Over on the Google+, I linked to an informal essay by John Norton, in which he recounts the activities of a workshop on QFT at the Center for the Philosophy of Science at the University of Pittsburgh last October. In Norton’s telling, the important conceptual divide was between those who want to study “axiomatic” QFT on the one hand, and those who want to study “heuristic” QFT on the other. Axiomatic QFT is an attempt to make everything absolutely perfectly mathematically rigorous. It is severely handicapped by the fact that it is nearly impossible to get results in QFT that are both interesting and rigorous. Heuristic QFT, on the other hand, is what the vast majority of working field theorists actually do — putting aside delicate questions of whether series converge and integrals are well defined, and instead leaping forward and attempting to match predictions to the data. Philosophers like things to be well-defined, so it’s not surprising that many of them are sympathetic to the axiomatic QFT program, tangible results be damned.

The question of whether or not the interesting parts of QFT can be made rigorous is a good one, but not one that keeps many physicists awake at night. All of the difficulty in making QFT rigorous can be traced to what happens at very short distances and very high energies. And that’s certainly important to understand. But the great insight of Ken Wilson and the effective field theory approach is that, as far as particle physics is concerned, it just doesn’t matter. Many different things can happen at high energies, and we can still get the same low-energy physics at the end of the day. So putting great intellectual effort into “doing things right” at high energies might be misplaced, at least until we actually have some data about what is going on there.

Something like that attitude is defended here by our former guest blogger David Wallace. (Hat tip to Cliff Harvey on G+.) Not the best video quality, but here is David trying to convince his philosophy colleagues to concentrate on “Lagrangian QFT,” which is essentially what Norton called “heuristic QFT,” rather than axiomatic QFT. His reasoning very much follows the Wilsonian effective field theory approach.

The concluding quote says it all:

LQFT is the most successful, precise scientific theory in human history. Insofar as philosophy of physics is about drawing conclusions about the world from our best physical theories, LQFT is the place to look.

Working out the numbers, the chances of 14 coin flips in a row being equal is 1 in 8,192. (The linked article says 1 in 16,000, which comes from 2^14; but that first coin flip has to be something, so the chances of 14 in a row are really 1 in 2^13. The anomaly would be just as strange if the AFC had won every time.) That’s a better than 3.8-sigma effect! Enough to call a press conference, if this were particle physics.

The question is … is this really a signal, or did we just get lucky? Is it a fair coin and the NFC has just been the happy recipient of a statistical fluctuation, or is there something fishy about the coin? Remember Barry Greenstein’s parable about how different people compute probabilities.

And let it be a lesson the next time you’re excited about 3-sigma anomalies.

I was invited to contribute, but wasn’t feeling very imaginative, so I moved quickly and picked one of the most obvious elegant explanations of all time: Einstein’s explanation for the universality of gravitation in terms of the curvature of spacetime. Steve Giddings and Roger Highfield had the same idea, although Steve rightly points out that Einstein won’t really end up having the final word on spacetime. Lenny Susskind picks Boltzmann’s explanation of why entropy increases as his favorite explanation, and mentions the puzzle of why entropy was lower in the past as his favorite unsolved problem — couldn’t have said it better myself. For those of you how prefer a little provocation, Martin Rees picks the anthropic principle.

But as usual, the most interesting responses to me are those from far outside physics. What’s your favorite?

Full text of my entry below the fold.

Einstein Explains Why Gravity Is Universal

The ancient Greeks believed that heavier objects fall faster than lighter ones. They had good reason to do so; a heavy stone falls quickly, while a light piece of paper flutters gently to the ground. But a thought experiment by Galileo pointed out a flaw. Imagine taking the piece of paper and tying it to the stone. Together, the new system is heavier than either of its components, and should fall faster. But in reality, the piece of paper slows down the descent of the stone.

Galileo argued that the rate at which objects fall would actually be a universal quantity, independent of their mass or their composition, if it weren’t for the interference of air resistance. Apollo 15 astronaut Dave Scott once illustrated this point by dropping a feather and a hammer while standing in vacuum on the surface of the Moon; as Galileo predicted, they fell at the same rate.

Subsequently, many scientists wondered why this should be the case. In contrast to gravity, particles in an electric field can respond very differently; positive charges are pushed one way, negative charges the other, and neutral particles not at all. But gravity is universal; everything responds to it in the same way.

Thinking about this problem led Albert Einstein to what he called “the happiest thought of my life.” Imagine an astronaut in a spaceship with no windows, and no other way to peer at the outside world. If the ship were far away from any stars or planets, everything inside would be in free fall, there would be no gravitational field to push them around. But put the ship in orbit around a massive object, where gravity is considerable. Everything inside will still be in free fall: because all objects are affected by gravity in the same way, no one object is pushed toward or away from any other one. Sticking just to what is observed inside the spaceship, there’s no way we could detect the existence of gravity.

Einstein, in his genius, realized the profound implication of this situation: if gravity affects everything equally, it’s not right to think of gravity as a “force” at all. Rather, gravity is a feature of spacetime itself, through which all objects move. In particular, gravity is the curvature of spacetime. The space and time through which we move are not fixed and absolute, as Newton would have had it; they bend and stretch due to the influence of matter and energy. In response, objects are pushed in different directions by spacetime’s curvature, a phenomenon we call “gravity.” Using a combination of intimidating mathematics and unparalleled physical intuition, Einstein was able to explain a puzzle that had been unsolved since Galileo’s time.

We’ve discussed this question before, and the idea of reproduction looms large in many people’s definitions of life. But I don’t think it really belongs. If you built an organism from scratch, that was as complicated and organic and lifelike as any living thing currently walking this Earth, except that it had no reproductive capacity, it would be silly to exclude it from “life” just because it was non-reproducing. Even worse, I realized that I myself wouldn’t even qualify as alive under Trifonov’s definition, since I don’t have kids and don’t plan on having any. (And no, those lawsuits were frivolous and the court records were sealed.)

It’s the yellow-taxi problem: in a city where all cars are blue except for taxis, which are yellow, it’s tempting to define “taxi” as “a yellow car.” But that doesn’t get anywhere near the essence of taxi-ness. Likewise, living species generally reproduce themselves; but that’s not really what makes them alive. Not that I have the one true definition (and maybe there shouldn’t be one). But any such definition better capture the idea of an ongoing complex material process far from equilibrium, or it’s barking up the wrong Tree.

Stephen Hawking is celebrating his 70th birthday today. That in itself is an amazing fact, just as it was amazing when he celebrated his 40th, and 50th, and 60th birthdays, as well as every other day he’s lived and thrived with a debilitating neuron disease. The extra fact that he continues to make contributions to science pushes beyond amazing to practically unbelievable.

Everyone likes to tell Hawking stories, and this blog is no exception. So here is mine, meagre as it is. I’ve gotten more than enough mileage out of this one in person, I might as well put it on the blog so I won’t be tempted to tell it any more.

At the end of 1992 I was a finishing grad student, applying for postdocs. One of the places I applied was Cambridge, to Hawking’s group at DAMTP. There is a slight potential barrier for American students to travel to the UK for postdocs, so they like to get out ahead of things and offer jobs early. Unfortunately I was out of my office the day Hawking called to offer me a position. Fortunately, my future-Nobel-Laureate officemate was there, and he took the call. He explained that Stephen Hawking had called to offer me a job — I was thrilled about the offer, but understood “Hawking called” as metaphorical. But no, Brian later convinced me that it actually was Hawking on the other end of the line, which he described as a somewhat surreal experience. Of course after the initial introduction the phone gets handed over to someone else, but still.

Cambridge is one of the world’s best places to do theoretical physics, and I was sorely tempted, but I ended up going to MIT instead. Three years later, I went through the process again, as postdocs typically do. And again Cambridge offered me the job — and again, after a very tough decision, I said no, heading of the the ITP in Santa Barbara instead.

Up to this point I had never actually met Hawking in person, although I had been in the audience for one of his lectures. But every year he visits Caltech and Santa Barbara, so I finally got to be with him in the same place. The first time he visited he brought along a young grad student named Raphael Bousso, who has gone on to do quite well for himself in his own right. As a group of us went to lunch, I mentioned to Raphael that I had never said hi to Stephen in person, so I’d appreciate it if he would introduce us. But, I cautioned, I hope he wasn’t upset with me, because he had offered me a postdoc and I turned it down.

Raphael just laughed and said, “Don’t worry, there’s this one guy who he offered a postdoc to twice, and he turned it down both times!” So I had to explain that this guy was actually me. At which point Raphael ran up to Hawking, exclaiming “Stephen! Stephen, this is the guy — the one who turned down DAMTP for postdocs twice in a row!”

That was my personal introduction to Stephen. He just smiled, no big deal — life goes on for him whether or not some callow American student wants to fly across the puddle to work as a postdoc.

Since then I’ve had the privilege of interacting with Hawking more substantively a few times. Once a long conversation just after the discovery of the acceleration of the universe, when he was interested in hearing more about the supernova observations. And once at a whisky tasting organized at an international cosmology conference. Handicaps notwithstanding, Hawking never misses a chance to experience life to its fullest. Another time when I picked up him and his retinue at the airport — which gave me a tiny glimpse of the massive logistical operation it is to move Hawking from place to place. The simplest things that we take for granted are for him an elaborate production.

Happy birthday, Stephen. I know I won’t make the contributions you have to science, but I hope I can live as long, and approach life with your gusto and good humor.

The machinery can’t force Prof Haggard to do anything really complicated – “You can’t make me sign my name,” he says, almost ruefully – but at one point, Christina is able to waggle his index finger slightly, like a schoolmaster. It’s very fine control, a part of the brain specifically in command of a part of the body. “There’s quite a detailed map of the brain’s wiring to the body that you can build,” he tells me.

We sometimes say “the Large Hadron Collider is the most complex machine ever built,” but I’m not sure how it would directly compare to a human being. All part of the great bootstrap up to greater complexity, which will continue for a while until it all inevitably deteriorates into empty space.

Tuesday December 13 has been a very exciting day for particle physics. The ATLAS and CMS experiments at the Large Hadron Collider (LHC) announced today that they are both seeing hints of a Higgs boson with properties that are close to those expected for the Standard Model (SM) Higgs boson as originally proposed by Peter Higgs and others. While the “significance” of the signals has not yet reached “discovery level” (5 sigma in technical language) the two experiments both see signals that exceed 2 sigma so that there is less than a 5% chance that they are simply statistical fluctuations. Most persuasively, the signals in the channels with excellent mass determination (the photon-photon final decay state and the 4-lepton final state) are all consistent with a a Higgs boson mass of around 125 GeV IN BOTH EXPERIMENTS. This coincidence in mass between two totally independent experiments (as well as independent final states) is persuasive evidence that the photon-photon and 4-lepton excesses seen near 125 GeV are not mere statistical fluctuations.

