I Wanna Live Forever

If you’re one of those people who look the universe in the eyeball without flinching, choosing to accept uncomfortable truths when they are supported by the implacable judgment of Science, then you’ve probably acknowledged that sitting is bad for you. Like, really bad. If you’re not convinced, the conclusions are available in helpful infographic form; here’s an excerpt.


And, you know, I sit down an awful lot. Doing science, writing, eating, playing poker — my favorite activities are remarkably sitting-based.

So I’ve finally broken down and done something about it. On the good advice of Carl Zimmer, I’ve augmented my desk at work with a Varidesk on top. The desk itself was formerly used by Richard Feynman, so I wasn’t exactly going to give that up and replace it with a standing desk. But this little gizmo lets me spend most of my time at work on my feet instead of sitting on my butt, while preserving the previous furniture.


It’s a pretty nifty device, actually. Room enough for my laptop, monitor, keyboard, mouse pad, and the requisite few cups for coffee. Most importantly for a lazybones like me, it doesn’t force you to stand up absolutely all the time; gently pull some handles and the whole thing gently settles down to desktop level, ready for your normal chair-bound routine.


We’ll see how the whole thing goes. It’s one thing to buy something that allows you to stand while working, it’s another to actually do it. But at least I feel like I’m trying to be healthier. I should go have a sundae to celebrate.

Posted in Health, Personal | 34 Comments

The Wrong Objections to the Many-Worlds Interpretation of Quantum Mechanics

Longtime readers know that I’ve made a bit of an effort to help people understand, and perhaps even grow to respect, the Everett or Many-Worlds Interpretation of Quantum Mechanics (MWI) . I’ve even written papers about it. It’s a controversial idea and far from firmly established, but it’s a serious one, and deserves serious discussion.

Which is why I become sad when people continue to misunderstand it. And even sadder when they misunderstand it for what are — let’s face it — obviously wrong reasons. The particular objection I’m thinking of is:

MWI is not a good theory because it’s not testable.

It has appeared recently in this article by Philip Ball — an essay whose snidely aggressive tone is matched only by the consistency with which it is off-base. Worst of all, the piece actually quotes me, explaining why the objection is wrong. So clearly I am either being too obscure, or too polite.

I suspect that almost everyone who makes this objection doesn’t understand MWI at all. This is me trying to be generous, because that’s the only reason I can think of why one would make it. In particular, if you were under the impression that MWI postulated a huge number of unobservable worlds, then you would be perfectly in your rights to make that objection. So I have to think that the objectors actually are under that impression.

An impression that is completely incorrect. The MWI does not postulate a huge number of unobservable worlds, misleading name notwithstanding. (One reason many of us like to call it “Everettian Quantum Mechanics” instead of “Many-Worlds.”)

Now, MWI certainly does predict the existence of a huge number of unobservable worlds. But it doesn’t postulate them. It derives them, from what it does postulate. And the actual postulates of the theory are quite simple indeed:

  1. The world is described by a quantum state, which is an element of a kind of vector space known as Hilbert space.
  2. The quantum state evolves through time in accordance with the Schrödinger equation, with some particular Hamiltonian.

That is, as they say, it. Notice you don’t see anything about worlds in there. The worlds are there whether you like it or not, sitting in Hilbert space, waiting to see whether they become actualized in the course of the evolution. Notice, also, that these postulates are eminently testable — indeed, even falsifiable! And once you make them (and you accept an appropriate “past hypothesis,” just as in statistical mechanics, and are considering a sufficiently richly-interacting system), the worlds happen automatically.

Given that, you can see why the objection is dispiritingly wrong-headed. You don’t hold it against a theory if it makes some predictions that can’t be tested. Every theory does that. You don’t object to general relativity because you can’t be absolutely sure that Einstein’s equation was holding true at some particular event a billion light years away. This distinction between what is postulated (which should be testable) and everything that is derived (which clearly need not be) seems pretty straightforward to me, but is a favorite thing for people to get confused about.

