Cosmic inflation, a period of super-fast accelerated expansion in the early universe, was initially invented to help explain why the universe is so flat and smooth. (Whether this is a good motivation is another issue, one I hope to talk about soon.) In order to address these problems, the universe has to inflate by a sufficiently large amount. In particular, we have to have enough inflation so that the universe expands by a factor of more than 10^{22}, which is about *e*^{50}. Since physicists think in exponentials and logarithms, we usually just say “inflation needs to last for over 50 *e*-folds” for short.

So Grant Remmen, a grad student here at Caltech, and I have been considering a pretty obvious question to ask: if we assume that there was cosmic inflation, how much inflation do we actually expect to have occurred? In other words, given a certain inflationary model (some set of fields and potential energies), is it most likely that we get lots and lots of inflation, or would it be more likely to get just a little bit? Everyone who invents inflationary models is careful enough to check that their proposals *allow* for sufficient inflation, but that’s a bit different from asking whether it’s likely.

The result of our cogitations appeared on arxiv recently:

How Manye-Folds Should We Expect from High-Scale Inflation?

Grant N. Remmen, Sean M. CarrollWe address the issue of how many

e-folds we would naturally expect if inflation occurred at an energy scale of order 10^{16}GeV. We use the canonical measure on trajectories in classical phase space, specialized to the case of flat universes with a single scalar field. While there is no exact analytic expression for the measure, we are able to derive conditions that determine its behavior. For a quadratic potential V(ϕ)=m^{2}ϕ^{2}/2 withm=2×10^{13}GeV and cutoff atM_{Pl}=2.4×10^{18}GeV, we find an expectation value of 2×10^{10}e-folds on the set of FRW trajectories. For cosine inflation V(ϕ)=Λ^{4}[1−cos(ϕ/f)] withf=1.5×10^{19}GeV, we find that the expected total number ofe-folds is 50, which would just satisfy the observed requirements of our own Universe; iffis larger, more than 50e-folds are generically attained. We conclude that one should expect a large amount of inflation in large-field models and more limited inflation in small-field (hilltop) scenarios.

As should be evident, this builds on the previous paper Grant and I wrote about cosmological attractors. We have a technique for finding a measure on the space of cosmological histories, so it is natural to apply that measure to different versions of inflation. The result tells us — at least as far as the classical dynamics of inflation are concerned — how much inflation one would “naturally” expect in a given model.

The results were interesting. For definiteness we looked at two specific simple models: quadratic inflation, where the potential for the inflaton ϕ is simply a parabola, and cosine (or “natural“) inflation, where the potential is — wait for it — a cosine. There are many models one might consider (one recent paper looks at 193 possible versions of inflation), but we weren’t trying to be comprehensive, merely illustrative. And these are two nice examples of two different kinds of potentials: “large-field” models where the potential grows without bound (or at least until you reach the Planck scale), and “small-field” models where the inflaton can sit near the top of a hill.

Think for a moment about how much inflation *can* occur (rather than “probably does”) in these models. Remember that the inflaton field acts just like a ball rolling down a hill, except that there is an effective “friction” from the expansion of the universe. That friction becomes greater as the expansion rate is higher, which for inflation happens when the field is high on the potential. So in the quadratic case, even though the slope of the potential (and therefore the force pushing the field downwards) grows quite large when the field is high up, the friction is also very large, and it’s actually the increased friction that dominates in this case. So the field rolls slowly (and the universe inflates) at large values, while inflation stops at smaller values. But there is a cutoff, since we can’t let the potential grow larger than the Planck scale. So the quadratic model allows for a *large but finite* amount of inflation.

The cosine model, on the other hand, allows for a *potentially infinite* amount of inflation. That’s because the potential has a maximum at the top of the hill. In principle, there is a trajectory where the field simply sits there and inflates forever. Slightly more realistically, there are other trajectories that start (with zero velocity) very close to the top of the hill, and take an arbitrarily long time to roll down. There are also, of course, trajectories that have a substantial velocity near the top of the hill, which would inflate for a relatively short period of time. (Inflation only happens when the energy density is mostly in the form of the potential itself, with a minimal contribution from the kinetic energy caused by the field velocity.) So there is an interesting tradeoff, and we would like to know which effect wins.