Observation of the Higgs with approximately the SM-like rate suggests that to first approximation the Higgs is being produced as expected in the SM and that it also decays as predicted in the SM. Many theorists, including myself, have suggested that a Higgs might be produced as in the SM but might have extra decays that would have decreased the photon-photon and 4-lepton decay frequencies to an unobservable level, making the Higgs boson much harder to detect at the LHC. The level of the observed excesses argues against such extra decays being very important. The photon-photon and 4-lepton detection modes were originally proposed and shown to be viable for a SM-like Higgs boson by myself and collaborators (in particular, Gordy Kane and Jose Wudka) way back in 1986-1987. It has taken a long time (25 years) for the technology and funding to reach the point where these detection modes could be examined. I often joked that I was personally responsible for forcing each of the LHC collaborations to spend the 30 million dollars or so needed to build a photon detector with the energy resolution required. Fortunately, it seems that the money was well-spent and the ATLAS and CMS detectors both found ways to build the needed detectors, a real triumph of international collaboration and technical expertise. Also key is the very successful operation of the LHC that has produced the enormously large number of collision events needed to dig out the Higgs signal from uninteresting ‘background’ events. Until this summer produced the first very weak signs of the Higgs, I was beginning to wonder if the Higgs would be discovered during my lifetime. Fortunately, simplicity (i.e. a very conventional SM-like Higgs boson) seems to have prevailed and ended my wait.

Going forward, by the end of 2012 the levels of these excesses should reach the 5 sigma “discovery” level if the SM-like Higgs really does have the mass and decays indicated by current results. Further, we will begin to have some moderately precise (20%-30% or so?) measurements of the individual decay modes of the Higgs boson that might indicate just how precisely SM-like it is. Many theories beyond the Standard Model predict the possibility of deviations from the predictions of the purely SM Higgs boson. As data accumulate, looking for such deviations will be a major focus. Current data (weakly) hint at the possibility that the Higgs production rate might turn out to be modestly larger than predicted if the Higgs is that of the Standard Model and has mass of 125 GeV — the best-fit ATLAS cross section is about 1.5 times as large as the SM prediction, whereas the best-fit CMS cross section is very close to the SM prediction. Time (i.e. more accumulated data) will tell.

Of course, the current run of the LHC will be halted at the end of 2012, followed by a lengthy shut down for upgrades to the accelerator and to the detectors. After this upgrade, the LHC will operate at a much higher energy (14 TeV) compared to the current energy of 7 TeV, and, if all goes according to plan, have a much higher collision rate. At this point, precision studies of the Higgs boson will certainly be possible. If deviations are observed, then we will strongly suspect that the SM is incomplete. Even before that time, we are hoping that by the end of 2012 we will have seen new types of particles that do not fit into the Standard Model. This, for example, is predicted if the universe is supersymmetric. Supersymmetric models tend to predict a light Higgs boson that is fairly, but not completely, SM-like with mass in the range 110 GeV to 140 GeV and so are very consistent with what is being observed. If some supersymmetric particles (so called sparticles) are observed then their masses and properties constrain the Higgs mass in a given model and consistency of the entire theory can be nicely tested. Hopefully, some version of physics beyond the Standard Model will be directly observed at the LHC, in which case there will be at least a decade of exciting observations and analyses to determine the precise beyond-the-Standard-Model theory.

Still, the pure Standard Model with no new physics cannot be totally discarded. Although this would not allow a so-called “natural” explanation of the Z boson and Higgs boson masses, the pure SM is still internally consistent even at energies close to the Planck scale for a Higgs mass greater than or equal to about 125-130 GeV. In other words, the observed Higgs mass is on a borderline. Above 125-130 GeV, new physics below the Planck mass scale is not required for internal consistency of the SM. But, had the Higgs boson mass been significantly below this (the precise border being somewhat uncertain theoretically), the SM would necessarily break down at some energy scale below the Planck scale and at this lower energy scale new physics would have to enter. In short, a SM-like Higgs boson with mass near 125 GeV is maximally interesting from many theoretical perspectives.

In any case, theorists and experimentalists are all very relieved that the LHC appears to be observing a Higgs boson thereby ensuring an extremely interesting program of physics at the LHC for decades to come. Further, such a light SM-like Higgs boson provides strong motivation for a linear electron-positron collider of low center-of-mass energy. Studies suggest that only such a collider can easily measure the properties of such a light Higgs boson at the few percent level, although the LHC might not do that much worse depending upon future improvements and upgrades.

It’s like rushing to the tree on Christmas morning, ripping open a giant box, and finding a small note that says “Santa is on his way! Hang in there!” The LHC is real and Santa is not, but you know what I mean.

Here are the technical write-ups from ATLAS and CMS. For stories and some live-blogs, check out Philip Gibbs, Matt Strassler, Aidan Randle-Conde, Ken Bloom, or Jester. Or if you just want the bottom line sigmas, Jim Rohlf provides them. ATLAS gives 3.6 sigma local significance, 2.3 sigma global significance; CMS gives 2.6 sigma local significance, 1.9 sigma global significance (although CMS points to about 124 GeV, while ATLAS points to about 126, which might be important). The difference between “local” and “global” is that the first asks “if I were only looking at this one point in parameter space, how surprising would the result be?”, while the latter asks “what is the chance I would find this kind of deviation somewhere in parameter space?” Nominally the global significance is obviously more relevant, although one could argue that we have good reasons to expect that the Higgs is actually lurking right there, so the local significance isn’t completely cheating.

Let’s put it this way: if we were testing a theory that everyone thought was wrong, rather than one that everyone thinks is right, nobody would take these results as strong indications that the idea was correct. We have a strong theoretical bias that the Higgs exists and is somewhere close to this mass range, so it’s completely reasonable to think that we are seeing hints (tantalizing ones!) that it’s there, but wait-and-see is still the right attitude.

Here are the simplest plots I could find. First the full analysis from ATLAS (zoomed in on the interesting region), via Philip Gibbs’s blog.:

Then from CMS, via Ken Bloom:

These plots are complicated because they’re trying to tell you two things at once. The black curve is the data, the green/yellow bands are the expected ranges of the data at 1 sigma and 2 sigma. If all you want to do is ask whether we can exclude the Higgs in a certain range, just check whether the black band is below the value 1. But if you want to say you have evidence for the Higgs, you need the black line to wander above the yellow band (or higher, if you want more than 2 sigma [and you do]). So ATLAS sees something at 126 GeV, CMS is at least consistent with 123-124 GeV (although it doesn’t see much at 126).

As Sarah Kavassalis puts it, the real message today is that the LHC is working great. 2012 will bring another year of data, hopefully at even higher luminosity (so many more total events). The Higgs has been around for 13.7 billion years, it will still be there tomorrow.

If you want to know why it’s exciting, after you’ve read John’s description of life in the trenches and Matt Strassler’s post about the multiple stages of hunting the Higgs and mine about why we need something like it, see even more recent posts by Matt, Jester, and Pauline Gagnon. Reader’s Digest version: not only are we being updated on the status of the search, there are believable rumors that the searches are actually seeing something — hints of a Higgs near 125 GeV, with better than 3-sigma significance from ATLAS and better than 2-sigma significance from CMS. But obviously rumors are no match for what actually happens.

All I’m here to tell you is: you should not expect to hear anyone announcing that we have discovered the Higgs boson. This will, at best, be a hint — “evidence for” something, not “discovery of” that thing. The collaborations realistically can’t claim to have actually discovered the Higgs, even if it’s there — they don’t have enough data. (CERN even issued a press release to drive home the point.) And in the real world, hints are sometimes misleading. That is: the experimenters will give us their absolute best judgment about what they are seeing, but at this stage of the game that judgment is necessarily extremely preliminary. If they say “we have 3.5-sigma evidence, which is quite suggestive,” do not think that they are just being coy and what they really mean is “oh, we know it’s there, we just have to follow the protocols.” The protocols are there for a reason! Mostly, that many 3-sigma findings eventually go away. This is one step on a journey, not the culmination of anything. (For Americans out there: it’s like a bill has been passed by the House, but not yet passed by the Senate, and certainly not signed by the President. Much can go wrong along the way.)

The journey of a thousand miles begins with a single step. It’s possible that tomorrow’s announcement means that we’re nearing the end of the journey, say at the mile-990 marker. But we can’t be sure, and there are no royal roads to particle physics. Patience! The excitement of not knowing for sure is what makes science one of the most compelling human stories.

But, following on Matt Strassler’s excellent post about the physics, I thought it might be interesting to tell you what it’s been like this past year getting to this stage in this search. As you probably know, each of the two big experiments has over 3000 physicists participating, from all over the world. Many, but by no means the majority, are resident at CERN; most are at their home institutions in Europe, North America, and Asia and elsewhere.

The main thing that allows us to collaborate on a global scale like this is video conferencing. We used a system called EVO, developed at Caltech, which allows us to schedule meetings and connect to them from a laptop or desktop computer, or even dial in by phone. Sometimes it’s clear that people are connected by phone from the oddest places: once I heard the clear sounds of someone participating in the meeting from a train ride in Italy, oftentimes you hear people speak while they are driving the (hopefully with a hands-free device), and often one hears the sounds of children in the background (including my own). The issue is that meetings can be at any time of day for different people in different continents. Fortunately the experiments have gravitated toward having meetings in the late afternoon, Europe time, which makes it early morning for people like me in California.

A good thing about our videoconferencing system is that you actually have a choice whether or not to transmit or receive the video part of the meeting, which tends to be less useful than looking at material under discussion, which is usually in the form of PowerPoint slides. That makes it even easier to participate early in the morning! No one has to see you in your pajamas, and they probably don’t want to. I did it this morning, in fact, at 6 am.

So what are all these meetings? In CMS, our whole system of producing physics results has a sort of pyramidal structure. Each experiment has a number of physics analysis groups which meet a weekly or biweekly, typically, and have two “conveners” who set the agenda and run the meetings. These convener positions are typically held by senior people in the collaboration such as professors or senior lab scientists, for two years at a stretch, one convener changing out each year. They report to an overall physics coordinator and his or her deputies. Within the physics analysis groups are subgroups devoted to sets of analyses which share common themes, common tools, or similar approaches. Each of these subgroups in turn is led by a pair of conveners who establish the ongoing analyses and guide them to eventual approval within physics analysis group.

We have what I think is a pretty impressive internal website devoted to tracking the progress of each physics analysis. From a single website you can drill down into a particular physics group find the analysis you want get links to all the documentation, and follow what’s happening. In parallel, there is a web system for recording the material presented at every meeting.

The goal of every analysis is to be approved by its physics group, so it can be shown in public at conferences and seminars. This requires having complete documentation including internal notes with full details of the analysis, and a “public analysis summary” which is available to the public, and which often serves as the basis for a peer–reviewed paper which soon follows.

Every analysis is assigned an analysis review committee of three to five people with experience in the topic, who act as a sort of hit squad, keeping the analyzers on their toes with questions and comments at every stage of the analysis, both on the actual analysis details and on the documentation. After all, if we are not our own worst critics, someone else will gladly fill the role!

The process from initially recording data from proton–proton collisions to ultimate physics results can take months. By now the basic algorithms which are run on every collision in order to reconstruct what happened are well-established. But during the year the running conditions of the accelerator changed with ever–increasing rates of proton–proton collisions happening. In every 25 nanosecond “bunch crossing”, by the end of running this year we were recording an average of up to 10 proton–proton collisions. Typically only one of these is of interest and the rest are “minimum bias” events in which the protons strike glancing blows. Nevertheless, these additional interactions caused us a lot of trouble this year because they result in additional energy recorded by the detectors, additional charged tracks, and skew various quantities which we are trying to measure in each collision. This was one of the major challenges of 2011.

In parallel with processing the data that we record, we run full simulations of well–known standard model collision processes which represent our background when we are doing searches for new particles. There is a big organizational challenge in doing these simulations, which run on a worldwide grid of computers devoted to CMS data analysis. We make use of the Open Science Grid for this in the US, the EuroGrid in Europe, and other clusters scattered all around the world, comprising tens of thousands of computing nodes.