Ah, but the MWI-naysayers say (as Ball actually does say), but every version of quantum mechanics has those two postulates or something like them, so testing them doesn’t really test MWI. So what? If you have a different version of QM (perhaps what Ted Bunn has called a “disappearing-world” interpretation), it must somehow differ from MWI, presumably by either changing the above postulates or adding to them. And in that case, if your theory is well-posed, we can very readily test those proposed changes. In a dynamical-collapse theory, for example, the wave function does not simply evolve according to the Schrödinger equation; it occasionally collapses (duh) in a nonlinear and possibly stochastic fashion. And we can absolutely look for experimental signatures of that deviation, thereby testing the relative adequacy of MWI vs. your collapse theory. Likewise in hidden-variable theories, one could actually experimentally determine the existence of the new variables. Now, it’s true, any such competitor to MWI probably has a limit in which the deviations are very hard to discern — it had better, because so far every experiment is completely compatible with the above two axioms. But that’s hardly the MWI’s fault; just the opposite.

The people who object to MWI because of all those unobservable worlds aren’t really objecting to MWI at all; they just don’t like and/or understand quantum mechanics. Hilbert space is big, regardless of one’s personal feelings on the matter.

Which saddens me, as an MWI proponent, because I am very quick to admit that there are potentially quite good objections to MWI, and I would much rather spend my time discussing those, rather than the silly ones. Despite my efforts and those of others, it’s certainly possible that we don’t have the right understanding of probability in the theory, or why it’s a theory of probability at all. Similarly, despite the efforts of Zurek and others, we don’t have an absolutely airtight understanding of why we see apparent collapses into certain states and not others. Heck, you might be unconvinced that the above postulates really do lead to the existence of distinct worlds, despite the standard decoherence analysis; that would be great, I’d love to see the argument, it might lead to a productive scientific conversation. Should we be worried that decoherence is only an approximate process? How do we pick out quasi-classical realms and histories? Do we, in fact, need a bit more structure than the bare-bones axioms listed above, perhaps something that picks out a preferred set of observables?

All good questions to talk about! Maybe someday the public discourse about MWI will catch up with the discussion that experts have among themselves, evolve past self-congratulatory sneering about all those unobservable worlds, and share in the real pleasure of talking about the issues that matter.

Posted in Science | 112 Comments

Problem Book in Relativity and Gravitation: Free Online!

If I were ever to publish a second edition of Spacetime and Geometry — unlikely, but check back in another ten years — one thing I would like to do would be to increase the number of problems at the end of each chapter. I like the problems that are there, but they certainly could be greater in number. And there is no solutions manual, to the chagrin of numerous professors over the last decade.

What I usually do, when people ask for solutions and/or more problems, is suggest that they dig up a copy of the Problem Book in Relativity and Gravitation by Lightman, Press, Price, and Teukolsky. It’s a wonderful resource, with twenty chapters chock-full of problems, all with complete solutions in the back. A great thing to have for self-study. The book is a bit venerable, dating from 1975, and the typesetting isn’t the most modern; but the basics of GR haven’t changed in that time, and the notation and level are a perfect fit for my book.

And now everyone can have it for free! Where by “now” I mean “for the last five years,” although somehow I never heard of this. Princeton University Press, the publisher, gave permission to put the book online, for which students everywhere should be grateful.

Problem Book in Relativity and Gravitation

If you’re learning (or teaching) general relativity, you owe yourself to check it out.

Posted in Science | 9 Comments

New Course: The Higgs Boson and Beyond

Happy to announce that I have a new course out with The Great Courses (produced by The Teaching Company). This one is called The Higgs Boson and Beyond, and consists of twelve half-hour lectures. I previously have done two other courses for them: Dark Matter and Dark Energy, and Mysteries of Modern Physics: Time. Both of those were 24 lectures each, so this time we’re getting to the good stuff more quickly.

The inspiration for the course was, naturally, the 2012 discovery of the Higgs, and you’ll be unsurprised to learn that there is some overlap with my book The Particle at the End of the Universe. It’s certainly not just me reading the book, though; the lecture format is very different than the written word, and I’ve adjusted the topics and order appropriately. Here’s the lineup:


  1. The Importance of the Higgs Boson
  2. Quantum Field Theory
  3. Atoms to Particles
  4. The Power of Symmetry
  5. The Higgs Field
  6. Mass and Energy
  7. Colliding Particles
  8. Particle Accelerators and Detectors
  9. The Large Hadron Collider
  10. Capturing the Higgs Boson
  11. Beyond the Standard Model
  12. Frontiers: Higgs in Space

Because it is a course, the presentation here is in a more strictly logical order than it is in the book, starting from quantum field theory and working our way up. It’s still aimed at a completely non-expert audience, though a bit of enthusiasm for physics will be helpful for grappling with the more challenging material. And it’s available in both audio-only or video — but I have to say they did a really nice job with the graphics this time around, so the video is worth having.