The answer that Grant and I derived is: in the quadratic potential, we generically expect a huge amount of inflation, while in the cosine potential, we expect barely enough (fairly close to the hoped-for 50 *e*-folds). Although you can in principle inflate forever near a hilltop, such behavior is non-generic; you really need to fine-tune the initial conditions to make it happen. In the quadratic potential, by contrast, getting a buttload of inflation is no problem at all.

Of course, this entire analysis is done using the classical measure on trajectories through phase space. The actual world is not classical, nor is there any strong reason to expect that the initial conditions for the universe are randomly chosen with respect to the Liouville measure. (Indeed, we’re pretty sure that they are not.)

So this study has a certain aspect of looking-under-the-lamppost. We consider the naturalness of cosmological histories with respect to the conserved measure on classical phase space because that’s what we can do. If we had a finished theory of quantum gravity and knew the wave function of the universe, we would just look at that.

We’re not looking for blatant contradictions with data, we’re looking for clues that can help us move forward. The way to interpret our result is to say that, *if* the universe has a field with a potential like the quadratic inflation model (and the initial conditions are sufficiently smooth to allow inflation at all), *then* it’s completely natural to get more than enough inflation. If we have the cosine potential, on the other hand, than getting enough inflation is possible, but far from a sure thing. Which might be very good news — it’s generally thought that getting precisely “just enough” inflation is annoying finely-tuned, but here it seems quite plausible. That might suggest that we could observe remnants of the pre-inflationary universe at very large scales today.

Attractor Solutions in Scalar-Field Cosmology

Grant N. Remmen, Sean M. CarrollModels of cosmological scalar fields often feature “attractor solutions” to which the system evolves for a wide range of initial conditions. There is some tension between this well-known fact and another well-known fact: Liouville’s theorem forbids true attractor behavior in a Hamiltonian system. In universes with vanishing spatial curvature, the field variables () specify the system completely, defining an effective phase space. We investigate whether one can define a unique conserved measure on this effective phase space, showing that it exists for

m^{2}φ^{2}potentials and deriving conditions for its existence in more general theories. We show that apparent attractors are places where this conserved measure diverges in the () variables and suggest a physical understanding of attractor behavior that is compatible with Liouville’s theorem.

This paper investigates a well-known phenomenon in inflationary cosmology: the existence of purported “attractor” solutions. There is a bit of lore that says that an inflationary scalar field might start off doing all sorts of things, but will quickly settle down to a preferred kind of evolution, known as the attractor. But that lore is nominally at odds with a mathematical theorem: in classical mechanics, closed systems never have attractor solutions! That’s because “attractor” means “many initial conditions are driven to the same condition,” while Liouville’s theorem says “a set of initial conditions maintains its volume as it evolves.” So what’s going on?

Let’s consider the simplest kind of model: you just have a single scalar field φ, and a potential energy function *V*(φ), in the context of an expanding universe with no other forms of matter or energy. That fully specifies the model, but then you have to specify the actual trajectory that the field takes as it evolves. Any trajectory is fixed by giving certain initial data in the form of the value of the field φ and its “velocity” . For a very simple potential like *V*(φ) ~ φ^{2}, the trajectories look like this:

This is the “effective phase space” of the model — in a spatially flat universe (and only there), specifying φ and its velocity uniquely determines a trajectory, shown as the lines on the plot. See the dark lines that start horizontally, then spiral toward the origin? Those are the attractor solutions. Other trajectories (dashed lines) basically zoom right to the attractor, then stick nearby for the rest of their evolution. Physically, the expansion of the universe acts as a kind of friction; away from the attractor the friction is too small to matter, but once you get there friction begins to dominate and the the field rolls very slowly. So the idea is that there aren’t *really* that many different kinds of possible evolution; a “generic” initial condition will just snap onto the attractor and go from there.

This story seems to be in blatant contradiction with Liouville’s Theorem, which roughly says that there *cannot be* true attractors, because volumes in phase space (the space of initial conditions, i.e. coordinates and momenta) remain constant under time-evolution. Whereas in the picture above, volumes get squeezed to zero because every trajectory flows to the 1-dimensional attractor, and then of course eventually converges to the origin. But we know that the above plot really does show what the trajectories do, and we also know that Liouville’s theorem is correct and does apply to this situation. Our goal for the paper was to show how everything actually fits together.