The basic idea of any new particle search is simple: you make a selection which retains as many collision events potentially coming from the new particle, while retaining as few background events as possible. Then you predict the number of background events from well-known processes, and see if any excess remains. Almost all analyses these days use the distribution of some final quantity (such as the estimated mass of the new particle) to look for these excesses. At this point one can then use statistical techniques to estimate the largest contribution that could be possible given the observed spectrum, or if there is an access, calculate the probability that the background alone could give rise to such an excess. This is how we quote the statistical significance of the results.

The details, though, boggle the mind. The graphic here shows the complexity of the statistical procedure for correctly keeping track of all the little correlations that can occur among and between the search channels. (This graphic is courtesy Kyle Cranmer of NYU, one of the main Higgs combiners in the ATLAS experiment.)

A great deal of our meetings is devoted to studying the level of agreement between our observed spectra and predicted spectra based on full simulation or clever techniques using the actual observed data to predict the background. In the case of the search for the Higgs boson, there are a couple dozen “channels” in which to search, which reflect how the Higgs is produced and how it decays. The results from these individual channels are then combined into one final statistical analysis which essentially answers the question: is there evidence of Higgs boson production if the Higgs masses thus-and-such a value? What will be presented next week at CERN is in fact the result of that analysis and as much detail as possible about the results feeding into the answer.

This year has seen a dramatic leap in our knowledge about where the Higgs boson isn’t, and as of a few weeks ago the combined results from CMS And ATLAS left open only a small mass window from 115–140 GeV where it could exist. As luck would have it, the remaining mass region is the most difficult to explore with the LHC, but it has been clear for some time that with the data set anticipated this year, and now recorded and analyzed, the LHC could close the window even further, and perhaps all the way after combining the results from both experiments and with the data from the Tevatron. But the window will only close completely if the Higgs is not there.

For a while, earlier this fall, there was rampant speculation in the science media about the possibility that “there is no Higgs boson” but as the allowed mass window has shrunk, it’s shrunk down right down to the region where we would expect the Higgs boson to exist, if it does. So we shouldn’t give up yet! We’ve known it will take a lot more data to establish the existence of the Higgs boson at the golden five sigma level and begin to measure its mass, etc., but by next summer I think the it should be clear one way or the other.

My own role in this whole process started years ago when I worked with my students and postdoc to create a new algorithms for identifying tau lepton decays (the tau is the heaviest partner of the electron), and helped develop new methods for calculating the Higgs boson mass from its decays to pairs of taus. By last year, before we had an appreciably large sample of physics data, we had established within the physics analysis groups the methods we wanted to deploy in this search. We teamed up with groups from other institutions and, a year ago, another professor (Sridhara Dasu from University of Wisconsin-Madison) and I led a team of about 10 students and postdocs in getting the first version of this analysis through the full process. It took months, but we eventually published the results in is a Physical Review Letter in the spring, as the LHC started to deliver much higher luminosity.

With new data in hand, we “turned the crank” on the same analysis, more or less, for the summer conferences adding a few embellishments, and then improved it again this fall. This pattern was repeated in parallel by a dozen other teams in the Higgs search. I would reckon there are at least 200 people involved in the search in a serious way in CMS. It’s been more of a marathon than a sprint for all concerned, and now my former student is now a postdoc at Wisconsin and my former postdoc is now a scientist at a Ecole Polytechnique in France. Our analysis group, I can tell you, has some of the most talented physicists with whom I’ve ever had the privilege to work. For me, that’s one of the great joys of being in this field: you are surrounded by really smart people.

It will be interesting to see how the media spin the results that emerge next week. Physicists still smart from the sting of an article in the New York Times back in 1992 with the title “300 Physicists Fail to Find Supersymmetry” and have become much more media-savvy.

If you believe the rumors, then perhaps a more apt metaphor is that of a tiny, growing new plant, two leaves reaching above the soil. With more water and light, it will grow. And grow. And grow.

So we’re very happy to welcome a guest post by Matt Strassler, who is an expert particle theorist, to help explain what’s at stake and where the search for the Higgs might lead. Matt has made numerous important contributions, from phenomenology to string theory, and has recently launched the website Of Particular Significance, aimed at making modern particle physics accessible to a wide audience. Go there for a treasure trove of explanatory articles, growing at an impressive pace.

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After this year’s very successful run of the Large Hadron Collider (LHC), the world’s most powerful particle accelerator, a sense of great excitement is beginning to pervade the high-energy particle physics community. The search for the Higgs particle… or particles… or whatever appears in its place… has entered a crucial stage.

We’re now deep into Phase 1 of this search, in which the LHC experiments ATLAS and CMS are looking for the simplest possible Higgs particle. This unadorned version of the Higgs particle is usually called the Standard Model Higgs, or “SM Higgs” for short. The end of Phase 1 looks to be at most a year away, and possibly much sooner. Within that time, either the SM Higgs will show up, or it will be ruled out once and for all, forcing an experimental search for more exotic types of Higgs particles. Either way, it’s a turning point in the history of our efforts to understand nature’s elementary laws.

This moment has been a long time coming. I’ve been working as a scientist for over twenty years, and for a third decade before that I was reading layperson’s articles about particle physics, and attending public lectures by my predecessors. Even then, the Higgs particle was a profound mystery. Within the Standard Model (the equations that used at the LHC to describe all the particles and forces of nature we know about so far, along with the SM Higgs field and particle) it stood out as a bit different, a bit ad hoc, something not quite like the others. It has always been widely suspected that the full story might be more complicated. Already in the 1970s and 1980s there were speculative variants of the Standard Model’s equations containing several types of Higgs particles, and other versions with a more complicated Higgs field and no Higgs particle — with a key role of the Higgs particle being played by other new particles and forces.

But everyone also knew this: you could not simply take the equations of the Standard Model, strip the Higgs particle out, and put nothing back in its place. The resulting equations would not form a complete theory; they would be self-inconsistent. Though still effective in many contexts, they would become useless for predicting certain high-energy processes, including ones that the LHC, a few years from now, will directly study. So it was widely known, over thirty years ago, that something like a Higgs particle had to be in those equations to make them sensible. In fact, the condition is even stronger than that. The equations of the Standard Model will require significant and historic modifications unless the Standard Model Higgs particle is found with a mass below about 800 GeV/c². (For scale, the mass of a hydrogen atom is about 1 GeV/c².)

It’s this last point that explains why the current moment is such a critical one. Sure, previous experiments have looked for the Higgs particle too. And they were able to sweep some areas clean; we know, from these experiments, that the mass of the SM Higgs cannot lie below 115 GeV/c². But the LHC is special; it is the first accelerator capable of finding the SM Higgs over the entire allowed mass range still remaining, from 115 up to and beyond 800 GeV/c².

Phase 1, the search for the SM Higgs, is the easy part of the quest for the Higgs particle (or particles or whatever). What makes it easy? The Standard Model equations are so detailed and well-specified that everything about the SM Higgs particle is already known, except for one thing: its mass. More precisely, if you told me the SM Higgs particle’s mass, I could tell you how it is produced at the LHC and at what rate, and what it decays to, and how often it decays to one set of particles rather than another. This makes life relatively simple for the experimenters at ATLAS and CMS, because all they have to do is this: pick a mass in the allowed range, ask theorists to calculate the properties of an SM Higgs particle of that mass, figure out the best way to seek it in their data, and look at the data: is there evidence for or against its presence? They must then repeat this across the entire range of possible masses systematically, until they’ve covered all the allowed territory. (Actually they do all of these searches simultaneously, not sequentially.) When their coverage is complete — or when they find something — Phase 1 is over.

It is useful to think of Phase 1 as three subprojects, going on all at the same time but proceeding at different rates, involving the search strategies for a lightweight, middleweight and heavyweight SM Higgs. We’re almost done with the middleweight case, the easy one, in which one looks mainly for a Higgs decaying (i.e., disintegrating) to two W particles or two Z particles. The entire range from 141 to 470 GeV/c² is now excluded (according to the combination of the summer’s data from ATLAS and CMS that was announced a few weeks ago). The lightweight range, down to 115 GeV/c², is dominated by a search for a Higgs particle decaying to two photons. To observe this decay, a very rare process, requires a lot of data, so exploring this range fully will take another six months to a year. But we should already learn more about the lightweight Higgs on December 13th, when CERN will be providing an update on the Higgs search.

The heavyweight range — above 450 GeV/c² or so — is a little more subtle. Many theorists argue this window is already closed, by indirect experimental evidence. There are processes, carefully measured over the past 20 years, that are indirectly sensitive to the mass of the SM Higgs, and that strongly suggest it should be on the lighter side… below something like 300-400 GeV/c², though reasonable people might disagree on where exactly to set this bound. But even if you didn’t buy this powerful argument, it wouldn’t trouble the experiments. Depending upon exactly how much data the LHC takes in 2012, we should see most of the heavyweight range explored experimentally by late next year. The experimental results, combined with the theoretical arguments, should allow Phase 1 to conclude, to the satisfaction of almost all experts, once the 2012 data is fully analyzed.

So what are the possible outcomes of Phase 1?

1) The SM Higgs particle, already known with substantial confidence not to be in the middleweight range, might turn up in the lightweight or heavyweight range.
2) The SM Higgs particle might be entirely excluded, from 115 up to 800 GeV/c² or so. (Remember, though, that this would not mean there is no Higgs particle of any type — it would mean only that the simplest type is not found in nature.)
3) A Higgs-like particle that is clearly not a Standard Model Higgs particle (because it has the wrong production rates, or the wrong decay rates, given its mass) might be found instead.
3a) Some other great discovery at the LHC might move the SM Higgs search off the front pages for a while.

What would be the pros and cons of these different scenarios?

1) If the SM Higgs is found, that will be a historic discovery by the LHC, provisionally confirming the Standard Model’s simplest Higgs. That said, in some ways it will be a bit disappointing, since the Standard Model leaves many important questions in particle physics unanswered, and only by finding flaws in its equations do we have much hope of answering those questions.

2) If the SM Higgs is excluded, that will be an even more historic discovery, implying that the Standard Model’s equations are not the complete story at the LHC. For most particle physicists, this will be a much more exciting outcome! There will be a great opportunity for the LHC to teach us something profound about nature that we may not currently even suspect, although we’ll be on tenderhooks, potentially for quite a while, wondering whether the LHC’s data will provide clear guidance as to how to modify the Standard Model, or only give us some suggestive hints.

3) If a new non-SM Higgs particle is found, that will be the best possible outcome! Not only will we see the Standard Model’s equations fail, we’ll have a direct clue, in the form of the new particle, as to how to begin modifying them. In this case the LHC will immediately help us to start writing the new chapter in particle physics textbooks.

3a) And if something else unexpected is found in LHC data in the meantime, no one will complain! This of course would also mean the failure of the Standard Model’s equations, and new clues into nature’s mysteries.

You may have noticed that on this list there’s really no bad outcome. That’s right: as long as there are no technical problems at the LHC that limit the amount of data it collects in 2012, we are in a no-lose situation in the short term. This does not happen very often! And this is why there’s so much excitement right now in the field. We’re not wondering if we’ll get some historic information over the next year or so, or whether it will change the field of particle physics. We’re just wondering what it will be. (Needless to say, we’re all wishing the accelerator physicists and engineers over at the LHC, who keep the machine running efficiently, the very, very best — and while we’re at it, let’s hear a big round of applause for them right now, for what they’ve achieved in 2011!)