And it’s on sale! Don’t know how long that will last, but there’s a big difference between regular prices at The Great Courses and the sale prices. A bargain either way!

Posted in Higgs, Personal | 27 Comments

The State of the Early Universe

Well hello, blog. It’s been too long! Feels good to be back.

The big cosmological excitement this week was the announcement of new cosmic microwave background measurements. These include a big release of new papers from the Planck satellite, as well as a joint polarization analysis combining data from BICEP2, the Keck array, and Planck.

Polarization measurements from Planck superimposed on CMB temperature anisotropies. From the Planckoscope, h/t Bob McNees and Raquel Ribeiro.

Polarization measurements from Planck superimposed on CMB temperature anisotropies. From the Planckoscope, h/t Bob McNees and Raquel Ribeiro.

The good news is: we understand the current universe pretty darn well! So much so, in fact, that even an amazingly high-precision instrument such as Planck has a hard time discovering truly new and surprising things about cosmology. Hence, the Planck press releases chose to highlight the finding that the earliest stars formed about 0.1 billion years later than had previously been thought. Which is an awesome piece of science, but doesn’t quite rise to the level of excitement that other possible discoveries might have reached.

Power spectrum of CMB temperature fluctuations, from Planck. Now that is some agreement between theory and experiment!

Power spectrum of CMB temperature fluctuations, from Planck. Now that is some agreement between theory and experiment!

For example, the possibility that we had seen primordial gravitational waves from inflation, as the original announcement of the BICEP2 results suggested back in March. If you’ll remember, the polarization of the CMB can be mathematically decomposed into “E-modes,” which look like gradients and arise naturally from the perturbations in density that we all know and love, and “B-modes,” which look like curls and are not produced (in substantial amounts) from density perturbations. They could be produced by gravitational waves, which in turn could be generated during cosmic inflation — so finding them is a very big deal, indeed.

A big deal that apparently hasn’t happened. As has been suspected for a while now, while BICEP2 did detect B-modes, they seem to have been generated by dust in our galaxy, rather than by gravitational waves during inflation. That is the pretty definitive conclusion from the new Planck/BICEP2/Keck joint analysis.

And therefore, what we had hoped was a detection of primordial gravitational waves now turns into a less-thrilling (but equally scientifically crucial) upper limit. Here’s one way of looking at the situation now. On the horizontal axis we have ns, the “tilt” in the power spectrum of perturbations, i.e. the variation in the amplitude of those perturbations on different distances across space. And on the vertical axis we have r, the ratio of the gravitational waves to the ordinary density perturbations. The original BICEP2 interpretation was that we had discovered r = 0.2; now we see that r is less than 0.15, probably less than 0.10, depending on which pieces of information you combine to get your constraint. No sign that it’s anything other than zero.

Current constraints on the "tilt" of the primordial perturbations (horizontal axis) and the contribution from gravitational waves (vertical axis).

Current constraints on the “tilt” of the primordial perturbations (horizontal axis) and the contribution from gravitational waves (vertical axis).

So what have we learned? Here are some take-away messages. Continue reading

Posted in Science | 26 Comments

We Are All Machines That Think

My answer to this year’s Edge Question, “What Do You Think About Machines That Think?”

Active_brainJulien de La Mettrie would be classified as a quintessential New Atheist, except for the fact that there’s not much New about him by now. Writing in eighteenth-century France, La Mettrie was brash in his pronouncements, openly disparaging of his opponents, and boisterously assured in his anti-spiritualist convictions. His most influential work, L’homme machine (Man a Machine), derided the idea of a Cartesian non-material soul. A physician by trade, he argued that the workings and diseases of the mind were best understood as features of the body and brain.

As we all know, even today La Mettrie’s ideas aren’t universally accepted, but he was largely on the right track. Modern physics has achieved a complete list of the particles and forces that make up all the matter we directly see around us, both living and non-living, with no room left for extra-physical life forces. Neuroscience, a much more challenging field and correspondingly not nearly as far along as physics, has nevertheless made enormous strides in connecting human thoughts and behaviors with specific actions in our brains. When asked for my thoughts about machines that think, I can’t help but reply: Hey, those are my friends you’re talking about. We are all machines that think, and the distinction between different types of machines is eroding.