Obviously (when you think about it, and know a little bit about phase space), the problem is with the coordinates on the above graph. In particular, might be the “velocity” of the field, but it definitely isn’t its “momentum,” in the strict mathematical sense. The canonical momentum is actually , where *a* is the scale factor that measures the size of the universe. And the scale factor changes with time, so there is no simple translation between the nice plot we saw above and the “true” phase space — which should, after all, also include the scale factor itself as well as *its* canonical momentum.

So there are good reasons of convenience to draw the plot above, but it doesn’t really correspond to phase space. As a result, it looks like there are attractors, although there really aren’t — at least not by the strict mathematical definition. It’s just a convenient, though possibly misleading, nomenclature used by cosmologists.

Still, there is something physically relevant about these cosmological attractors (which we will still call “attractors” even if they don’t match the technical definition). If it’s not “trajectories in phase space focus onto them,” what is it? To investigate this, Grant and I turned to a formalism for defining the measure on the space of *trajectories* (rather than just points in phase space), originally studied by Gibbons, Hawking, and Stewart and further investigated by Heywood Tam and me a couple of years ago.

The interesting thing about the “GHS measure” on the space of trajectories is that it diverges — becomes infinitely big — for cosmologies that are spatially flat. That is, *almost all universes are spatially flat* — if you were to pick a homogeneous and isotropic cosmology out of a hat, it would have zero spatial curvature with probability unity. (Which means that the flatness problem you were taught as a young cosmologist is just a sad misunderstanding — more about that later in another post.) That’s fine, but it makes it mathematically tricky to study those flat universes, since the measure is infinity there. Heywood and I proposed a way to regulate this infinity to get a finite answer, but that was a mistake on our part — upon further review, our regularization was not invariant under time-evolution, as it should have been.

That left an open problem — what is the correct measure on the space of flat universes? This is what Grant and I tackled, and basically solved. Long story short, we studied the necessary and sufficient conditions for there to be the right kind of measure on the effective phase space shown in the plot above, and argued that such a measure (1) exists, and (2) is apparently unique, at least in the simple case of a quadratic potential (and probably more generally). That is, we basically reverse-engineered the measure from the requirement that Liouville’s theorem be obeyed!

So there is such a measure, but it’s very different from the naïve “graph-paper measure” that one is tempted to use for the effective phase space plotted above. (A temptation to which almost everyone in the field gives in.) Unsurprisingly, the measure blows up on the attractor, and near the origin. That is, what looks like an attractor when you plot it in these coordinates is really a sign that the density of trajectories grows very large there — which is the least surprising thing in the world, really.

At the end of the day, despite the fact that we mildly scold fellow cosmologists for their sloppy use of the word “attractor,” the physical insights connected to this idea go through essentially unaltered. The field and its velocity are the variables that are most readily observable (or describable) by us, and in terms of these variables the apparent attractor behavior is definitely there. The real usefulness of our paper would come when we wanted to actually use the measure we constructed, for example to calculate the expected amount of inflation in a given model — which is what we did in our more recent paper, to be described later.

This paper, by the way, was one from which I took equations for the blackboards in an episode of *Bones*. It was fun to hear Richard Schiff, famous as Toby from *The West Wing*, play a physicist who explains his alibi by saying “I was constructing an invariant measure on the phase space of cosmological spacetimes.”

The episode itself is great, you should watch it if you can. But I warn you — you will cry.

]]>What immediately strikes me are (1) how nominally unobjectionable all the suggestions are, and (2) how so many of them are routinely violated even in situations that wouldn’t strike us as relatively civil and respectful. Here are some cherry-picked examples from the list:

- Don’t interrupt.
- Don’t present objections as flat dismissals (leave open the possibility that there’s a response).
- Don’t dominate the discussion (partial exception for the speaker here!).
- Unless you’re speaker, existing questioner, or chair, don’t speak without being called on (limited exceptions for occasional jokes and other very brief interjections, not to be abused).
- The chair should attempt to balance the discussion among participants, prioritizing those who have not spoken before.
- Prioritize junior people in calling on questions (modified version: don’t prioritize senior people).