No matter what, there will be a very important Phase 2 to the Higgs search, which will extend for perhaps ten years beyond Phase 1. If Phase 1 finds something that looks like an SM Higgs particle, Phase 2 will be all about checking its details with high precision. The new particle may look at first as though it is just what the simplest version of the Higgs story would predict, but if even the slightest detail is out of place, it would show the Higgs is not so simple after all, which would be an exciting turn of events. If instead Phase 1 rules out the SM Higgs, then a great host of new search strategies will be brought to bear, and the experimentalists will (figuratively) fan out like a massive search party, looking for all varieties of exotic types of Higgs particles. Also — remembering that there may be no Higgs particle, but if so, the Standard Model’s equations can’t be entirely right — they’ll step up their efforts to look for the many other types of particles and forces that we might need to add to the Standard Model to make its equations sensible again, absent any type of Higgs particle in nature. And if a non-SM Higgs particle is found in Phase 1, Phase 2 will involve all of these strategies at once, producing the new Higgs particle in large quantities and studying it in detail, while looking for more clues as to what else is missing from the Standard Model’s equations.

It has been decades since a moment in particle physics looked as bright as do the next couple of years (healthy LHC operations permitting.) We’ve seen turning points in other fields: the human genome project was guaranteed to revolutionize genetics and genomics, and the study of the cosmic microwave background radiation was guaranteed to change our understanding of the early universe, as long as the experimental methods worked. These were great historic achievements that gave new life to those subjects; but we do not yet know whether the same is in store for particle physics. Will the flash of new understanding provided by Phase 1 quickly fade, or will it brighten into a dawn of a new age? Will the Standard Model’s equations work perfectly at the LHC, giving us a sense of satisfaction but no clues for future understanding? Will the equations fail, but in an obscure fashion, leaving us uncertain as to how to fix them? Or will their failure be clear and instructive, as has been the case for so many sets of equations before them, allowing the LHC, along perhaps with other experiments, to provide us with the insights we need to proceed to a more profound understanding of nature?

There is only one way to find out: run the experiment, and let nature speak.

So keep your eyes on Phase 1 of the Higgs search as it progresses toward a conclusion over the coming weeks and months. If the LHC works as hoped, the year ahead will be a memorable one.

Massimo Pigliucci has written a primer for skeptics of determinism, in part spurred by reading (and taking issue with) Alex Rosenberg’s new book The Atheist’s Guide to Reality, which I mentioned here. And Jerry Coyne responds, mostly to say that none of this amounts to “free will” over and above the laws of physics. (Which is true, even if, as I’ll mention below, quantum indeterminacy can propagate upward to classical behavior.) I wanted to give my own two cents, partly as a physicist and partly as a guy who just can’t resist giving his two cents.

Echoing Massimo’s structure, here are some talking points:

* There are probably many notions of what determinism means, but let’s distinguish two. The crucial thing is that the universe can be divided up into different moments of time. (The division will generally be highly non-unique, but that’s okay.) Then we can call “global determinism” the claim that, if we know the exact state of the whole universe at one time, the future and past are completely determined. But we can also define “local determinism” to be the claim that, if we know the exact state of some part of the universe at one time, the future and past of a certain region of the universe (the “domain of dependence”) is completely determined. Both are reasonable and relevant.

* It makes sense to be interested, as Massimo seems to be, in whether or not the one true correct ultimate set of laws of physics are deterministic or not. He argues that we don’t know, and that’s obviously right, since we don’t know what the final theory is. But that’s a rather defeatist attitude all by itself; we can look at the theories we do understand and try to draw lessons from them.

* Classical mechanics, which you might have thought was deterministic if anything was, actually has some loopholes. We can think of certain situations where more than one future obeys the equations of motion starting from the same past. This is discussed a bit in the Stanford Encyclopedia of Philosophy article on causal determinism. But I personally don’t find the examples that impressive. For one thing, they are highly non-generic; you have to work really hard to find these kinds of solutions, and they certainly aren’t stable under small perturbations. More importantly, classical mechanics isn’t right; it’s just an approximation to quantum mechanics, and these finely-tuned classical solutions would be dramatically altered by quantum effects.

* General relativity is a classical theory, so it’s also not correct, but we don’t have the final theory of quantum gravity so it’s worth a look. As Massimo points out, there are good examples in GR where traditional global determinism breaks down; naked singularities would be an example. (Basically, determinism breaks down when information can in principle “flow in” from a singularity or boundary that isn’t included in “the whole universe at one moment of time.”) We might sidestep this problem by arguing that naked singularities aren’t physical, which is quite reasonable. But there are much more benign examples, such as anti-de Sitter space — a maximally symmetric spacetime with a negative cosmological constant. This universe has no singularities, but does have a boundary at infinity, so a single moment of time only determines part of the universe, not the whole thing. On the other hand, like the classical-mechanics examples alluded to above, this seems like a technicality that can be cleared up with a slight change of definition, e.g. by imposing some simple boundary condition at infinity.

Much more importantly, these kinds of GR phenomena are very far away from our everyday lives; there’s really no relevance to discussions of free will. GR violates global determinism in the strict sense, but certainly obeys local determinism; that’s all that should be required for this kind of discussion.

* Quantum mechanics is where things get interesting. When a quantum state is happily evolving along according to the Schrödinger equation, everything is perfectly deterministic; indeed, more so than classical mechanics, because the space of states (Hilbert space) doesn’t allow for the kind of non-generic funny business that let non-deterministic classical solutions sneak in. But when we make an observation, we are unable to deterministically predict what its outcome will be. (And Bell’s theorem at least suggests that this inability is not just because we’re not smart enough; we never will be able to make such predictions.) At this point, opinions become split about whether the loss of determinism is real, or merely apparent. This is a crucial question for both physicists and philosophers, but not directly relevant for the question of free will.

The traditional (“Copenhagen”) view is that QM is truly non-deterministic, and that probability plays a central role in the measurement process when wave functions collapse. Unfortunately, this process is extremely unsatisfying, not just because it runs contrary to our philosophical prejudices but because what counts as a “measurement” and the quantum/classical split are extremely ill-defined. Almost everyone agrees we should do better, despite the fact that we still teach this approach in textbooks. Someone like Tom Banks would try to eliminate the magical process of wave function collapse, but keep probability (and thus a loss of determinism) as a central feature. There is a whole school of thought along these lines, which treats the quantum state as a device for tracking probabilities; see this excellent post by Matt Leifer for more details.

The other way to go is many-worlds, which says that the ordinary deterministic evolution of the Schrödinger equation is all that ever happens. The problem there is comporting such a claim with the reality of our experience — we see Schrödinger’s cat to be alive or dead, not ever in a live/dead superposition as QM would seem to imply. The resolution is that “we” are not described by the entire quantum state; rather, we live in one branch of the wave function, which also includes numerous other branches where different outcomes were observed. This approach (which I favor) restores determinism at the level of the fundamental equations, but sacrifices it for the observational predictions made by real observers. If I were keeping a tally, I would certainly put this one in the non-determinism camp, for anyone interested in questions of free will.

* Then there is the question of whether or not the lack of determinism in QM plays any role at all in our everyday lives. When we flip a coin or play the lottery, one might think that the relevant probabilities are “purely classical” — i.e. they stem from our lack of knowledge about the state of the muscles and nerves in my hand and the wind and the coin that is about to be flipped, but if I knew all of those things I could make a perfectly deterministic prediction about what would happen to the coin. (Indeed, a well-trained magician can flip a coin and get whatever result they want.)

This is actually a tricky problem, to which the answers aren’t clear. Yes, there may be a level of classical description in terms of a probability distribution; but where does that probability distribution come from? Physicists disagree about whether or not quantum mechanics plays a crucial role here. Since I have friends in high places, this weekend I emailed Andy Albrecht, who answered and brought David Deutsch into the conversation. They both argue — plausibly, although I’m not really qualified to pass judgment — that essentially all classical probabilities can ultimately traced down to the quantum wave function. And indeed, that this reasoning provides the only sensible basis for talking about probabilities at all! (David mentions that Lev Vaidman seems to disagree, so it’s not uncontroversial by any means.) They are both, in other words, firmly anti-Bayesian in their view on probability. A good Bayesian thinks that probabilities are always statements about our fundamental ignorance concerning what is “really” going on. Albrecht and Deutsch would argue that’s not true, probabilities are ultimately always statements about the wave function of the universe. If they’re right — and again, it looks plausible, but I need to think about it more — then QM effects are indeed of crucial importance in accounting for our inability to predict the future in the everyday world.

* I should say something about chaos, which always comes up in these discussions. In classical mechanics, even when the underlying model is perfectly deterministic, it can often be the case that a small uncertainty in our knowledge of the initial state can lead to large uncertainty in the future/past evolution. (E.g. for the tumbling of Hyperion.) This is sometimes brought up as if it causes problems for determinism: “since tiny mistakes propagate, you couldn’t realistically predict the future anyway.” This is about as irrelevant as it is possible to be irrelevant. The Laplacian viewpoint was always that if you had perfect information, you could predict the past and future. But that was always a statement of principle, not of practice. Of course, in practice, you have nowhere near enough information to make the kinds of calculation that Laplace’s vast intellect likes to do. That was perfectly obvious long before the advent of chaos theory. The correct statement is “in a classical deterministic system, with perfect information and arbitrary computing power you can predict the future in principle, but not in practice,” and that statement is completely unaltered by an understanding of chaos.

So where does that leave us? My personal suspicion is that the ultimate laws of physics will embody something like the many-worlds philosophy: the underlying laws are perfectly deterministic, but what happens along any specific history is irreducibly probabilistic. (In a better understanding of quantum gravity, our notion of “time” might be altered, and therefore our notion of “determinism” might be affected; but I suspect that there will still be some underlying equations that are rigidly obeyed.) But that’s just a suspicion, not anything worth taking to the bank. For everyday-life purposes, we can’t get around the fact that quantum mechanics makes it impossible to predict the future robustly.

Of course, this is all utterly irrelevant for questions of free will. (I’m sure Massimo knows this, but he didn’t discuss it in his blog post.) We can imagine four different possibilities: determinism + free will, indeterminism + free will, determinism + no free will, and indeterminism + no free will. All of these are logically possible, and in fact beliefs that some people actually hold! Bringing determinism into discussions of free will is a red herring.

It matters, of course, how one defines “free will.” The usual strategy in these discussions is to pick your own definition, and then argue on that basis, no matter what definition is being used by the person you’re arguing with. It’s not a strategy that advances human knowledge, but it makes for an endless string of debates.

A better question is, if we choose to think of human beings as collections of atoms and particles evolving according to the laws of physics, is such a description accurate and complete? Or is there something about human consciousness — some strong sense of “free will” — that allows us to deviate from the predictions that such a purely mechanistic model would make?

If that’s your definition of free will, then it doesn’t matter whether the laws of physics are deterministic or not — all that matters is that there are laws. If the atoms and particles that make up human beings obey those laws, there is no free will in this strong sense; if there is such a notion of free will, the laws are violated. In particular, if you want to use the lack of determinism in quantum mechanics to make room for supra-physical human volition (or, for that matter, occasional interventions by God in the course of biological evolution, as Francis Collins believes), then let’s be clear: you are not making use of the rules of quantum mechanics, you are simply violating them. Quantum mechanics doesn’t say “we don’t know what’s going to happen, but maybe our ineffable spirit energies are secretly making the choices”; it says “the probability of an outcome is the modulus squared of the quantum amplitude,” full stop. Just because there are probabilities doesn’t mean there is room for free will in that sense.