We pay a lot of attention these days, with good reason, to “artificial” machines and intelligences — ones constructed by human ingenuity. But the “natural” ones that have evolved through natural selection, like you and me, are still around. And one of the most exciting frontiers in technology and cognition is the increasingly permeable boundary between the two categories.

Artificial intelligence, unsurprisingly in retrospect, is a much more challenging field than many of its pioneers originally supposed. Human programmers naturally think in terms of a conceptual separation between hardware and software, and imagine that conjuring intelligent behavior is a matter of writing the right code. But evolution makes no such distinction. The neurons in our brains, as well as the bodies through which they interact with the world, function as both hardware and software. Roboticists have found that human-seeming behavior is much easier to model in machines when cognition is embodied. Give that computer some arms, legs, and a face, and it starts acting much more like a person.

From the other side, neuroscientists and engineers are getting much better at augmenting human cognition, breaking down the barrier between mind and (artificial) machine. We have primitive brain/computer interfaces, offering the hope that paralyzed patients will be able to speak through computers and operate prosthetic limbs directly.

What’s harder to predict is how connecting human brains with machines and computers will ultimately change the way we actually think. DARPA-sponsored researchers have discovered that the human brain is better than any current computer at quickly analyzing certain kinds of visual data, and developed techniques for extracting the relevant subconscious signals directly from the brain, unmediated by pesky human awareness. Ultimately we’ll want to reverse the process, feeding data (and thoughts) directly to the brain. People, properly augmented, will be able sift through enormous amounts of information, perform mathematical calculations at supercomputer speeds, and visualize virtual directions well beyond our ordinary three dimensions of space.

Where will the breakdown of the human/machine barrier lead us? Julien de La Mettrie, we are told, died at the young age of 41, after attempting to show off his rigorous constitution by eating an enormous quantity of pheasant pâte with truffles. Even leading intellects of the Enlightenment sometimes behaved irrationally. The way we think and act in the world is changing in profound ways, with the help of computers and the way we connect with them. It will be up to us to use our new capabilities wisely.

Posted in Technology | 141 Comments

Dark Matter, Explained

If you’ve ever wondered about dark matter, or been asked puzzled questions about it by your friends, now you have something to point to: this charming video by 11-year-old Lucas Belz-Koeling. (Hat tip Sir Harry Kroto.)

The title references “Draw My Life style,” which is (the internet informs me) a label given to this kind of fast-motion photography of someone drawing on a white board.

You go, Lucas. I doubt I would have been doing anything quite this good at that age.

Posted in Science | 38 Comments

A Simple Form of Poker “Essentially” Solved

You know it’s a good day when there are refereed articles in Science about poker. (Enthusiasm slightly dampened by the article being behind a paywall, but some details here.)

Poker, of course, is a game of incomplete information. You don’t know your opponent’s cards, they don’t know yours. Part of your goal should be to keep it that way: you don’t want to give away information that would let your opponent figure out what you have.

As a result, the best way to play poker (against a competent opponent) is to use a mixed strategy: in any given situation, you want to have different probabilities for taking various actions, rather than a deterministic assignment of the best thing to do. If, for example, you always raise with certain starting hands, and always call with others, an attentive player will figure that out, and thereby gain a great deal of information about your hand. It’s much better to sometimes play weak hands as if they are strong (bluffing) and strong hands as if they are weak (slow-playing). The question is: how often should you be doing that?

Now researchers at a University of Alberta group that studies computerized poker has offered and “essentially” perfect strategy for a very simple form of poker: Heads-Up Limit Hold’em. In Hold’em, each player has two “hole” cards face down, and there are five “board” cards face-up in the middle of the table; your hand is the best five-card combination you can form from your hole cards and the board. “Heads-up” means that only two players are playing (much simpler than a multi-player game), and “limit” means that there is any bet comes in a single pre-specified amount (much simpler than “no-limit,” where you can bet anything from a fixed minimum up to the size of your stack or your opponent’s, whichever is smaller).

A simple game, but not very simple. Bets occur after each player gets their hole cards, and again after three cards (the “flop”) are put on the board, again after a fourth card (the turn), and finally after the last board card (the river) is revealed. If one player bets, the other can raise, and then the initial better can re-raise, up to a number of bets (typically four) that “caps” the betting.