Not that I’m saying we *shouldn’t* strive to be as respectful as David’s lists suggests — just that we don’t even try, really. How many times have you been in a seminar in which a senior person in the audience interrupted and dominated discussion? I can imagine someone defending that kind of environment, on the grounds that it leads to more fun and feisty give-and-take, and perhaps even a more rapid convergence to the truth. I can testify that I was once at a small workshop — moderated by David Chalmers — where the “hand-and-finger system” was employed, according to which you raise a hand to ask a new question, and a finger to ask a follow-up to someone else’s question, and the chair keeps a list of who is next in the queue. Highly structured, but it actually worked quite well, carving out some space for everyone’s questions to be treated equally, regardless of their natural degree of assertiveness or their social status within the room. (If you’ve ever been at a “Russian” physics seminar, imagine exactly the opposite of that.) I will leave it as an exercise for the reader to judge whether enforcing these norms more actively would create a more hospitable academic environment overall.

David also suggests some related resources:

- Rules for philosophy classes (Ned Markosian)
- Pledges for a professional philosopher (Carrie Jenkins)
- Seminar chairing policy suggestions (British Philosophical Association)
- How to criticize with kindness (Daniel Dennett)

Any other suggestions? Comments are open, and you don’t have to wait to be called on.

]]>Self-indulgently long post below. **Short version:** Steven Salaita, an associate professor of English at Virginia Tech who had been offered and accepted a faculty job at the University of Illinois Urbana-Champaign, had his offer rescinded when the administration discovered that he had posted inflammatory tweets about Israel, such as “At this point, if Netanyahu appeared on TV with a necklace made from the teeth of Palestinian children, would anybody be surprised? #Gaza.” Many professors in a number of disciplines, without necessarily agreeing with Salaita’s statements, believe strongly that academic norms give him the right to say them without putting his employment in jeopardy, and have organized a boycott of UIUC in response. Alan Sokal of NYU is supporting the boycott, and has written a petition meant specifically for science and engineering faculty, who are welcome to sign if they agree.

Everyone agrees that “free speech” is a good thing. We live in a society where individual differences are supposed to be respected, and we profess admiration for the free market of ideas, where competing claims are discussed and subjected to reasonable critique. (Thinking here of the normative claim that free speech is a good thing, not legalistic issues surrounding the First Amendment and government restrictions.) We also tend to agree that such freedom is not absolute; you don’t have the right to come into my house (or the comment section of my blog) and force me to listen to your new crackpot theory of physics. A newspaper doesn’t have an obligation to print something just because you wrote it. Biology conferences don’t feel any need to give time to young-Earth creationists. In a classroom, teachers don’t have to sit quietly if a student wants to spew blatantly racist invective (and likewise for students while teachers do so).

So there is a line to be drawn, and figuring out where to draw it isn’t an easy task. It’s not hard to defend people’s right to say things we agree with; the hard part is defending speech we disagree with. And some speech, in certain circumstances, really isn’t worth defending — organizations have the right to get rid of employees who are (for example) consistently personally abusive to their fellow workers. The hard part — and it honestly is difficult — is to distinguish between “speech that I disagree with but is worth defending” and “speech that is truly over the line.”

To complicate matters, people who disagree often become — how to put this delicately? — emotional and polemical rather than dispassionate and reasonable. People are very people-ish that way. Consequently, we are often called upon to defend speech that we not only disagree with, but whose tone and connotation we find off-putting or even offensive. Those who would squelch disagreeable speech therefore have an easy out: “I might not agree with *what* they said, but what I really can’t countenance is *the way* they said it.” If we really buy the argument that ideas should be free and rational discourse between competing viewpoints is an effective method of discovering truth and wisdom, we have to be especially willing to defend speech that is couched in downright objectionable terms.

As an academic and writer, in close cases I will almost always fall on the side of defending speech even if I disagree with it (or how it is said). Recently several different cases have illustrated just how tricky this is — but in each case I think that the people in question have been unfairly punished for things they have said.