On the other hand, if you use a weak sense of free will, along the lines of “a useful theory of macroscopic human behavior models people as rational agents capable of making choices,” then free will is completely compatible with the underlying laws of physics, whether they are deterministic or not. That is the (fairly standard) compatibilist position, as defended by me in Free Will is as Real as Baseball. I would argue that this is the most useful notion of free will, the one people have in mind as they contemplate whether to go right to law school or spend a year hiking through Europe. It is not so weak as to be tautological: we could imagine a universe in which there were simple robust future boundary conditions, such that a model of rational agents would not be sufficient to describe the world. E.g. a world in which there were accurate prophesies of the future: “You will grow up to marry a handsome prince.” (Like it or not.) For better or for worse, that’s not the world we live in. What happens to you in the future is a combination of choices you make and forces well beyond your control — make the best of it!

Error bars are a simple and convenient way to characterize the expected uncertainty in a measurement, or for that matter the expected accuracy of a prediction. In a wide variety of circumstances (though certainly not always), we can characterize uncertainties by a normal distribution — the bell curve made famous by Gauss. Sometimes the measurements are a little bigger than the true value, sometimes they’re a little smaller. The nice thing about a normal distribution is that it is fully specified by just two numbers — the central value, which tells you where it peaks, and the standard deviation, which tells you how wide it is. The simplest way of thinking about an error bar is as our best guess at the standard deviation of what the underlying distribution of our measurement would be if everything were going right. Things might go wrong, of course, and your neutrinos might arrive early; but that’s not the error bar’s fault.

Now, there’s much more going on beneath the hood, as any scientist (or statistician!) worth their salt would be happy to explain. Sometimes the underlying distribution is not expected to be normal. Sometimes there are systematic errors. Are you sure you want the standard deviation, or perhaps the standard error? What are the error bars on your error bars?

While these are important issues, we’re in a holiday mood and aren’t trying to be so picky. What we’re celebrating is not the concept of statistical uncertainty, but the elegant shortcut provided by the concept of the error bar. Sure, many things can be going on, and ultimately we want to be more careful; nevertheless, there’s no question that the ability to sum up our rough degree of precision in a single number is enormously useful. That’s the genius of the error bar: it lets you decide at a glance whether a result is possibly worth believing or not. The power spectrum of the cosmic microwave background is a pretty plot, but it only becomes convincing when we see the error bars. Then you have a right to go, “Aha, I see three peaks there!”

And the error bar isn’t just pretty, it provides some quantitative oomph. An error bar is basically the standard deviation — “sigma,” as the scientists like to call it. So if your distribution really is normal you know that an individual measurement should be within one sigma of the expected value about 68% of the time; within two sigma 95% of the time, and within three sigma 99.7% of the time. So if you’re not within three sigma, you begin to think your expectation was wrong — something fishy is going on. (Like maybe a Nobel-prize-worthy discovery?) Once you’re out at five sigma, you’re outside the 99.9999% range — in normal human experience, that’s pretty unlikely.

Error bars aren’t the last word on statistical significance, they’re the first word. But we can all be thankful that so much meaning can be compressed into one little quantity.

To help understand the lay of the land, we’re very happy to host this guest post by David Wallace, a philosopher of science at Oxford. David has been one of the leaders in trying to make sense of the many-worlds interpretation of quantum mechanics, in particular the knotty problem of how to get the Born rule (“the wave function squared is the probability”) out of the this formalism. He was also a participant at our recent time conference, and the co-star of one of the videos I posted. He’s a very clear writer, and I think interested parties will get a lot out of reading this.

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Why the quantum state isn’t (straightforwardly) probabilistic

In quantum mechanics, we routinely talk about so-called “superposition states” – both at the microscopic level (“the state of the electron is a superposition of spin-up and spin-down”) and, at least in foundations of physics, at the macroscopic level (“the state of Schrodinger’s cat is a superposition of alive and dead”). Rather a large fraction of the “problem of measurement” is the problem of making sense of these superposition states, and there are basically two views. On the first (“state as physical”), the state of a physical system tells us what that system is actually, physically, like, and from that point of view, Schrodinger’s cat is seriously weird. What does it even mean to say that the cat is both alive and dead? And, if cats can be alive and dead at the same time, how come when we look at them we only see definitely-alive cats or definitely-dead cats? We can try to answer the second question by invoking some mysterious new dynamical process – a “collapse of the wave function” whereby the act of looking at half-alive, half-dead cats magically causes them to jump into alive-cat or dead-cat states – but a physical process which depends for its action on “observations”, “measurements”, even “consciousness”, doesn’t seem scientifically reputable. So people who accept the “state-as-physical” view are generally led either to try to make sense of quantum theory without collapses (that leads you to something like Everett’s many-worlds theory), or to modify or augment quantum theory so as to replace it with something scientifically less problematic.

On the second view, (“state as probability”), Schrodinger’s cat is totally unmysterious. When we say “the state of the cat is half alive, half dead”, on this view we just mean “it has a 50% probability of being alive and a 50% probability of being dead”. And the so-called collapse of the wavefunction just corresponds to us looking and finding out which it is. From this point of view, to say that the cat is in a superposition of alive and dead is no more mysterious than to say that Sean is 50% likely to be in his office and 50% likely to be at a conference.

Now, to be sure, probability is a bit philosophically mysterious. It’s not uncontroversial what it means to say that something is 50% likely to be the case. But we have a number of ways of making sense of it, and for all of them, the cat stays unmysterious. For instance, perhaps we mean that if we run the experiment many times (good luck getting that one past PETA), we’ll find that half the cats live, and half of them die. (This is the Frequentist view.) Or perhaps we mean that we, personally, know that that the cat is alive or dead but we don’t know which, and the 50% is a way of quantifying our lack of knowledge. (This is the Bayesian view.) But on either view, the weirdness of the cat still goes away.

So, it’s awfully tempting to say that we should just adopt the “state-as-probability” view, and thus get rid of the quantum weirdness. But This doesn’t work, for just as the “state-as-physical” view struggles to make sense of macroscopic superpositions, so the “state-as-probability” view founders on microscopic superpositions.

Consider, for instance, a very simple interference experiment. We split a laser beam into two beams (Beam 1 and Beam 2, say) with a half-silvered mirror. We bring the beams back together at another such mirror and allow them to interfere. The resultant light ends up being split between (say) Output Path A and Output Path B, and we see how much light ends up at each. It’s well known that we can tune the two beams to get any result we like – all the light at A, all of it at B, or anything in between. It’s also well known that if we block one of the beams, we always get the same result – half the light at A, half the light at B. And finally, it’s well known that these results persist even if we turn the laser so far down that only one photon passes through at a time.

According to quantum mechanics, we should represent the state of each photon, as it passes through the system, as a superposition of “photon in Beam 1″ and “Photon in Beam 2″. According to the “state as physical” view, this is just a strange kind of non-local state a photon is. But on the “state as probability” view, it seems to be shorthand for “the photon is either in beam 1 or beam 2, with equal probability of each”. And that can’t be correct. For if the photon is in beam 1 (and so, according to quantum physics, described by a non-superposition state, or at least not by a superposition of beam states) we know we get result A half the time, result B half the time. And if the photon is in beam 2, we also know that we get result A half the time, result B half the time. So whichever beam it’s in, we should get result A half the time and result B half the time. And of course, we don’t. So, just by elementary reasoning – I haven’t even had to talk about probabilities – we seem to rule out the “state-as-probability” rule.

Indeed, we seem to be able to see, pretty directly, that something goes down each beam. If I insert an appropriate phase factor into one of the beams – either one of the beams – I can change things from “every photon ends up at A” to “every photon ends up at B”. In other words, things happening to either beam affect physical outcomes. It’s hard at best to see how to make sense of this unless both beams are being probed by physical “stuff” on every run of the experiment. That seems pretty definitively to support the idea that the superposition is somehow physical.

There’s an interesting way of getting around the problem. We could just say that my “elementary reasoning” doesn’t actually apply to quantum theory – it’s a holdover of old, bad, classical ways of thinking about the world. We might, for instance, say that the kind of either-this-thing-happens-or-that-thing-does reasoning I was using above isn’t applicable to quantum systems. (Tom Banks, in his post, says pretty much exactly this.)

There are various ways of saying what’s problematic with this, but here’s a simple one. To make this kind of claim is to say that the “probabilities” of quantum theory don’t obey all of the rules of probability. But in that case, what makes us think that they are probabilities? They can’t be relative frequencies, for instance: it can’t be that 50% of the photons go down the left branch and 50% go down the right branch. Nor can they quantify our ignorance of which branch it goes down – because we don’t need to know which branch it goes down to know what it will do next. So to call the numbers in the superposition “probabilities” is question-begging. Better to give them their own name, and fortunately, quantum mechanics has already given us a name: amplitudes.

But once we make this move, we’ve lost everything distinctive about the “state-as-probability” view. Everyone agrees that according to quantum theory, the photon has some amplitude of being in beam A and some amplitude of being in beam B (and, indeed, that the cat has some amplitude of being alive and some amplitude of being dead); the question is, what does that mean? The “state-as-probability” view was supposed to answer, simply: it means that we don’t know everything about the photon’s (or the cat’s) state; but that now seems to have been lost. And the earlier argument that something goes down both beams remains unscathed.

Now, I’ve considered only the most straightforward kind of state-as-probability view you can think of – a view which I think is pretty decisively refuted by the facts. It’s possible to imagine subtler probabilistic theories – maybe the quantum state isn’t about the probabilities of each term in the superposition, but it’s still about the probabilities of something. But people’s expectations have generally been that the ubiquity of interference effects makes that hard to sustain, and a succession of mathematical results – from classic results like the Bell-Kochen-Specker theorem, to cutting-edge results like the recent theorem by Pusey, Barrett and Rudolph – have supported that expectation.

In fact, only one currently-discussed state-as-probability theory seems even half-way viable: the probabilities aren’t the probability of anything objective, they’re just the probabilities of measurement outcomes. Quantum theory, in other words, isn’t a theory that tells us about the world: it’s just a tool to predict the results of experiment. Views like this – which philosophers call instrumentalist – are often adopted as fall-back positions by physicists defending state-as-probability takes on quantum mechanics: Tom Banks, for instance, does exactly this in the last paragraph of his blog entry.

There’s nothing particularly quantum-mechanical about instrumentalism. It has a long and rather sorry philosophical history: most contemporary philosophers of science regard it as fairly conclusively refuted. But I think it’s easier to see what’s wrong with it just by noticing that real science just isn’t like this. According to instrumentalism, palaeontologists talk about dinosaurs so they can understand fossils, astrophysicists talk about stars so they can understand photoplates, virologists talk about viruses so they can understand NMR instruments, and particle physicists talk about the Higgs Boson so they can understand the LHC. In each case, it’s quite clear that instrumentalism is the wrong way around. Science is not “about” experiments; science is about the world, and experiments are part of its toolkit.

Tom’s last post was “technical” in the sense that it dug deeply into speculative ideas at the cutting edge of research. This one is technical in a different sense: the concepts are presented at a level that second-year undergraduate physics majors should have no trouble following, but there are explicit equations that might make it rough going for anyone without at least that much background. The translation from LaTeX to WordPress is a bit kludgy; here is a more elegant-looking pdf version if you’d prefer to read that.