So a finite number of things can possibly happen, which makes the game amenable to computer analysis. But it’s still a large number. There are about 3×1017 “states” that one can reach in the game, where a “state” is defined by a certain number of bets having been made as well as the configuration of cards that have already been dealt. Not easy to analyze! Fortunately (or not), as a player with incomplete information you won’t be able to distinguish between all of those states — i.e. you don’t know your opponent’s hole cards. So it turns out that there are about 3×1014 distinct “decision points” from which a player might end up having to act.

So all you need to do is: for each of those 300 trillion possibilities, assign the best possible mixed strategy — your probability to bet/check if there hasn’t already been a bet, fold/call/raise if there has — and act accordingly. Hey, nobody ever said being a professional poker player would be easy. (As you might know, human beings are very bad at randomness, so many professionals use the second hand on a wristwatch to generate pseudo-random numbers and guide their actions.)

Nobody is going to do that, of course. Continue reading

Posted in Miscellany | 20 Comments

Life Is the Flame of a Candle

Emperor Has No Clothes Award Last October I was privileged to be awarded the Emperor Has No Clothes award from the Freedom From Religion Foundation. The physical trophy consists of the dashing statuette here on the right, presumably the titular Emperor. It’s made by the same company that makes the Academy Award trophies. (Whenever I run into Meryl Streep, she’s just won’t shut up about how her Oscars are produced by the same company that does the Emperor’s New Clothes award.)

Part of the award-winning is the presentation of a short speech, and I wasn’t sure what to talk about. There are only so many things I have to say, but it’s boring to talk about the same stuff over and over again. More importantly, I have no real interest in giving religion-bashing talks; I care a lot more about doing the hard and constructive work of exploring the consequences of naturalism.

So I decided on a cheerful topic: Death and Physics. I talked about modern science gives us very good reasons to believe (not a proof, never a proof) that there is no such thing as an afterlife. Life is a process, not a substance, and it’s a process that begins, proceeds along for a while, and comes to an end. Certainly something I’ve said before, e.g. in my article on Physics and the Immortality of the Soul, and in the recent Afterlife Debate, but I added a bit more here about entropy, complexity, and what we mean by the word “life.”

If you’re in a reflective mood, here it is. I begin at around 3:50. One of the points I tried to make is that the finitude of life has its upside. Every moment is precious, and what we should value is what is around us right now — because that’s all there is. It’s a scary but exhilarating view of the world.

Posted in Humanity, Philosophy, Religion, Science | 79 Comments

Guest Post: Chip Sebens on the Many-Interacting-Worlds Approach to Quantum Mechanics

Chip Sebens I got to know Charles “Chip” Sebens back in 2012, when he emailed to ask if he could spend the summer at Caltech. Chip is a graduate student in the philosophy department at the University of Michigan, and like many philosophers of physics, knows the technical background behind relativity and quantum mechanics very well. Chip had funding from NSF, and I like talking to philosophers, so I said why not?

We had an extremely productive summer, focusing on our different stances toward quantum mechanics. At the time I was a casual adherent of the Everett (many-worlds) formulation, but had never thought about it carefully. Chip was skeptical, in particular because he thought there were good reasons to believe that EQM should predict equal probabilities for being on any branch of the wave function, rather than the amplitude-squared probabilities of the real-world Born Rule. Fortunately, I won, although the reason I won was mostly because Chip figured out what was going on. We ended up writing a paper explaining why the Born Rule naturally emerges from EQM under some simple assumptions. Now I have graduated from being a casual adherent to a slightly more serious one.

But that doesn’t mean Everett is right, and it’s worth looking at other formulations. Chip was good enough to accept my request that he write a guest blog post about another approach that’s been in the news lately: a “Newtonian” or “Many-Interacting-Worlds” formulation of quantum mechanics, which he has helped to pioneer.

In Newtonian physics objects always have definite locations. They are never in two places at once. To determine how an object will move one simply needs to add up the various forces acting on it and from these calculate the object’s acceleration. This framework is generally taken to be inadequate for explaining the quantum behavior of subatomic particles like electrons and protons. We are told that quantum theory requires us to revise this classical picture of the world, but what picture of reality is supposed to take its place is unclear. There is little consensus on many foundational questions: Is quantum randomness fundamental or a result of our ignorance? Do electrons have well-defined properties before measurement? Is the Schrödinger equation always obeyed? Are there parallel universes?