The **first case** is Ashutosh Jogalekar, who writes the Curious Wavefunction blog at *Scientific American*. Or “wrote,” I should say, since Ash has been fired, and is now blogging independently. (Full disclosure: I don’t know Ash, but he did write a nice review of my book; my wife Jennifer is also a blogger at SciAm, although I’m not privy to any inside info.)

The offenses for which he was let go amount to three controversial posts. One was actually a guest post by Chris Martin, arguing that Larry Summers was right when he said that ~~“innate ability”~~ “intrinsic aptitude” (my mistake — see comments) was a major determinant of women’s underrepresentation in science and math. It’s a dispiritingly self-righteous and sloppy argument, as well as one that has been thoroughly debunked; there may very well be innate differences, but the idea that they explain underrepresentation is laughably contradicted by the data, while the existence of discrimination is frighteningly demonstrable. The next post was a positive review of Nicholas Wade’s book *A Troublesome Inheritance*. Again, not very defensible on the merits; Wade’s book lurches incoherently from “genetic markers objectively correlate with geographical populations” to “therefore Chinese people may be clever, but they’ll never really understand democracy.” In a great Marshall McLuhan moment, over a hundred population geneticists — many of whom Wade relied on for the “scientific” parts of his book — wrote a scathing letter to the *New York Times* to make sure everyone understood “there is no support from the field of population genetics for Wade’s conjectures.” Finally, a post on Richard Feynman chronicled Ash’s feelings about the physicist, from young hero-worshiper to the eventual realization that Feynman could be quite disturbingly sexist, to ultimately feeling that we should understand Feynman’s foibles in the context of his time and not let his personal failings detract from our admiration for his abilities as a scientist. I didn’t really object to this one myself; I read it as someone grappling in good faith with the contradictions of a complex human being, even if in spots it came of as offering excuses for Feynman’s bad behavior. Others disagreed.

So SciAm decided to deal with the problem by letting Jogalekar go. In my mind, a really dumb decision. I disagree very strongly with some of the stuff Ash (or his guest poster) has said, but I never thought it came close to some standard of horribleness and offensiveness that would countenance firing him. I want to be challenged by people I disagree with, not just surrounded by fellow-travelers. I didn’t find much of interest in Ash’s three controversial posts, but overall his blog was often thought-provoking and enjoyable. SciAm had every “right” to fire him, as a legal matter. They are under no obligation to stand by their employees when those employees take controversial stances. But it was still the wrong thing to do; nothing Ash said was anywhere close to falling outside the realm of reasonable things to talk about, disagree with them though I may.

It breaks my heart. In the interminable arguments about gender and IQ and genetics, a favorite strategy of people who like to promote lazy arguments in favor of genetic determinism is to bemoan their victimhood status, claiming that even *asking* such questions is deemed unacceptable by the liberal thought police. (Wade’s defenders, for example, eagerly jumped on a rumor that he had been fired from the *Times* because of his book, when the truth is he had left the paper some years earlier.) Usually I have just laughed in response, pointing out that these questions are investigated all the time; the only real danger these people face is that others point out how superficial their arguments are, not that they are punished or lose their jobs for reaching the wrong conclusions. But I was wrong, and they were right, at least to some extent. You really can lose your job for holding the wrong view of these issues. (Sometimes the attitude is completely out in the open, as in this *Harvard Crimson* op-ed urging that we “give up on academic freedom in favor of justice.”) As a liberal and a feminist myself, I think we should be the ones who protect speech rights most vociferously, the ones who are happy to counter arguments with which we disagree with better arguments rather than blunt instruments of punishment. It’s a difficult standard to live up to.

The **second case** is less specific: the growing penchant for disinviting speakers with whom we disagree. It’s been bugging me for a while, but I won’t say too much about it here, especially since Massimo Pigliucci has already done a good job. I’m not a big fan of Condoleezza Rice’s contributions to US foreign policy, and I can understand that it might be disillusioning to hear that she was scheduled to be the featured speaker at your commencement at Rutgers. But I would advocate putting up with this mild inconvenience — unless you think that conservative students should also have the right to veto commencement talks by Democratic politicians.

