[Update: I have fixed some broken links, and modified the discussion of QBism and the recent paper by Chris Fuchs— thanks to Chris himself for taking the time to read and find my mistakes!]
For some reason, I’ve come across an idea called “Knightian Uncertainty” quite a bit lately. Frank Knight was an economist of the free-market conservative “Chicago School”, who considered various concepts related to probability in a book called Risk, Uncertainty, and Profit. He distinguished between “risk”, which he defined as applying to events to which we can assign a numerical probability, and “uncertainty”, to those events about which we know so little that we don’t even have a probability to assign, or indeed those events whose possibility we didn’t even contemplate until they occurred. In Rumsfeldian language, “risk” applies to “known unknowns”, and “uncertainty” to “unknown unknowns”. Or, as Nicholas Taleb put it, “risk” is about “white swans”, while “uncertainty” is about those unexpected “black swans”.
(As a linguistic aside, to me, “uncertainty” seems a milder term than “risk”, and so the naming of the concepts is backwards.)
Actually, there are a couple of slightly different concepts at play here. The black swans or unknown-unknowns are events that one wouldn’t have known enough about to even include in the probabilities being assigned. This is much more severe than those events that one knows about, but for which one doesn’t have a good probability to assign.
And the important word here is “assign”. Probabilities are not something out there in nature, but in our heads. So what should a Bayesian make of these sorts of uncertainty? By definition, they can’t be used in Bayes’ theorem, which requires specifying a probability distribution. Bayesian theory is all about making models of the world: we posit a mechanism and possible outcomes, and assign probabilities to the parts of the model that we don’t know about.
So I think the two different types of Knightian uncertainty have quite a different role here. In the case where we know that some event is possible, but we don’t really know what probabilities to assign to it, we at least have a starting point. If our model is broad enough, then enough data will allow us to measure the parameters that describe it. For example, in recent years people have started to realise that the frequencies of rare, catastrophic events (financial crashes, earthquakes, etc.) are very often well described by so-called power-law distributions. These assign much greater probabilities to such events than more typical Gaussian (bell-shaped curve) distributions; the shorthand for this is that power-law distributions have much heavier tails than Gaussians. As long as our model includes the possibility of these heavy tails, we should be able to make predictions based on data, although very often those predictions won’t be very precise.
But the “black swan” problem is much worse: these are possibilities that we don’t even know enough about to consider in our model. Almost by definition, one can’t say anything at all about this sort of uncertainty. But what one must do is be open-minded enough to adjust our models in the face of new data: we can’t predict the black swan, but we should expand the model after we’ve seen the first one (and perhaps revise our model for other waterfowl to allow more varieties!). In more traditional scientific settings, involving measurements with errors, this is even more difficult: a seemingly anomalous result, not allowed in the model, may be due to some mistake in the experimental setup or in our characterisation of the probabilities of those inevitable errors (perhaps they should be described by heavy-tailed power laws, rather than Gaussian distributions as above).
I first came across the concept as an oblique reference in a recent paper by Chris Fuchs, writing about his idea of QBism (or see here for a more philosophically-oriented discussion), an interpretation of quantum mechanics that takes seriously the Bayesian principle that all probabilities are about our knowledge of the world, rather than the world itself (which is a discussion for another day). He tentatively opined that the probabilities in quantum mechanics are themselves “Knightian”, referring not to a reading of Knight himself but to some recent, and to me frankly bizarre, ideas from Scott Aaronson, discussed in his paper, The Ghost in the Quantum Turing Machine, and an accompanying blog post, trying to base something like “free will” (a term he explicitly does not apply to this idea, however) on the possibility of our brains having so-called “freebits”, quantum states whose probabilities are essentially uncorrelated with anything else in the Universe. This arises from what is to me a mistaken desire to equate “freedom” with complete unpredictability. My take on free will is instead aligned with that of Daniel Dennett, at least the version from his Consciousness Explained from the early 1990s, as I haven’t yet had the chance to read his recent From Bacteria to Bach and Back: a perfectly deterministic (or quantum mechanically random, even allowing for the statistical correlations that Aaronson wants to be rid of) version of free will is completely sensible, and indeed may be the only kind of free will worth having.
