How Much Cosmic Inflation Probably Occurred?

Nothing focuses the mind like a hanging, and nothing focuses the science like an unexpected experimental result. The BICEP2 claimed discovery of gravitational waves in the cosmic microwave background — although we still don’t know whether it will hold up — has prompted cosmologists to think hard about the possibility that inflation happened at a very high energy scale. The BICEP2 paper has over 600 citations already, or more than 3/day since it was released. And hey, I’m a cosmologist! (At times.) So I am as susceptible to being prompted as anyone.

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 1022, which is about e50. 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 Many e-Folds Should We Expect from High-Scale Inflation?
Grant N. Remmen, Sean M. Carroll

We address the issue of how many e-folds we would naturally expect if inflation occurred at an energy scale of order 1016 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)=m2ϕ2/2 with m=2×1013 GeV and cutoff at MPl=2.4×1018 GeV, we find an expectation value of 2×1010 e-folds on the set of FRW trajectories. For cosine inflation V(ϕ)=Λ4[1−cos(ϕ/f)] with f=1.5×1019 GeV, we find that the expected total number of e-folds is 50, which would just satisfy the observed requirements of our own Universe; if f is larger, more than 50 e-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.

2potentials

Think for a moment about how much inflation can occur (rather than “probably does”) in these models. Continue reading

Posted in arxiv, Science | 16 Comments

Cosmological Attractors

I want to tell you about a paper I recently wrote with grad student Grant Remmen, about how much inflation we should expect to have occurred in the early universe. But that paper leans heavily on an earlier one that Grant and I wrote, about phase space and cosmological attractor solutions — one that I never got around to blogging about. So you’re going to hear about that one first! It’s pretty awesome in its own right. (Sadly “cosmological attractors” has nothing at all to do with the hypothetical notion of attractive cosmologists.)

Attractor Solutions in Scalar-Field Cosmology
Grant N. Remmen, Sean M. Carroll

Models 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 (\phi, \dot\phi) 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 m2φ2 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 (\phi, \dot\phi) 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” \dot\phi. For a very simple potential like V(φ) ~ φ2, the trajectories look like this:

attractors

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, \dot\phi might be the “velocity” of the field, but it definitely isn’t its “momentum,” in the strict mathematical sense. The canonical momentum is actually a^3\dot\phi, 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.” 

richard-schiff-on-bones

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

Posted in arxiv, Science | 18 Comments

Norms for Respectful Classroom/Seminar Discussion

David Chalmers is compiling a useful set of guidelines for respectful, constructive, and inclusive philosophical discussion. It makes sense to concentrate on a single field, like philosophy, since customs often vary wildly from one discipline to the other — but there’s really nothing specifically “philosophical” about the list, it could easily be adopted in just about any classroom or seminar environment I can think of. (Online, alas, is another story.)

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:

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

Posted in Academia | 16 Comments

Troublesome Speech and the UIUC Boycott

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. Continue reading

Posted in Academia, Humanity | 76 Comments

Should Scientific Progress Affect Religious Beliefs?

Sure it should. Here’s a new video from Closer to Truth, in which I’m chatting briefly with Robert Lawrence Kuhn about the question. “New” in the sense that it was just put on YouTube, although we taped it back in 2011. (Now my formulations would be considerably more sophisticated, given the wisdom that comes with age).

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!

Posted in Philosophy, Religion | 65 Comments

Effective Field Theory MOOC from MIT

Faithful readers are well aware of the importance of effective field theory in modern physics. EFT provides, in a nutshell, the best way we have to think about the fundamental dynamics of the universe, from the physics underlying everyday life to structure formation in the universe.

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.)

edx-eft

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.

Posted in Academia, Science | 15 Comments

Single Superfield Inflation: The Trailer

This is amazing. (Via Bob McNees and Michael Nielsen on Twitter.)

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 Terada

We 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.

Posted in arxiv, Humor, Science | 28 Comments

Quantum Foundations of a Classical Universe

Greetings from sunny (for the moment) Yorktown Heights, NY, home of IBM’s Watson Research Center. I’m behind on respectable blogging (although it’s been nice to see some substantive conversation on the last couple of comment threads), and I’m at a conference all week here, so that situation is unlikely to change dramatically in the next few days.

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. Continue reading

Posted in Science | 26 Comments

Quantum Sleeping Beauty and the Multiverse

Hidden in my papers with Chip Sebens on Everettian quantum mechanics is a simple solution to a fun philosophical problem with potential implications for cosmology: the quantum version of the Sleeping Beauty Problem. It’s a classic example of self-locating uncertainty: knowing everything there is to know about the universe except where you are in it. (Skeptic’s Play beat me to the punch here, but here’s my own take.)

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?”

Modified from a figure by Stuart Armstrong.

Modified from a figure by Stuart Armstrong.

(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. Continue reading

Posted in Philosophy, Science | 245 Comments

Why Probability in Quantum Mechanics is Given by the Wave Function Squared

One of the most profound and mysterious principles in all of physics is the Born Rule, named after Max Born. In quantum mechanics, particles don’t have classical properties like “position” or “momentum”; rather, there is a wave function that assigns a (complex) number, called the “amplitude,” to each possible measurement outcome. The Born Rule is then very simple: it says that the probability of obtaining any possible measurement outcome is equal to the square of the corresponding amplitude. (The wave function is just the set of all the amplitudes.)

Born Rule:     \mathrm{Probability}(x) = |\mathrm{amplitude}(x)|^2.

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:

Born Rule

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:

  1. Quantum states are represented by wave functions, which are vectors in a mathematical space called Hilbert space.
  2. Wave functions evolve in time according to the Schrödinger equation.
  3. The act of measuring a quantum system returns a number, known as the eigenvalue of the quantity being measured.
  4. The probability of getting any particular eigenvalue is equal to the square of the amplitude for that eigenvalue.
  5. 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:

  1. Quantum states are represented by wave functions, which are vectors in a mathematical space called Hilbert space.
  2. 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: Continue reading

Posted in arxiv, Philosophy, Science | 96 Comments