I’ve expressed similar feelings before, in the even more straightforward case where Larry Summers (he keeps popping up in these conversations) was disinvited from giving a talk to the Regents of the University of California. That seemed completely wrong to me — the idea apparently being that Summers, having once said incorrect things about one topic, should be prevented from speaking to any audience about any topic. At the same time, I had no problem at all with Harvard faculty working to remove Summers as President of the university after his problematic speech. The difference being that what he said had a direct bearing on his performance in the office. He clearly misunderstood the situation of many women in modern academia, especially the sciences, and that’s something a modern university president really needs to understand. And it turns out that — shockingly, I know — the number of women hired as senior faculty under Summers was in fact noticeably smaller than it had been under his predecessors. But that, of course, doesn’t mean he should have been fired from his position as a tenured professor of economics — as indeed he was not. (Although I have to say his teaching load seems pretty light.)

The **third case**, most recent and newsworthy, is Salaita. The specifics were listed up at the top: he was offered a position, accepted it, and had it withdrawn before he could actually move, when the administration learned about some inflammatory tweets. I am not completely conversant with all of the details of his contract — apparently the job had been offered, and he had resigned from Virginia Tech, but the step of having his contract approved by the Board of Trustees (usually a formality) hadn’t yet gone through, giving the UIUC Chancellor an opportunity to step in by deciding not to submit the appointment to the board. (I had read about the issue in various places, but special thanks to Paul Boghossian at NYU for nudging me to pay attention to it.)

There’s little question in my mind that some of Salaita’s remarks were ugly. In addition to the Netanyahu tweet at the top, he said things like like “Let’s cut to the chase: If you’re defending Israel right now you’re an awful human being” and “You may be too refined to say it, but I’m not: I wish all the fucking West Bank settlers would go missing.” Statements like this don’t have anything very useful to offer in terms of rational discourse and the free market of ideas. (Even if, as always, context matters.) But I’m perfectly willing to believe that his other work has something to offer. We don’t judge academics by their least-academic utterances. And one-liners like this, as off-putting as they might be when read in isolation, shouldn’t disqualify someone from participating in the wider discourse. (Salaita has also tweeted things like “I refuse to conceptualize #Israel/#Palestine as Jewish-Arab acrimony. I am in solidarity with many Jews and in disagreement with many Arabs” and “#ISupportGaza because I believe that Jewish and Arab children are equal in the eyes of God.”)

When a professor has already been vetted and approved by a department and essentially offered a position, there should be an extremely high bar indeed for the administration to step in at the last minute and attempt to reverse the decision. It would be one thing if new evidence had come to light that indicated the person would be incompetent at their job; in Salaita’s case there was nothing of the sort, and indeed he received excellent teaching evaluations at Virginia Tech. And what would be really bad would be if administrators were making decisions for non-academic reasons, in ways that threaten to truly undermine academic freedom. That seems to be the case here. It is clear, at least, that UIUC Chancellor Phyllis Wise was directly contacted by prominent donors who threatened to stop donating if Salaita were hired.

Most worrisome of all was Chacellor Wise’s statement about the controversy, which included remarks such as this:

What we cannot and will not tolerate at the University of Illinois are personal and disrespectful words or actions that demean and abuse either viewpoints themselves or those who express them.

It’s easy to let your eyes glaze over that, but the statement itself is clear: the UIUC administration thinks it is not permissible to “*demean and abuse … viewpoints themselves*.” At Urbana-Champaign, you can be fired if you make fun of creationism, racism, or sexism. Those are viewpoints, and viewpoints cannot be demeaned or abused!

Perhaps Wise, or whoever drafted the statement, dashed something off and wasn’t thinking about it too carefully. If that’s their defense, it’s not much of one; these are crucially important issues for a university, which warrant some careful thought and precise formulation. And if they stand by it, a statement like this is straightforwardly antithetical to everything that universities are supposed to stand for.