Fuchs himself tentatively uses Aaronson’s “Knightian Freedom” to refer to his own idea
that nature does what it wants, without a mechanism underneath, and without any “hidden hand” of the likes of Richard von Mises’s Kollective or Karl Popper’s propensities or David Lewis’s objective chances, or indeed any conception that would diminish the autonomy of nature’s events,
which I think is an attempt (and which I admit I don’t completely understand) to remove the probabilities of quantum mechanics entirely from any mechanistic account of physical systems, despite the incredible success of those probabilities in predicting the outcomes of experiments and other observations of quantum mechanical systems. I’m not quite sure this is what either Knight nor Aaronson had in mind with their use of “uncertainty” (or “freedom”), since at least in quantum mechanics, we do know what probabilities to assign, given certain other personal (as Fuchs would have it) information about the system. My Bayesian predilections make me sympathetic with this idea, but then I struggle to understand what, exactly, quantum mechanics has taught us about the world: why do the predictions of quantum mechanics work?
When I’m not thinking about physics, for the last year or so my mind has been occupied with politics, so I was amused to see Knightian Uncertainty crop up in a New Yorker article about Trump’s effect on the stock market:
Still, in economics there’s a famous distinction, developed by the great Chicago economist Frank Knight, between risk and uncertainty. Risk is when you don’t know exactly what will happen but nonetheless have a sense of the possibilities and their relative likelihood. Uncertainty is when you’re so unsure about the future that you have no way of calculating how likely various outcomes are. Business is betting that Trump is risky but not uncertain—he may shake things up, but he isn’t going to blow them up. What they’re not taking seriously is the possibility that Trump may be willing to do things—like start a trade war with China or a real war with Iran—whose outcomes would be truly uncertain.
It’s a pretty low bar, but we can only hope.
I recently finished my last term lecturing our second-year Quantum Mechanics course, which I taught for five years. It’s a required class, a mathematical introduction to one of the most important set of ideas in all of physics, and really the basis for much of what we do, whether that’s astrophysics or particle physics or almost anything else. It’s a slightly “old-fashioned” course, although it covers the important basic ideas: the Schrödinger Equation, the postulates of quantum mechanics, angular momentum, and spin, leading almost up to what is needed to understand the crowning achievement of early quantum theory: the structure of the hydrogen atom (and other atoms).
A more modern approach might start with qubits: the simplest systems that show quantum mechanical behaviour, and the study of which has led to the revolution in quantum information and quantum computing.
Moreover, the lectures rely on the so-called Copenhagen interpretation, which is the confusing and sometimes contradictory way that most physicists are taught to think about the basic ontology of quantum mechanics: what it says about what the world is “made of” and what happens when you make a quantum-mechanical measurement of that world. Indeed, it’s so confusing and contradictory that you really need another rule so that you don’t complain when you start to think too deeply about it: “shut up and calculate”. A more modern approach might also discuss the many-worlds approach, and — my current favorite — the (of course) Bayesian ideas of QBism.
The students seemed pleased with the course as it is — at the end of the term, they have the chance to give us some feedback through our “Student On-Line Evaluation” system, and my marks have been pretty consistent. Of the 200 or so students in the class, only about 90 bother to give their evaluations, which is disappointingly few. But it’s enough (I hope) to get a feeling for what they thought.
So, most students Definitely/Mostly Agree with the good things, although it’s clear that our students are most disappointed in the feedback that they receive from us (this is a more general issue for us in Physics at Imperial and more generally, and which may partially explain why most of them are unwilling to feed back to us through this form).
But much more fun and occasionally revealing are the “free-text comments”. Given the numerical scores, it’s not too surprising that there were plenty of positive ones:
Excellent lecturer - was enthusiastic and made you want to listen and learn well. Explained theory very well and clearly and showed he responded to suggestions on how to improve.
Possibly the best lecturer of this term.
Thanks for providing me with the knowledge and top level banter.
One of my favourite lecturers so far, Jaffe was entertaining and cleary very knowledgeable. He was always open to answering questions, no matter how simple they may be, and gave plenty of opportunity for students to ask them during lectures. I found this highly beneficial. His lecturing style incorporates well the blackboards, projectors and speach and he finds a nice balance between them. He can be a little erratic sometimes, which can cause confusion (e.g. suddenly remembering that he forgot to write something on the board while talking about something else completely and not really explaining what he wrote to correct it), but this is only a minor fix. Overall VERY HAPPY with this lecturer!
But some were more mixed:
One of the best, and funniest, lecturers I’ve had. However, there are some important conclusions which are non-intuitively derived from the mathematics, which would be made clearer if they were stated explicitly, e.g. by writing them on the board.
I felt this was the first time I really got a strong qualitative grasp of quantum mechanics, which I certainly owe to Prof Jaffe’s awesome lectures. Sadly I can’t quite say the same about my theoretical grasp; I felt the final third of the course less accessible, particularly when tackling angular momentum. At times, I struggled to contextualise the maths on the board, especially when using new techniques or notation. I mostly managed to follow Prof Jaffe’s derivations and explanations, but struggled to understand the greater meaning. This could be improved on next year. Apart from that, I really enjoyed going to the lectures and thought Prof Jaffe did a great job!