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Rabbi Eliezer ben Yaakov of Nahariya said in the 6th century, “He who has not said three things to his students, has not conveyed the true essence of quantum mechanics. And these are Probability, Intrinsic Probability, and Peculiar Probability”.

Probability first entered the teachings of men through the work of that dissolute gambler Pascal, who was willing to make a bet on his salvation. It was a way of quantifying our risk of uncertainty. Implicit in Pascal’s thinking, and all who came after him was the idea that there was a certainty, even a predictability, but that we fallible humans may not always have enough data to make the correct predictions. This implicit assumption is completely unnecessary and the mathematical theory of probability makes use of it only through one crucial assumption, which turns out to be wrong in principle but right in practice for many actual events in the real world.

For simplicity, assume that there are only a finite number of things that one can measure, in order to avoid too much math. List the possible measurements as a sequence

The a_N are the quantities being measured and each could have a finite number of values. Then a probability distribution assigns a number P(A) between zero and one to each possible outcome. The sum of the numbers has to add up to one. The so called frequentist interpretation of these numbers is that if we did the same measurement a large number of times, then the fraction of times or frequency with which we’d find a particular result would approach the probability of that result in the limit of an infinite number of trials. It is mathematically rigorous, but only a fantasy in the real world, where we have no idea whether we have an infinite amount of time to do the experiments. The other interpretation, often called Bayesian, is that probability gives a best guess at what the answer will be in any given trial. It tells you how to bet. This is how the concept is used by most working scientists. You do a few experiments and see how the finite distribution of results compares to the probabilities, and then assign a confidence level to the conclusion that a particular theory of the data is correct. Even in flipping a completely fair coin, it’s possible to get a million heads in a row. If that happens, you’re pretty sure the coin is weighted but you can’t know for sure.

Physical theories are often couched in the form of equations for the time evolution of the probability distribution, even in classical physics. One introduces “random forces” into Newton’s equations to “approximate the effect of the deterministic motion of parts of the system we don’t observe”. The classic example is the Brownian motion of particles we see under the microscopic, where we think of the random forces in the equations as coming from collisions with the atoms in the fluid in which the particles are suspended. However, there’s no a priori reason why these equations couldn’t be the fundamental laws of nature. Determinism is a philosophical stance, an hypothesis about the way the world works, which has to be subjected to experiment just like anything else. Anyone who’s listened to a geiger counter will recognize that the microscopic process of decay of radioactive nuclei doesn’t seem very deterministic.

The place where the deterministic hypothesis and the laws of classical logic are put into the theory of probability is through the rule for combining probabilities of independent alternatives. A classic example is shooting particles through a pair of slits. One says, “the particle had to go through slit A or slit B and the probabilities are independent of each other, so,

It seems so obvious, but it’s wrong, as we’ll see below. The probability sum rule, as the previous equation is called, allows us to define conditional probabilities. This is best understood through the example of hurricane Katrina. The equations used by weather forecasters are probabilistic in nature. Long before Katrina made landfall, they predicted a probability that it would hit either New Orleans or Galveston. These are, more or less, mutually exclusive alternatives. Because these weather probabilities, at least approximately, obey the sum rule, we can conclude that the prediction for what happens after we make the observation of people suffering in the Superdome, doesn’t depend on the fact that Katrina could have hit Galveston. That is, that observation allows us to set the probability that it could have hit Galveston to zero, and re-scale all other probabilities by a common factor so that the probability of hitting New Orleans was one.

Note that if we think of the probability function P(x,t) for the hurricane to hit a point x and time t to be a physical field, then this procedure seems non-local or a-causal. The field changes instantaneously to zero at Galveston as soon as we make a measurement in New Orleans. Furthermore, our procedure “violates the weather equations”. Weather evolution seems to have two kinds of dynamics. The deterministic, local, evolution of P(x,t) given by the equation, and the causality violating projection of the probability of Galveston to zero and rescaling of the probability of New Orleans to one, which is mysteriously caused by the measurement process. Recognizing P to be a probability, rather than a physical field, shows that these objections are silly.

Nothing in this discussion depends on whether we assume the weather equations are the fundamental laws of physics of an intrinsically uncertain world, or come from neglecting certain unmeasured degrees of freedom in a completely deterministic system.

The essence of QM is that it forces us to take an intrinsically probabilistic view of the world, and that it does so by discovering an unavoidable probability theory underlying the mathematics of classical logic. In order to describe this in the simplest possible way, I want to follow Feynman and ask you to think about a single ammonia molecule, NH₃. A classical picture of this molecule is a pyramid with the nitrogen at the apex and the three hydrogens forming an equilateral triangle at the base. Let’s imagine a situation in which the only relevant measurement we could make was whether the pyramid was pointing up or down along the z axis. We can ask one question Q, “Is the pyramid pointing up?” and the molecule has two states in which the answer is either yes or no. Following Boole, we can assign these two states the numerical values 1 and 0 for Q, and then the “contrary question” 1 − Q has the opposite truth values. Boole showed that all of the rules of classical logic could be encoded in an algebra of independent questions, satisfying

where the Kronecker symbol δ_ij = 1 if i = j and 0 otherwise. i,j run from 1 to N, the number of independent questions. We also have ∑Q_i = 1, meaning that one and only one of the questions has the answer yes in any state of the system. Our ammonia molecule has only two independent questions, Q and 1 − Q. Let me also define s_z = 2Q − 1 = ±1, in the two different states. Computer aficionadas will recognize our two question system as a bit.

We can relate this discussion of logic to our discussion of probability of measurements by introducing observables A = ∑a_i Q_i , where the a_i are real numbers, specifying the value of some measurable quantity in the state where only Q_i has the answer yes. A probability distribution is then just a special case ρ = ∑p_i Q_i, where p_i is non-negative for each i and ∑p_i = 1.

Restricting attention to our ammonia molecule, we denote the two states as | ±_z 〉 and summarize the algebra of questions by the equation

We say that ” the operator s_z acting on the states | ±_z 〉 just multiplies them by (the appropriate ) number”. Similarly, if A = a₊ Q + a₋ (1 − Q) then

The expected value of the observable Aⁿ in the probability distribution ρ is

In the last equation we have used the fact that all of our “operators” can be thought of as two by two matrices acting on a two dimensional space of vectors whose basis elements are |±_z 〉. The matrices can be multiplied by the usual rules and the trace of a matrix is just the sum of its diagonal elements. Our matrices are

They’re all diagonal, so it’s easy to multiply them.

So far all we’ve done is rewrite the simple logic of a single bit as a complicated set of matrix equations, but consider the operation of flipping the orientation of the molecule, which for nefarious purposes we’ll call s_x,

This has matrix

Note that s_z² = s_x² = 1, and s_x s_z = − s_z s_x = − i s_y , where the last equality is just a definition. This definition implies that s_y s_a = − s_a s_y, for a = x or a = z, and it follows that s_y² = 1. You can verify these equations by using matrix multiplication, or by thinking about how the various operations operate on the states (which I think is easier). Now consider for example the quantity B ≡ b_x s_x + b_z s_z . Then B² = b_x² + b_z² , which suggests that B is a quantity which takes on the possible values ±√{b₊² + b₋²}. We can calculate

for any choice of probability distribution. If n = 2k it’s just

whereas if n = 2k + 1 it’s

This is exactly the same result we would get if we said that there was a probability P₊ (B) for B to take on the value √{b_z² + b_x²} and probability P₋ (B) = 1 − P₊ (B), to take on the opposite value, if we choose

The most remarkable thing about this formula is that even when we know the answer to Q with certainty (p₊ = 1 or 0), B is still uncertain.

We can repeat this exercise with any linear combination b_x s_x + b_y s_y + b_z s_z. We find that in general, if we force one linear combination to be known with certainty, that all linear combinations where the vector (c_x, c_y, c_z) is not parallel to (b_x , b_y, b_z) are uncertain. This is the same as the condition guaranteeing that the two linear combinations commute as matrices.

Pursuing the mathematics of this further would lead us into the realm of eigenvalues of Hermitian matrices, complete ortho-normal bases and other esoterica. But the main point to remember is that any system we can think about in terms of classical logic inevitably contains in it an infinite set of variables in addition to the ones we initially thought about as the maximum set of things we thought could be measured. When our original variables are known with certainty, these other variables are uncertain but the mathematics gives us completely determined formulas for their probability distributions.

Another disturbing fact about the mathematical probability theory for non-compatible observables that we’ve discovered, is that it does NOT satisfy the probability sum rule. This is because, once we start thinking about incompatible observables, the notion of either this or that is not well defined. In fact we’ve seen that when we know “definitely for sure” that s_z is 1, the probability for B to take on its positive value could be any number between zero and one, depending on the ratio of b_z and b_x.

Thus QM contains questions that are neither independent nor dependent and the probability sum rule P(s_z or B ) = P(s_z) + P(B) does not make sense because the word or is undefined for non-commuting operators. As a consequence we cannot apply the conditional probability rule to general QM probability predictions. This appears to cause a problem when we make a measurement that seems to give a definite answer. We’ll explain below that the issue here is the meaning of the word measurement. It means the interaction of the system with macroscopic objects containing many atoms. One can show that conditional probability is a sensible notion, with incredible accuracy, for such objects, and this means that we can interpret QM for such objects as if it were a classical probability theory. The famous “collapse of the wave function” is nothing more than an application of the rules of conditional probability, to macroscopic objects, for which they apply.

The double slit experiment famously discussed in the first chapter of Feynman’s lectures on quantum mechanics, is another example of the failure of the probability sum rule. The question of which slit the particle goes through is one of two alternative histories. In Newton’s equations, a history is determined by an initial position and velocity, but Heisenberg’s famous uncertainty relation is simply the statement that position and velocity are incompatible observables, which don’t commute as matrices, just like s_z and s_x. So the statement that either one history or another happened does not make sense, because the two histories interfere.

Before leaving our little ammonia molecule, I want to tell you about one more remarkable fact, which has no bearing on the rest of the discussion, but shows the remarkable power of quantum mechanics. Way back at the top of this post, you could have asked me, “what if I wanted to orient the ammonia along the x axis or some other direction”. The answer is that the operator n_x s_x + n_y s_y + n_z s_z, where (n_x , n_y, n_z) is a unit vector, has definite values in precisely those states where the molecule is oriented along this unit vector. The whole quantum formalism of a single bit, is invariant under 3 dimensional rotations. And who would have ever thought of that? (Pauli, that’s who).

The fact that QM was implicit in classical physics was realized a few years after the invention of QM, in the 1930s, by Koopman. Koopman formulated ordinary classical mechanics as a special case of quantum mechanics, and in doing so introduced a whole set of new observables, which do not commute with the (commuting) position and momentum of a particle and are uncertain when the particle’s position and momentum are definitely known. The laws of classical mechanics give rise to equations for the probability distributions for all these other observables. So quantum mechanics is inescapable. The only question is whether nature is described by an evolution equation which leaves a certain complete set of observables certain for all time, and what those observables are in terms of things we actually measure. The answer is that ordinary positions and momenta are NOT simultaneously determined with certainty.