I am therefore in support of the call for a boycott, until Salaita’s position is restored, even if (and in fact, especially because of the fact that) I don’t agree with his positions. I don’t really like boycotts in general — again, always preferring to err on the side of engagement rather than disengagement. I think the idea of an academic boycott of Israel is silly and counterproductive, for example. But a boycott is one of the few things that academics can do to put pressure on a university administration. And in this case there are signs it might actually be having an effect on UIUC — it seems that the Chancellor has forwarded the case to the Board of Trustees after all, although they have yet to actually vote on it. So if you are a science/engineering faculty member and inclined to support it, feel free to sign the petition. (There are also petitions organized by other disciplines, of course.)

Having said all that, it’s not like I’m very gleeful in my support of all these people with whom I disagree. It’s necessary and, I think, honorable, but also uncomfortable. Some people disagree with my own stance, of course: while the American Association of University Professors is firmly in Salaita’s corner, former AAUP president Cary Nelson takes a dissenting view.

One of the reasons why these issues are difficult is because of the emotional component I’ve already mentioned. In particular, it’s easy for *me* to say “You people who are being denigrated by these statements should just buck up and take it in the name of free speech and inquiry.” I have never personally had to suffer from sexism, racism, or anti-Semitism (or having my neighborhood bombed, for that matter). I’m a straight, white, upper-class, lapsed-Episcopalian Anglo-American male — in the sweepstakes of being privileged, I just about hit the jackpot. It’s no problem for me to sit back from this position of comfort and extoll the virtues of unfettered speech. The way our society is set up, it would be very difficult for anyone to write a blog post or series of tweets that would call into question my self-worth simply because of a group to which I happened to belong.

And yet, I don’t think that my position disqualifies me from having an opinion. It just means that I should try to be cognizant of my biases, be thoughtful about how other people might feel, and try especially hard to actually *listen* to what they have to say.

So to me, the most effective statements I’ve read on the Salaita controversy have been those from Jonathan Judaken and Bonny Honig. Here is Judaken:

As a scholar familiar with Judeophobic imagery, Salaita’s one-liner veered dangerously close to the myth of blood-libel. For a thousand years, Jews have been accused of desiring the blood of non-Jewish children. If the depiction of Netanyahu as savage and barbaric was applied to President Obama (as it has been) the racism would be patent.

Having grown up as a Jewish person in South Africa under apartheid — a dominant racial group and a religious minority — Judaken understands this language when he hears it. But here’s the thing: he *supports* the boycott, “on the basis of the principles of faculty governance, academic freedom, and freedom of speech.” Protecting the right of scholars to have an express unpopular opinions is too important to compromise. And here’s Honig, a political theorist at Brown:

I found that tweet painful and painfully funny. It struck home with me, a Jew raised as a Zionist. Too many of us are too committed to being uncritical of Israel. Perhaps tweets like Prof. Salaita’s, along with images of violence from Gaza and our innate sense of fair play, could wake us from our uncritical slumbers. It certainly provoked ME, and I say “provoked” in the best way – awakened to thinking.

That’s such a great statement of the true academic mindset. Provoke me! Say something I disagree with, even in an intentionally disagreeable way. Make me think, force me to re-examine my cherished presuppositions. Probably I will come out with my basic opinions intact — as an empirical matter, that’s usually what happens. But if you provoke me well, I’ll understand those presuppositions even better, and be more prepared to defend them next time. And who knows? You might even change my mind. One way or another, let the disagreements fly.

**Very short version:** I wish I lived in a world where I could spend my time disagreeing with people whose views I found disagreement-worthy, rather than fighting for their right to say disagreeable things.

It’s interesting that the “religious beliefs are completely independent of evidence and empirical investigation” meme has enjoyed such success in certain quarters that people express surprise to learn of the existence of theologians and believers who still think we can find evidence for the existence of God in our experience of the world. In reality, there are committed believers (“sophisticated” and otherwise) who feel strongly that we have evidence for God in the same sense that we have evidence for gluons or dark matter — because it’s the best way to make sense of the data — just as there are others who think that our knowledge of God is of a completely different kind, and therefore escapes scientific critique. It’s part of the problem that theism is not well defined.