The course was inevitably very difficult to follow.
And several students explicitly commented on my attempts to get students to ask questions in as public a way as possible, so that everyone can benefit from the answers and — this really is true! — because there really are no embarrassing questions!
Really good at explaining and very engaging. Can seem a little abrasive at times. People don’t like asking questions in lectures, and not really liking people to ask questions in private afterwards, it ultimately means that no questions really get answered. Also, not answering questions by email makes sense, but no one really uses the blackboard form, so again no one really gets any questions answered. Though the rationale behind not answering email questions makes sense, it does seem a little unnecessarily difficult.
We are told not to ask questions privately so that everyone can learn from our doubts/misunderstandings, but I, amongst many people, don’t have the confidence to ask a question in front of 250 people during a lecture.
Forcing people to ask questions in lectures or publically on a message board is inappropriate. I understand it makes less work for you, but many students do not have the confidence to ask so openly, you are discouraging them from clarifying their understanding.
Inevitably, some of the comments were contradictory:
Would have been helpful to go through examples in lectures rather than going over the long-winded maths to derive equations/relationships that are already in the notes.
Professor Jaffe is very good at explaining the material. I really enjoyed his lectures. It was good that the important mathematics was covered in the lectures, with the bulk of the algebra that did not contribute to understanding being left to the handouts. This ensured we did not get bogged down in unnecessary mathematics and that there was more emphasis on the physics. I liked how Professor Jaffe would sometimes guide us through the important physics behind the mathematics. That made sure I did not get lost in the maths. A great lecture course!
And also inevitably, some students wanted to know more about the exam:
- It is a difficult module, however well covered. The large amount of content (between lecture notes and handouts) is useful. Could you please identify what is examinable though as it is currently unclear and I would like to focus my time appropriately?
And one comment was particularly worrying (along with my seeming “a little abrasive at times”, above):
- The lecturer was really good in lectures. however, during office hours he was a bit arrogant and did not approach the student nicely, in contrast to the behaviour of all the other professors I have spoken to
If any of the students are reading this, and are willing to comment further on this, I’d love to know more — I definitely don’t want to seem (or be!) arrogant or abrasive.
But I’m happy to see that most students don’t seem to think so, and even happier to have learned that I’ve been nominated “multiple times” for Imperial’s Student Academic Choice Awards!
Finally, best of luck to my colleague Jonathan Pritchard, who will be taking over teaching the course next year.
[Update: The bug seems fixed in the latest version, 10.0.2.]
I am in my third year teaching a course in Quantum Mechanics, and we spend a lot of time working with a very simple system known as the harmonic oscillator — the physics of a pendulum, or a spring. In fact, the simple harmonic oscillator (SHO) is ubiquitous in almost all of physics, because we can often represent the behaviour of some system as approximately the motion of an SHO, with some corrections that we can calculate using a technique called perturbation theory.
It turns out that in order to describe the state of a quantum SHO, we need to work with the Gaussian function, essentially the combination exp(-y²/2), multiplied by another set of functions called Hermite polynomials. These latter functions are just, as the name says, polynomials, which means that they are just sums of terms like ayⁿ where a is some constant and n is 0, 1, 2, 3, … Now, one of the properties of the Gaussian function is that it dives to zero really fast as y gets far from zero, so fast that multiplying by any polynomial still goes to zero quickly. This, in turn, means that we can integrate polynomials, or the product of polynomials (which are just other, more complicated polynomials) multiplied by our Gaussian, and get nice (not infinite) answers.
Unfortunately, Wolfram Inc.’s Mathematica (the most recent version 10.0.1) disagrees:
The details depend on exactly which Hermite polynomials I pick — 7 and 16 fail, as shown, but some combinations give the correct answer, which is in fact zero unless the two numbers differ by just one. In fact, if you force Mathematica to split the calculation into separate integrals for each term, and add them up at the end, you get the right answer.
I’ve tried to report this to Wolfram, but haven’t heard back yet. Has anyone else experienced this?
The academic year has begun, and I’m teaching our second-year Quantum Mechanics course again. I was pretty happy with last year’s version, and the students didn’t completely disagree.
This year, there have been a few changes to the structure of the course — although not as much to the content as I might have liked (“if it ain’t broke, don’t fix it”, although I’d still love to use more of the elegant Dirac notation and perhaps discuss quantum information a bit more). We’ve moved some of the material to the first year, so the students should already come into the course with at least some exposure to the famous Schrödinger Equation which describes the evolution of the quantum wave function. But of course all lecturers treat this material slightly differently, so I’ve tried to revisit some of that material in my own language, although perhaps a bit too quickly.