Which raises the question of why it took us so long to notice this, and why it’s so hard for us to think about and accept. The answers to these questions also resolve “the problem of quantum measurement theory”. The answer lies essentially in the definition of a macroscopic object. First of all it means something containing a large number N of microscopic constituents. Let me call them atoms, because that’s what’s relevant for most everyday objects. For even a very tiny piece of matter weighing about a thousandth of a gram, the number N  ∼ 10²⁰. There are a few quantum states of the system per atom, let’s say 10 to keep the numbers round. So the system has 10¹⁰²⁰ states. Now consider the motion of the center of mass of the system. The mass of the system is proportional to N, so Heisenberg’s uncertainty relation tells us that the mutual uncertainty of the position and velocity of the system is of order [1/N]. Most textbooks stop at this point and say this is small and so the center of mass behaves in a classical manner to a good approximation.

In fact, this misses the central point, which is that under most conditions, the system has of order 10^N different states, all of which have the same center of mass position and velocity (within the prescribed uncertainty). Furthermore the internal state of the system is changing rapidly on the time scale of the center of mass motion. When we compute the quantum interference terms between two approximately classical states of the center of mass coordinate, we have to take into account that the internal time evolution for those two states is likely to be completely different. The chance that it’s the same is roughly 10^−N, the chance that two states picked at random from the huge collection, will be the same. It’s fairly simple to show that the quantum interference terms, which violate the classical probability sum rule for the probabilities of different classical trajectories, are of order 10^−N. This means that even if we could see the [1/N] effects of uncertainty in the classical trajectory, we could model them by ordinary classical statistical mechanics, up to corrections of order 10^−N.

It’s pretty hard to comprehend how small a number this is. As a decimal, it’s a decimal point followed by 100 billion billion zeros and then a one. The current age of the universe is less than a billion billion seconds. So if you wrote one zero every hundredth of a second you couldn’t write this number in the entire age of the universe. More relevant is the fact that in order to observe the quantum interference effects on the center of mass motion, we would have to do an experiment over a time period of order 10^N. I haven’t written the units of time. The smallest unit of time is defined by Newton’s constant, Planck’s constant and the speed of light. It’s 10^{− 44} seconds. The age of the universe is about 10⁶¹ of these Planck units. The difference between measuring the time in Planck times or ages of the universe is a shift from N = 10²⁰ to N = 10²⁰ − 60, and is completely in the noise of these estimates. Moreover, the quantum interference experiment we’re proposing would have to keep the system completely isolated from the rest of the universe for these incredible lengths of time. Any coupling to the outside effectively increases the size of N by huge amounts.

Thus, for all purposes, even those of principle, we can treat quantum probabilities for even mildly macroscopic variables, as if they were classical, and apply the rules of conditional probability. This is all we are doing when we “collapse the wave function” in a way that seems (to the untutored) to violate causality and the Schrodinger equation. The general line of reasoning outlined above is called the theory of decoherence. All physicists find it acceptable as an explanation of the reason for the practical success of classical mechanics for macroscopic objects. Some physicists find it inadequate as an explanation of the philosophical “paradoxes” of QM. I believe this is mostly due to their desire to avoid the notion of intrinsic probability, and attribute physical reality to the Schrodinger wave function. Curiously many of these people think that they are following in the footsteps of Einstein’s objections to QM. I am not a historian of science but my cursory reading of the evidence suggests that Einstein understood completely that there were no paradoxes in QM if the wave function was thought of merely as a device for computing probability. He objected to the contention of some in the Copehagen crowd that the wave function was real and satisfied a deterministic equation and tried to show that that interpretation violated the principles of causality. It does, but the statistical treatment is the right one. Einstein was wrong only in insisting that God doesn’t play dice.

Once we have understood these general arguments, both quantum measurement theory and our intuitive unease with QM are clarified. A measurement in QM is, as first proposed by von Neumann, simply the correlation of some microscopic observable, like the orientation of an ammonia molecule, with a macro-observable like a pointer on a dial. This can easily be achieved by normal unitary evolution. Once this correlation is made, quantum interference effects in further observation of the dial are exponentially suppressed, we can use the conditional probability rule, and all the mystery is removed.

It’s even easier to understand why humans don’t “get” QM. Our brains evolved according to selection pressures that involved only macroscopic objects like fruit, tigers and trees. We didn’t have to develop neural circuitry that had an intuitive feel for quantum interference phenomena, because there was no evolutionary advantage to doing so. Freeman Dyson once said that the book of the world might be written in Jabberwocky, a language that human beings were incapable of understanding. QM is not as bad as that. We CAN understand the language if we’re willing to do the math, and if we’re willing to put aside our intuitions about how the world must be, in the same way that we understand that our intuitions about how velocities add are only an approximation to the correct rules given by the Lorentz group. QM is worse, I think, because it says that logic, which our minds grasp as the basic, correct formulation of rules of thought, is wrong. This is why I’ve emphasized that once you formulate logic mathematically, QM is an obvious and inevitable consequence. Systems that obey the rules of ordinary logic are special QM systems where a particular choice among the infinite number of complementary QM observables remains sharp for all times, and we insist that those are the only variables we can measure. Viewed in this way, classical physics looks like a sleazy way of dodging the general rules. It achieves a more profound status only because it also emerges as an exponentially good approximation to the behavior of systems with a large number of constituents.

To summarize: All of the so-called non-locality and philosophical mystery of QM is really shared with any probabilistic system of equations and collapse of the wave function is nothing more than application of the conventional rule of conditional probabilities. It is a mistake to think of the wave function as a physical field, like the electromagnetic field. The peculiarity of QM lies in the fact that QM probabilities are intrinsic and not attributable to insufficiently precise measurement, and the fact that they do not obey the law of conditional probabilities. That law is based on the classical logical postulate of the law of the excluded middle. If something is definitely true, then all other independent questions are definitely false. We’ve seen that the mathematical framework for classical logic shows this principle to be erroneous. Even when we’ve specified the state of a system completely, by answering yes or no to every possible question in a compatible set, there are an infinite number of other questions one can ask of the same system, whose answer is only known probabilistically. The formalism predicts a very definite probability distribution for all of these other questions.

Many colleagues who understand everything I’ve said at least as well as I do, are still uncomfortable with the use of probability in fundamental equations. As far as I can tell, this unease comes from two different sources. The first is that the notion of “expectation” seems to imply an expecter, and most physicists are reluctant to put intelligent life forms into the definition of the basic laws of physics. We think of life as an emergent phenomenon, which can’t exist at the level of the microscopic equations. Certainly, our current picture of the very early universe precludes the existence of any form of organized life at that time, simply from considerations of thermodynamic equilibrium.

The frequentist approach to probability is an attempt to get around this. However, its insistence on infinite limits makes it vulnerable to the question about what one concludes about a coin that’s come up heads a million times. We know that’s a possible outcome even if the coin and the flipper are completely honest. Modern experimental physics deals with this problem every day both for intrinsically QM probabilities and those that arise from ordinary random and systematic fluctuations in the detector. The solution is not to claim that any result of measurement is definitely conclusive, but merely to assign a confidence level to each result. Human beings decide when the confidence level is high enough that we “believe” the result, and we keep an open mind about the possibility of coming to a different conclusion with more work. It may not be completely satisfactory from a philosophical point of view, but it seems to work pretty well.

The other kind of professional dissatisfaction with probability is, I think, rooted in Einstein’s prejudice that God doesn’t play dice. With all due respect, I think this is just a prejudice. In the 18th century, certain theoretical physicists conceived the idea that one could, in principle, measure everything there was to know about the universe at some fixed time, and then predict the future. This was wild hubris. Why should it be true? It’s remarkable that this idea worked as well as it did. When certain phenomena appeared to be random, we attributed that to the failure to make measurements that were complete and precise enough at the initial time. This led to the development of statistical mechanics, which was also wildly successful. Nonetheless, there was no real verification of the Laplacian principle of complete predictability. Indeed, when one enquires into the basic physics behind much of classical statistical mechanics one finds that some of the randomness invoked in that theory has a quantum mechanical origin. It arises after all from the motion of individual atoms. It’s no surprise that the first hints that classical mechanics was wrong came from failures of classical statistical mechanics like the Gibbs paradox of the entropy of mixing, and the black body radiation laws.

It seems to me that the introduction of basic randomness into the equations of physics is philosophically unobjectionable, especially once one has understood the inevitability of QM. And to those who find it objectionable all I can say is “It is what it is”. There isn’t anymore. All one must do is account for the successes of the apparently deterministic formalism of classical mechanics when applied to macroscopic bodies, and the theory of decoherence supplies that account.

Perhaps the most important lesson for physicists in all of this is not to mistake our equations for the world. Our equations are an algorithm for making predictions about the world and it turns out that those predictions can only be statistical. That this is so is demonstrated by the simple observation of a
Geiger counter and by the demonstration by Bell and others that the statistical predictions of QM cannot be reproduced by a more classical statistical theory with hidden variables, unless we allow for grossly non-local interactions. Some investigators into the foundations of QM have concluded that we should expect to find evidence for this non-locality, or that QM has to be modified in some fundamental way. I think the evidence all goes in the other direction: QM is exactly correct and inevitable and “there are more things in heaven and earth than are conceived of in our naive classical philosophy”. Of course, Hamlet was talking about ghosts…

The latest hint of a new result comes from the Large Hadron Collider, in particular the LHCb experiment. Unlike the general-purpose CMS and ATLAS experiments, LHCb is specialized: it looks at the decays of heavy mesons (particles consisting of one quark and one antiquark) to search for CP violation. “C” is for “charge” and “P” is for “parity”; so “CP violation” means you measure something happening with some particles, and then you measure the analogous thing happening when you switch particles with antiparticles and take the mirror image. (Parity reverses directions in space.) We know that CP is a pretty good symmetry in nature, but not a perfect one — Cronin and Fitch won the Nobel Prize in 1980 for discovering CP violation experimentally.

While the existence of CP violation is long established, it remains a target of experimental particle physicists because it’s a great window onto new physics. What we’re generally looking for in these big accelerators are new particles that are just to heavy and short-lived to be easily noticed in our everyday low-energy world. One way to do that is to just make the new particles directly and see them decaying into something. But another way is more indirect — measure the tiny effect of heavy virtual particles on the interactions of known particles. That’s what’s going on here.

More specifically, we’re looking at the decay of D mesons in two different ways, into kaons and pions. If you like thinking in terms of quarks, here are the dramatis personae:

• D0 meson: charm quark + anti-up quark
• anti-D0: anti-charm quark + up quark
• K-: strange quark + anti-up quark
• K+: anti-strange quark + up quark
• π-: down quark + anti-up quark
• π+: anti-down quark + up quark

Let’s look at the D0 meson. What happens is the charm quark (much heavier than the anti-up) decays into three lighter quarks: either up + strange + anti-strange, or up + down + anti-down. If it’s the former, we get a K- and a K+; if it’s the latter, we get a π- and a π+. Here’s one example, where D0 goes to K- and K+.

Of course the anti-D0 can also decay, and the anti-charm will go to either anti-up plus strange plus anti-strange, or anti-up plus down plus anti-down (just the antiparticles of what the D0 could go to). But if you match up the quarks, you see that the decay products are exactly the same as they were in the case of the original D0: either K- and K+, or π- and a π+.

Here’s where the search for CP violation comes in. If you take a D0 meson and “do a CP transformation to it,” you get an anti-D0, and vice-versa. So we can test for CP violation by comparing the rate at which D0′s decay to the rate of anti-D0′s. That’s basically the way Cronin and Fitch discovered CP violation, except that they started with neutral kaons and anti-kaons and watched them decay.