One can go further than I did in the brief clip above, to argue that any notion of God that *can’t* be judged on the basis of empirical evidence isn’t much of a notion at all. If God exists but has no effect on the world whatsoever — the actual world we experience could be precisely the same even without God — then there is no reason to believe in it, and indeed one can draw no conclusions whatsoever (about right and wrong, the meaning of life, etc.) from positing it. Many people recognize this, and fall back on the idea that God is in some sense necessary; there is no possible world in which he doesn’t exist. To which the answer is: “No he’s not.” Defenses of God’s status as necessary ultimately come down to some other assertion of a purportedly-inviolable metaphysical principle, which can always simply be denied. (The theist could win such an argument by demonstrating that the naturalist’s beliefs are incoherent in the absence of such principles, but that never actually happens.)

I have more sympathy for theists who do try to ground their belief in evidence, rather than those who insist that evidence is irrelevant. At least they are playing the game in the right way, even if I disagree with their conclusions. Despite what Robert suggests in the clip above, the existence of disagreement among smart people does not imply that there is not a uniquely right answer!

]]>And now you can learn about the real thing! MIT is one of the many colleges and universities that is doing a great job putting top-quality lecture courses online, such as the introduction to quantum mechanics I recently mentioned. (See the comments of that post for other goodies.) Now they’ve announced a course at a decidedly non-introductory level: a graduate course in effective field theory, taught by Caltech alumn Iain Stewart. This is the real enchilada, the same stuff a second-year grad student in particle theory at MIT would be struggling with. If you want to learn how to really think about naturalness, or a good way of organizing what we learn from experiments at the LHC, this would be a great place to start. (Assuming you already know the basics of quantum field theory.)

Classes start Sept. 16. I would love to take it myself, but I have other things on my plate at the moment — anyone who does take it, chime in and let us know how it goes.

]]>Backstory for the puzzled: here is a nice paper that came out last month, on inflation in supergravity.

Inflation in Supergravity with a Single Chiral Superfield

Sergei V. Ketov, Takahiro TeradaWe propose new supergravity models describing chaotic Linde- and Starobinsky-like inflation in terms of a single chiral superfield. The key ideas to obtain a positive vacuum energy during large field inflation are (i) stabilization of the real or imaginary partner of the inflaton by modifying a Kahler potential, and (ii) use of the crossing terms in the scalar potential originating from a polynomial superpotential. Our inflationary models are constructed by starting from the minimal Kahler potential with a shift symmetry, and are extended to the no-scale case. Our methods can be applied to more general inflationary models in supergravity with only one chiral superfield.

Supergravity is simply the supersymmetric version of Einstein’s general theory of relativity, but unlike GR (where you can consider just about any old collection of fields to be the “source” of gravity), the constraints of supersymmetry place quite specific requirements on what counts as the “stuff” that creates the gravity. In particular, the allowed stuff comes in the form of “superfields,” which are combinations of boson and fermion fields. So if you want to have inflation within supergravity (which is a very natural thing to want), you have to do a bit of exploring around within the allowed set of superfields to get everything to work. Renata Kallosh and Andrei Linde, for example, have been examining this problem for quite some time.

What Ketov and Terada have managed to do is boil the necessary ingredients down to a minimal amount: just a single superfield. Very nice, and worth celebrating. So why not make a movie-like trailer to help generate a bit of buzz?

Which is just what Takahiro Terada, a PhD student at the University of Tokyo, has done. The link to the YouTube video appeared in an unobtrusive comment in the arxiv page for the revised version of their paper. iMovie provides a template for making such trailers, so it can’t be all that hard to do — but (1) nobody else does it, so, genius, and (2) it’s a pretty awesome job, with just the right touch of humor.

I wouldn’t have paid nearly as much attention to the paper without the trailer, so: mission accomplished. Let’s see if we can’t make this a trend.

]]>But the conference should be great — a small workshop, Quantum Foundations of a Classical Universe. We’re going to be arguing about how we’re supposed to connect wave functions and quantum observables to the everyday world of space and stuff. I will mostly be paying attention to the proceedings, but I might occasionally interject a tweet if something interesting/amusing happens. I’m told that some sort of proceedings will eventually be put online.

**Update:** Trying something new here. I’ve been tweeting about the workshop under the hashtag #quantumfoundations. So here I am using Storify to collect those tweets, making a quasi-live-blog on the cheap. Let’s see if it works.

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!

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