philosophers themselves sometimes get too defensive about this, trying to come up with reasons why philosophy is useful to physics. Who cares?

Nevertheless, there are some physics questions where philosophical input actually is useful. Foundational questions, such as the quantum measurement problem, the arrow of time, the nature of probability, and so on. Again, a huge majority of working physicists don’t ever worry about these problems. But some of us do! And frankly, if more physicists who wrote in these areas would make the effort to talk to philosophers, they would save themselves from making a lot of simple mistakes.

**“Philosophers care too much about deep-sounding meta-questions, instead of sticking to what can be observed and calculated.”**

Finally, the deeply depressing critique. Here we see the unfortunate consequence of a lifetime spent in an academic/educational system that is focused on taking ambitious dreams and crushing them into easily-quantified units of productive work. The idea is apparently that *developing a new technique for calculating a certain wave function* is an honorable enterprise worthy of support, while *trying to understand what wave functions actually are and how they capture reality* is a boring waste of time. I suspect that a substantial majority of physicists who use quantum mechanics in their everyday work are uninterested in or downright hostile to attempts to understand the quantum measurement problem.

This makes me sad. I don’t know about all those other folks, but personally I did not fall in love with science as a kid because I was swept up in the romance of finding slightly more efficient calculational techniques. Don’t get me wrong — finding more efficient calculational techniques is crucially important, and I cheerfully do it myself when I think I might have something to contribute. But it’s not *the point* — it’s a step along the way to the point.

The point, I take it, is to understand how nature works. Part of that is knowing how to do calculations, but another part is asking deep questions about what it all means. That’s what got me interested in science, anyway. And part of that task is understanding the foundational aspects of our physical picture of the world, digging deeply into issues that go well beyond merely being able to calculate things. It’s a shame that so many physicists don’t see how good philosophy of science can contribute to this quest. The universe is much bigger than we are and stranger than we tend to imagine, and I for one welcome all the help we can get in trying to figure it out.

]]>In the meantime, here’s a video to keep you occupied: a recording of the World Science Festival panel on quantum mechanics I had previously mentioned.

David Albert is defending dynamical collapse formulations, Sheldon Goldstein stands up for hidden variables, I am promoting the many-worlds formulation, and Rüdiger Schack is in favor of QBism, a psi-epistemic approach. Brian Greene is the moderator, and has brought along some fancy animations. It’s an hour and a half of quantal goodness, so settle in for quite a ride.

Just as the panel was happening, my first official forays into quantum foundations were appearing on the arxiv: a paper with Charles Sebens on deriving the Born Rule in Everettian quantum mechanics, as well as a shorter conference proceeding.

- Self-Locating Uncertainty and the Origin of Probability in Everettian Quantum Mechanics
- Many Worlds, the Born Rule, and Self-Locating Uncertainty

No time to delve into the details here, but I promise to do so soon!

]]>Personally, I’m completely in favor of national curricula and standards. Indeed, I’d like to go much further, and nationalize the schools, so that public spending on students in rural Louisiana is just as high as that in wealthy suburbs in the Northeast. I’m also not dead set against swift action by small groups of people who are willing to get things done, rather than sit around for decades trading white papers and town hall meetings. (I even helped a bit with such non-democratic action myself, and suffered the attendant abuse with stoic calm.)

What I don’t know, since I simply am completely unfamiliar with the details, is whether the actual Common Core initiative (as opposed to the general idea of a common curriculum) is a good idea. I know that some people are very much against it — so much so that it’s difficult to find actual information about it, since emotions run very high, and you are more likely to find either rampant boosterism or strident criticism. Of course you can look up what the standards are, both in English Language Arts and in Mathematics (there don’t seem to be standards for science, history, or social studies). But what you read is so vague as to be pretty useless. For example, the winningly-named “CCSS.ELA-LITERACY.CCRA.W.1” standard reads

Write arguments to support claims in an analysis of substantive topics or texts using valid reasoning and relevant and sufficient evidence.

That sounds like a good idea! But doesn’t translate unambiguously into something teachable. The devil is in the implementation.

So — anyone have any informed ideas about how it works in practice, and whether it’s helpful and realistic? (Early results seem to be mildly promising.) I worry from skimming some of the information that there seems to be an enormous emphasis on “assessment,” which presumably translates into standardized testing. I recognize the value of such testing in the right context, but also have the feeling that it’s already way overdone (in part because of No Child Left Behind), and the Common Core just adds another layer of requirements. I’d rather have students and schools spend more time on teaching and less time on testing, all else being equal.

]]>I’m participating in several WSF events, and one of them tonight will be live-streamed in this very internet. The title is Measure for Measure: Quantum Physics and Reality, and we kick off at 8pm Eastern, 5pm Pacific. The live-stream is here, but I’ll also try to embed it and see how that goes:

The other participants are David Albert, Sheldon Goldstein, and Rüdiger Schack, with the conversation moderated by Brian Greene. The group is not merely a randomly-selected collection of people who know and love quantum mechanics; each participant was carefully chosen to defend a certain favorite version of this most mysterious of physical theories.

- David Albert will propound the idea of dynamical collapse theories, such as the Ghirardi-Rimini-Weber (GRW) model. They posit that QM is truly stochastic, with wave functions really “collapsing” at unpredictable times, with a tiny rate that is negligible for individual particles but becomes rapid for macroscopic objects.
- Shelly Goldstein will support some version of hidden-variable theories such as Bohmian mechanics. It’s sometimes thought that hidden variables have been ruled out by experimental tests of Bell’s inequalities, but that’s not right; only local hidden variables have been excluded. Non-local hidden variables are still very viable!
- Rüdiger Schack will be telling us about a relatively new approach called Quantum Bayesianism, or QBism for short. (Don’t love the approach, but the nickname is awesome.) The idea here is that QM is really a theory about our ignorance of the world, similar to what Tom Banks defended here way back when.
- My job, of course, will be to defend the honor of the Everett (many-worlds) formulation. I’ve done a lot less serious research on this issue than the other folks, but I will make up for that disadvantage by supporting the theory that is actually true. And coincidentally, by the time we’ve started debating I should have my first official paper on the foundations of QM appear on the arxiv: new work on deriving the Born Rule in Everett with Chip Sebens.

(For what it’s worth, I cannot resist quoting David Wallace in this context: when faced with the measurement problem in quantum mechanics, philosophers are eager to change the physics, while physicists are sure it’s just a matter of better philosophy.)

(Note also that both Steven Weinberg and Gerard ‘t Hooft have proposed new approaches to thinking about quantum mechanics. Neither of them were judged to be sufficiently distinguished to appear on our panel.)

It’s not accidental that I call these “formulations” rather than “interpretations” of quantum mechanics. I’d like to see people abandon the phrase “interpretation of quantum mechanics” entirely (though I often slip up and use it myself). The options listed above are not different *interpretations* of the same underlying structure — they are legitimately different physical theories, with potentially different experimental consequences (as our recent work on quantum fluctuations shows).

Relatedly, I discovered this morning that celebrated philosopher Hilary Putnam has joined the blogosphere, with the whimsically titled “Sardonic Comment.” His very first post shares an email conversation he had about the measurement problem in QM, including my co-panelists David and Shelly, and also Tim Maudlin and Roderich Tumulka (but not me). I therefore had the honor of leaving the very first comment on Hilary Putnam’s blog, encouraging him to bring more Everettians into the discussion!

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