One problem is that the LHC itself doesn’t treat particles and anti-particles equally. It collides protons with protons, not protons with anti-protons. (It’s easier to make protons, so you get a higher luminosity [more events] if you stick with just protons.) So you end up making a lot more D0′s than anti-D0′s. In principle you can correct for that if you understand everything there is to understand about particle physics and your detector, but in practice we don’t. So the LHCb experimentalists did a clever thing: rather than just measuring the decay of D0′s and anti-D0′s into either kaons or pions, they measured them both, and then took the difference. This procedure is meant to cancel out all of the annoying experimental features, leaving only the pristine physics underneath. (If there is a nonzero difference in the CP violation rates between decays into kaons and decays into pions, at least one of those decays must itself violate CP.)

And the answer is: there is a noticeable difference! It’s -0.82%, plus or minus 0.24%, for a total of 3.5 sigma. (82 divided by 24 is about 3.5.) And the prediction from the Standard Model is that we should get almost zero for this quantity — maybe 0.01% or thereabouts.

So what could be going on? As Jester says, this is a surprising result — there aren’t a lot of models on the market that predict this level of CP violation in D0 decays but not in any of the other experiments we’ve already done. But the general idea, if you wanted to come up with such a model, would be to add new heavy particles that gently interfere with the process by which the charm quark in the above diagram decays into lighter quarks.

If I were to guess, I’d put my money on this result going away. But it stands a fighting chance! If it does hold up, to be honest it would be a bit frustrating — we would know that something new was going on, but not have too much of an idea what exactly it would be. But at least we’d know something about where to look, which is a huge advantage.

Truth in advertising notice: folks who write articles or press releases about CP violation are contractually obligated to say that this will help explain the matter-antimatter asymmetry in the universe. That might be true, or … it might not. My strong feeling is that we should be excited by discovering new particles of nature, and not rely on the crutch of relating everything to cosmology.

Previously I did one on dark energy. It came out right after the Nobel Prize announcement, but don’t let that trick you into thinking I won the Prize myself. (Some people were tricked.)

Meanwhile, in a parallel universe, instead of writing Spacetime and Geometry, I wrote a massive tome on Cosmology. This parallel universe was featured on this week’s episode of Fringe. Here’s Walter Bishop retrieving his copy from Peter.

I helped with some of the equations on the episode. Thanks to Glen Whitman and Rob Chiappetta for the shout-out.

Here’s neuroscientist David Eagleman, talking about how we perceive time.

Here’s physicist-turned-complexity-theorist Raissa D’Souza, talking about complexity.

Here’s another physicist-turned-complexity-theorist, Geoffrey West, taking the complexity story even further.

Here’s former guest-blogger, now Discover blogger, and engineer/roboticist/neuroscientist/philosopher Malcolm Maciver, talking about making choices and the evolution of consciousness.

And to top things off, here’s one of those mock debates (where participants attempt to defend the side they don’t believe in). This time it’s David Albert vs. David Wallace, on the many-worlds interpretation of quantum mechanics.

Seriously good stuff. There are still more talks not yet up, I’ll let you know.

Update: I didn’t realize my own talk was up. Here it is.

Don’s reply focuses less on details of eternal inflation and more on the general issue of how we should think about quantum gravity in a cosmological context, especially when it comes to counting the number of states. Don is (as he mentions below) an Evangelical Christian, but by no means a Young Earth Creationist!

Same rules apply as before: this is a technical discussion, which you are welcome to skip if it’s not your cup of tea.

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I tend to agree with Tom’s point that “it is extremely plausible, given the Bekenstein Hawking entropy formula for black holes, that the quantum theory of a space-time , which is dS in both the remote past and remote future, has a finite dimensional Hilbert space,” at least for four-dimensional spacetimes (excluding issues raised by Raphael Bousso, Oliver DeWolfe, and Robert Myers for higher dimensions in Unbounded entropy in space-times with positive cosmological constant) if the cosmological constant has a fixed finite value, or if there are a finite number of possible values that are all positive. The “conceptual error … that de Sitter (dS) space is a system with an ever increasing number of quantum degrees of freedom” seems to me to arise from considering perturbations of de Sitter when it is large (on a large compact Cauchy surface) that would evolve to a big bang or big crunch when the Cauchy surface gets small and hence would prevent the spacetime from having both a remote past and a remote future. As Tom nicely puts it, “In the remote past or future we can look at small amplitude wave packets. However, as we approach the neck of dS space, the wave packets are pushed together. If we put too much information into the space in the remote past, then the packets will collide and form a black hole whose horizon is larger than the neck. The actual solution is singular and does not resemble dS space in the future.”

So it seems to me that, for fixed positive cosmological constant, we can have an arbitrarily large number of quantum states if we allow big bangs or big crunches, but if we restrict to nonsingular spacetimes that expand forever in both the past and future, then the number of states may be limited by the value of the cosmological constant.

This reminds me of the 1995 paper by Gary Horowitz and Robert Myers, The value of singularities, which argued that the timelike naked singularity of the negative-mass Schwarzschild solution is important to be excluded in order to eliminate such states which would lead to energy unbounded below and instabilities from the presumably possible production (conserving energy) of arbitrarily many possible combinations of positive and negative energy. Perhaps in a similar way, big bang and big crunch singularities are important to be excluded, as they also would seem to allow infinitely many states with positive cosmological constant.

Now presumably we would want quantum gravity states to include the formation and evaporation of black holes (or of what phenomenologically appear similar to black holes, whether or not they actually have the causal structure of classical black holes), which in a classical approximation have singularities inside them, so presumably such `singularities’ should be allowed, even if timelike naked singularities and, I would suggest, big bang and big crunch singularities should be excluded. Perhaps one can postulate that one should restrict to states of immortal de Sitter spacetime, which has no timelike naked singularities anywhere, and which is asymptotically a single region in the very distant past that is locally de Sitter (though globally it can be highly distorted) and which is also asymptotically a single region in the very distant future that is likewise locally de Sitter, without any big crunch or big bang singularities in between, or between more than one asymptotic region (as one would expect to get from perturbations of the Nariai metric that lead to asymptotic de Sitter regions separated by big bang and big crunch singularities). Such singularities might be considered mortal wounds for de Sitter, allowing an infinite number of states to fester up from such wounds, killing the hope for a finite number of states and for unitarity. On the other hand, a localized black hole that forms within de Sitter could be considered a wound that is not mortal and which can be healed by Hawking evaporation without going outside the assumed finite number of quantum states for immortal de Sitter with a strictly positive cosmological constant. I have considered such a possibility in my paper No-Bang Quantum State of the Cosmos.

On the other hand, even if we find such a restriction for positive cosmological constant to a finite set of states, the number of such states for the observed tiny value of the cosmological constant seems so huge that it would seem surprising to me if most of them can explain our observations, so I now am skeptical of the maximally mixed state I proposed in No-Bang Quantum State of the Cosmos and now tend to prefer a pure state like what I proposed in Symmetric-Bounce Quantum State of the Universe. When coupled with a solution to the measure problem, such as what I proposed in Agnesi Weighting for the Measure Problem of Cosmology, this pure state seems to be consistent with all our observations, though of course these are highly preliminary proposals and are not yet fully precisely specified, nor are they nearly so simple and esthetically pleasing as I would hope for in a final complete theory of the universe. We would like a simple quantum state for the universe and simple rules for extracting the probabilities of observations from it. (In The Born Rule Dies and in related papers, I showed that the Born rule, interpreted mathematically in the form that probabilities of observations are expectation values of projection operators, does not work in a universe large enough for multiple copies of an observation, but the probabilities still could be expectation values of other quantum operators, though it is so far unknown what they should be; this knowledge would be a solution of the measure problem if the quantum state were also known.)

Some people seem to prefer the simple state that is the maximally mixed state out of a finite-dimensional Hilbert space, but if this does not explain our observations, I see no objection to postulating that the state is some other simple state, perhaps (but not necessarily) a pure state. By Occam’s razor, we scientists tend to ascribe higher prior probabilities to simpler theories, so we would tend to prefer a simpler quantum state, but I don’t think we should take so narrow a view that such a simple state has to be a thermal state, or a maximally mixed state.

Now even if the quantum state of the universe is chosen (perhaps as a simple mixed state, perhaps as a simple pure state) from a finite set of states, or even if it is a particular pure state, I’m agnostic as to whether or not this could be a state with eternal inflation. Eternal inflation may lead to a huge universe in which there are an arbitrarily large number of nearby states, but those may be states with mortal perturbations (evolving forward and/or backward in time to mortal singularities in a classical description) that would be excluded from the allowed states. So I don’t see that there is any problem with having disallowed states that at some time look nearby to the allowed states. Our universe looks very much as if it could have had a big bang in its past (which would allow an infinite number of states), and presumably a generic quasiclassical perturbation of the present state would evolve back to such a mortal singularity, but the actual state of our universe might have been one of the very large but finite number of states that did not have such a mortal singularity but instead had a bounce. If it did have a bounce, the size at the bounce seems that it might have been much smaller than the throat of de Sitter with the observed value of the cosmological constant, which suggests to me that the quantum state is much more restricted than just the restriction to the large but finite number of states that have bounces rather than mortal singularities.

The question seems to be open, even if the state is immortal de Sitter (by which I mean having a positive cosmological constant but no mortal singularities, not that it is metrically de Sitter or even close to de Sitter), whether this state consists of a superposition of quasiclassical spacetimes with most of them with significant amplitudes having a huge or infinite number of bubbles that keep forming and branching off (and presumably attaching on as well). Even from the considerations of Tom Banks and of my discussion above, I don’t see any obvious reason why it might not. One could presumably have a single simple pure state such that, if it were decomposed into quasiclassical components, would have components that are individually very complex, rather as the binary representation of the Bible, or of the Library of Congress, or of the eprint arXiv, or the entire Internet, or of all the words that have ever been written on earth, is each a very large and presumably very complex integer but is just one component of the set of all integers, which is a simple whole, of which almost all of the separate parts are individually much more complex.

I would guess that presumably at least some quasiclassical components of any simple quantum state that could describe our universe would be very complex and perhaps have a huge number of bubble or pocket subuniverses that could be described as having eternal inflation. Perhaps a more relevant question is whether the probabilities of our observations are given by something like a path integral that is dominated by such eternal inflating spacetimes, or whether the path integral (or whatever procedure gives the quantum probabilities) is dominated by simpler spacetimes, such as ones with only one region and a single bounce. For thermodynamic reasons I presently have some slight personal inclination toward the latter view, that for the dominant contribution it is sufficient to consider only one bounce in the past, at roughly five trillion days ago (an easily memorized value for the age of the universe that fits to four digits the middle of the range of the current measurements of 13.69(13) Gyr), rather than an infinite sequence of bubble formations in my past. Though I am not a Young Earth Creationist as some of my fellow Evangelical Christians are, perhaps I am still a Young Universe Creationist (for thermodynamic reasons rather than for theological reasons, since I personally do not see any theological reason that God could not have created an infinite eternally inflating universe) in the sense that I suspect that most of the quantum amplitude for our observations can be ascribed to quasiclassical universes (or spacetimes in a path integral) that each had its smallest spatial size (say in a foliation of Cauchy surfaces that are closed hypersurfaces of constant extrinsic curvature, to exclude nearly null hypersurfaces of arbitrarily tiny three-volume) not much more than about five trillion days ago. But so far as I can see, this question of whether or not eternally inflating spacetimes are important for the quantum probabilities of our observations is very much an open question, even if the set of quantum states is restricted to be finite or to be a single pure state.