Perhaps more importantly, we’ve also changed the tutorial system. We used to attempt an imperfect rendition of the Oxbridge small-group tutorial system, but we’ve moved to something with larger groups and (we hope) a more consistent presentation of the material. We’re only on the second term with this new system, so the jury is still out, both in terms of the students’ reactions, and our own. Perhaps surprisingly, they do like the fact that there is more assessed (i.e., explicitly graded, counting towards the final mark in the course) material — coming from the US system, I would like to see yet more of this, while those brought up on the UK system prefer the final exam to carry most (ideally all!) the weight.
So far I’ve given three lectures, including a last-minute swap yesterday. The first lecture — mostly content-free — went pretty well, but I’m not too happy with my performance on the last two: I’ve made a mistake in each of the last two lectures. I’ve heard people say that the students don’t mind a few (corrected) mistakes; it humanises the teachers. But I suspect that the students would, on the whole, prefer less-human, more perfect, lecturing…
Yesterday, we were talking about a particle trapped in a finite potential well — that is, a particle confined to be in a box, but (because of the weirdness of quantum mechanics) with some probability of being found outside. That probability depends upon the energy of the particle, and because of the details of the way I defined that energy (starting at a negative number, instead of the more natural value of zero), I got confused about the signs of some of the quantities I was dealing with. I explained the concepts (I think) completely correctly, but with mistakes in the math behind them, the students (and me) got confused about the details. But many, many thanks to the students who kept pressing me on the issue and helped us puzzle out the problems.
Today’s mistake was less conceptual, but no less annoying — I wrote (and said) “cotangent” when I meant “tangent” (and vice versa). In my notes, this was all completely correct, but when you’re standing up in front of 200 or so students, sometimes you miss the detail on the page in front of you. Again, this was in some sense just a mathematical detail, but (as we always stress) without the right math, you can’t really understand the concepts. So, thanks to the students who saw that I was making a mistake, and my apologies to the whole class.
A couple of weeks I received the Student On-Line Evaluation (SOLE) results for my Quantum Mechanics course.
There were only two specific questions, rating each of the following from “Very Good” through “Poor” (there’s a “no response” off to the right, as well):
- The structure and delivery of the teaching sessions
- The content of this module
The numerical results (at right) were pretty good. Note that 114 students — about half — responded.
The rest of the results are free-form comments. With such a big class it’s very difficult to find a style of teaching that suits everyone. Hence, the comments showed a split between the students who enjoyed the very mathematical approach of the course and those who wanted more physical examples from the beginning (not that easy in the context of an axiomatic approach to quantum mechanics — but there are a few simple system like quantum dots that exhibit some of the properties of the simplest systems we study in class; it’s clear that these should be highlighted more than I have). Similarly, some students wanted a more step-by-step approach to the mathematics, whereas others would prefer just a sketch of the proofs on the board (“put the algebra in the notes and let students work through it”).
But one set of comments especially hit home. Here’s a good example:
Frankly, I think that Prof. Jaffe has the potential to be an outstanding lecturer, one which he wastes by not being properly prepared. Just showing up to lectures and writing down on the board what was in (the previous lecturer’s) notes without thinking much about it in advance results in time spent staring at notes and board which could otherwise have been used to face the audience and explain what it is we’re doing. Maybe that sounded harsh, but he really is very good and could be outstanding if he put a little more into preparing for lectures and didn’t stick to his notes quite so much… If you actually perform the calculations, and think about the various steps yourself, then it all happens in a way, and at a pace, which suits us as students and allows us to follow.
(OK, I picked one that made an effort to heap on some praise along with the criticism.) I have to admit that this point, repeated by several students, seemed right on, for at least some of the lectures. I did always make an effort to go over the notes in detail beforehand. But these were notes indeed written by the previous lecturer, and this gives a few problems. Yes, I probably wasn’t always careful enough to go over the details of the mathematics beforehand. So sometimes I did spend too much effort trying to puzzle out exactly what I wanted to say (some students also complained about the occasional mistakes I made on the board, perhaps related to this). But sometimes the problem is more subtle: I might not always want to explain the concepts in the same way as the previous lecturer — and sometimes I might only realise this when actually doing the explaining! Either of these can happen in any lecture, but the combination of teaching this course for the first time, and doing so from someone else’s notes, certainly made it worse.