Some of us feel that the theory is understood well enough to be getting on with. Even though we might not know what electrons are up to when no one is looking, we know how to apply the theory to make predictions for the results of experiments. Much progress has been made―observe the wonder of the standard model―without answering these foundational questions. Perhaps one day with insight gained from new physics we can return to these basic questions. I will call those with such a mindset the doers. Richard Feynman was a doer:

“It will be difficult. But the difficulty really is psychological and exists in the perpetual torment that results from your saying to yourself, ‘But how can it be like that?’ which is a reflection of uncontrolled but utterly vain desire to see it in terms of something familiar. I will not describe it in terms of an analogy with something familiar; I will simply describe it. … I think I can safely say that nobody understands quantum mechanics. … Do not keep saying to yourself, if you can possibly avoid it, ‘But how can it be like that?’ because you will get ‘down the drain’, into a blind alley from which nobody has yet escaped. Nobody knows how it can be like that.”

-Feynman, The Character of Physical Law (chapter 6, pg. 129)

In contrast to the doers, there are the dreamers. Dreamers, although they may often use the theory without worrying about its foundations, are unsatisfied with standard presentations of quantum mechanics. They want to know “how it can be like that” and have offered a variety of alternative ways of filling in the details. Doers denigrate the dreamers for being unproductive, getting lost “down the drain.” Dreamers criticize the doers for giving up on one of the central goals of physics, understanding nature, to focus exclusively on another, controlling it. But even by the lights of the doer’s primary mission―being able to make accurate predictions for a wide variety of experiments―there are reasons to dream:

“Suppose you have two theories, A and B, which look completely different psychologically, with different ideas in them and so on, but that all consequences that are computed from each are exactly the same, and both agree with experiment. … how are we going to decide which one is right? There is no way by science, because they both agree with experiment to the same extent. … However, for psychological reasons, in order to guess new theories, these two things may be very far from equivalent, because one gives a man different ideas from the other. By putting the theory in a certain kind of framework you get an idea of what to change. … Therefore psychologically we must keep all the theories in our heads, and every theoretical physicist who is any good knows six or seven different theoretical representations for exactly the same physics.”

-Feynman, The Character of Physical Law (chapter 7, pg. 168)

In the spirit of finding alternative versions of quantum mechanics―whether they agree exactly or only approximately on experimental consequences―let me describe an exciting new option which has recently been proposed by Hall, Deckert, and Wiseman (in Physical Review X) and myself (forthcoming in Philosophy of Science), receiving media attention in: Nature, New Scientist, Cosmos, Huffington Post, Huffington Post Blog, FQXi podcast… Somewhat similar ideas have been put forward by Böstrom, Schiff and Poirier, and Tipler. The new approach seeks to take seriously quantum theory’s hydrodynamic formulation which was developed by Erwin Madelung in the 1920s. Although the proposal is distinct from the many-worlds interpretation, it also involves the postulation of parallel universes. The proposed multiverse picture is not the quantum mechanics of college textbooks, but just because the theory looks so “completely different psychologically” it might aid the development of new physics or new calculational techniques (even if this radical picture of reality ultimately turns out to be incorrect).

Let’s begin with an entirely reasonable question a dreamer might ask about quantum mechanics.

“I understand water waves and sound waves. These waves are made of particles. A sound wave is a compression wave that results from particles of air bunching up in certain regions and vacating other. Waves play a central role in quantum mechanics. Is it possible to understand these waves as being made of some things?”

There are a variety of reasons to think the answer is no, but they can be overcome. In quantum mechanics, the state of a system is described by a wave function Ψ. Consider a single particle in the famous double-slit experiment. In this experiment the one particle initially passes through both slits (in its quantum way) and then at the end is observed hitting somewhere on a screen. The state of the particle is described by a wave function which assigns a complex number to each point in space at each time. The wave function is initially centered on the two slits. Then, as the particle approaches the detection screen, an interference pattern emerges; the particle behaves like a wave.

Figure 1: The evolution of Ψ with the amount of color proportional to the amplitude (a.k.a. magnitude) and the hue indicating the phase of Ψ.

Figure 1: The evolution of Ψ with the amount of color proportional to the amplitude (a.k.a. magnitude) and the hue indicating the phase of Ψ.

There’s a problem with thinking of the wave as made of something: the wave function assigns strange complex numbers to points in space instead of familiar real numbers. This can be resolved by focusing on |Ψ|2, the squared amplitude of the wave function, which is always a positive real number.

Figure 2: The evolution of |Ψ|2.

Figure 2: The evolution of |Ψ|2.

We normally think of |Ψ|2 as giving the probability of finding the particle somewhere. But, to entertain the dreamer’s idea about quantum waves, let’s instead think of |Ψ|2 as giving a density of particles. Whereas figure 2 is normally interpreted as showing the evolution of the probability distribution for a single particle, instead understand it as showing the distribution of a large number of particles: initially bunched up at the two slits and later spread out in bands at the detector (figure 3). Although I won’t go into the details here, we can actually understand the way that wave changes in time as resulting from interactions between these particles, from the particles pushing each other around. The Schrödinger equation, which is normally used to describe the way the wave function changes, is then viewed as consequence of this interaction.

Figure 3: The evolution of particles with |Ψ|2 as the density. This animation is meant to help visualize the idea, but don’t take the precise motions of the particles too seriously. Although we know how the particles move en masse, we don’t know precisely how individual particles move.

Figure 3: The evolution of particles with |Ψ|2 as the density. This animation is meant to help visualize the idea, but don’t take the precise motions of the particles too seriously. Although we know how the particles move en masse, we don’t know precisely how individual particles move.

In solving the problem about complex numbers, we’ve created two new problems: How can there really be a large number of particles if we only ever see one show up on the detector at the end? If |Ψ|2 is now telling us about densities and not probabilities, what does it have to do with probabilities?

Removing a simplification in the standard story will help. Instead of focusing on the wave function of a single particle, let’s consider all particles at once. To describe the state of a collection of particles it turns out we can’t just give each particle its own wave function. This would miss out on an important feature of quantum mechanics: entanglement. The state of one particle may be inextricably linked to the state of another. Instead of having a wave function for each particle, a single universal wave function describes the collection of particles.

The universal wave function takes as input a position for each particle as well as the time. The position of a single particle is given by a point in familiar three dimensional space. The positions of all particles can be given by a single point in a very high dimensional space, configuration space: the first three dimensions of configuration space give the position of particle 1, the next three give the position of particle 2, etc. The universal wave function Ψ assigns a complex number to each point of configuration space at each time.  |Ψ|2 then assigns a positive real number to each point of configuration space (at each time). Can we understand this as a density of some things?

A single point in configuration space specifies the locations of all particles, a way all things might be arranged, a way the world might be. If there is only one world, then only one point in configuration space is special: it accurately captures where all the particles are. If there are many worlds, then many points in configuration space are special: each accurately captures where the particles are in some world. We could describe how densely packed these special points are, which regions of configuration space contain many worlds and which regions contain few. We can understand |Ψ|2 as giving the density of worlds in configuration space. This might seem radical, but it is the natural extension of the answer to the dreamer’s question depicted in figure 3.

Now that we have moved to a theory with many worlds, the first problem above can be answered: The reason that we only see one particle hit the detector in the double-slit experiment is that only one of the particles in figure 3 is in our world. When the particles hit the detection screen at the end we only see our own. The rest of the particles, though not part of our world, do interact with ours. They are responsible for the swerves in our particle’s trajectory. (Because of this feature, Hall, Deckert, and Wiseman have coined the name “Many Interacting Worlds” for the approach.)

Figure 4: The evolution of particles in figure 3 with the particle that lives in our world highlighted.

Figure 4: The evolution of particles in figure 3 with the particle that lives in our world highlighted.

No matter how knowledgeable and observant you are, you cannot know precisely where every single particle in the universe is located. Put another way, you don’t know where our world is located in configuration space. Since the regions of configuration space where |Ψ|2 is large have more worlds in them and more people like you wondering which world they’re in, you should expect to be in a region of configuration space where|Ψ|2 is large. (Aside: this strategy of counting each copy of oneself as equally likely is not so plausible in the old many-worlds interpretation.) Thus the connection between |Ψ|2 and probability is not a fundamental postulate of the theory, but a result of proper reasoning given this picture of reality.

There is of course much more to the story than what’s been said here. One particularly intriguing consequence of the new approach is that the three sentence characterization of Newtonian physics with which this post began is met. In that sense, this theory makes quantum mechanics look like classical physics. For this reason, in my paper I gave the theory the name “Newtonian Quantum Mechanics.”

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