Next year, things will at least be different: I’ll be teaching for the second time, and so have some idea of the pitfalls from this past year. Moreover, our department is making some significant changes to the overall structure of the curriculum, phasing out our system of tutorials and so-called classworks for a series of three medium-sized (20 student) group sessions each week. This is happening alongside some specific changes to the quantum mechanics curriculum, with more material in the first year (happening already). My course will be shortened by a full five lectures, but I suspect that this combination of changes will give me a bit more breathing room, as well as a few different ways to make sure the material gets said in different ways, appropriate for different students.
Further criticism, comments, ideas, etc., are always welcome.
A week ago, I finished my first time teaching our second-year course in quantum mechanics. After a bit of a taster in the first year, the class concentrates on the famous Schrödinger equation, which describes the properties of a particle under the influence of an external force. The simplest version of the equation is just
This relates the so-called wave function, ψ, to what we know about the external forces governing its motion, encoded in the Hamiltonian operator, Ĥ. The wave function gives the probability (technically, the probability amplitude) for getting a particular result for any measurement: its position, its velocity, its energy, etc. (See also this excellent public work by our department’s artist-in-residence.)
Over the course of the term, the class builds up the machinery to predict the properties of the hydrogen atom, which is the canonical real-world system for which we need quantum mechanics to make predictions. This is certainly a sensible endpoint for the 30 lectures.
But it did somehow seem like a very old-fashioned way to teach the course. Even back in the 1980s when I first took a university quantum mechanics class, we learned things in a way more closely related to the way quantum mechanics is used by practicing physicists: the mathematical details of Hilbert spaces, path integrals, and Dirac Notation.
Today, an up-to-date quantum course would likely start from the perspective of quantum information, distilling quantum mechanics down to its simplest constituents: qbits, systems with just two possible states (instead of the infinite possibilities usually described by the wave function). The interactions become less important, superseded by the information carried by those states.
Really, it should be thought of as a full year-long course, and indeed much of the good stuff comes in the second term when the students take “Applications of Quantum Mechanics” in which they study those atoms in greater depth, learn about fermions and bosons and ultimately understand the structure of the periodic table of elements. Later on, they can take courses in the mathematical foundations of quantum mechanics, and, yes, on quantum information, quantum field theory and on the application of quantum physics to much bigger objects in “solid-state physics”.
Despite these structural questions, I was pretty pleased with the course overall: the entire two-hundred-plus students take it at the beginning of their second year, thirty lectures, ten ungraded problem sheets and seven in-class problems called “classworks”. Still to come: a short test right after New Year’s and the final exam in June. Because it was my first time giving these lectures, and because it’s such an integral part of our teaching, I stuck to to the same notes and problems as my recent predecessors (so many, many thanks to my colleagues Paul Dauncey and Danny Segal).
Once the students got over my funny foreign accent, bad board handwriting, and worse jokes, I think I was able to get across both the mathematics, the physical principles and, eventually, the underlying weirdness, of quantum physics. I kept to the standard Copenhagen Interpretation of quantum physics, in which we think of the aforementioned wavefunction as a real, physical thing, which evolves under that Schrödinger equation — except when we decide to make a measurement, at which point it undergoes what we call collapse, randomly and seemingly against causality: this was Einstein’s “spooky action at a distance” which seemed to indicate nature playing dice with our Universe, in contrast to the purely deterministic physics of Newton and Einstein’s own relativity. No one is satisfied with Copenhagen, although a more coherent replacement has yet to be found (I won’t enumerate the possibilities here, except to say that I find the proliferating multiverse of Everett’s Many-Worlds interpretation ontologically extravagant, and Chris Fuchs’ Quantum Bayesianism compelling but incomplete).
I am looking forward to getting this year’s SOLE results to find out for sure, but I think the students learned something, or at least enjoyed trying to, although the applause at the end of each lecture seemed somewhat tinged with British irony.
Somehow I’ve managed to forget my usual end-of-term post-mortem of the year’s lecturing. I think perhaps I’m only now recovering from 11 weeks of lectures, lab supervision, tutoring alongside a very busy time analysing Planck satellite data.
But a few weeks ago term ended, and I finished teaching my undergraduate cosmology course at Imperial, 27 lectures covering 14 billion years of physics. It was my fourth time teaching the class (I’ve talked about my experiences in previous years here, here, and here), but this will be the last time during this run. Our department doesn’t let us teach a course more than three or four years in a row, and I think that’s a wise policy. I think I’ve arrived at some very good ways of explaining concepts such as the curvature of space-time itself, and difficulties with our models like the 122-or-so-order-of-magnitude cosmological constant problem, but I also noticed that I wasn’t quite as excited as in previous years, working up from the experimentation of my first time through in 2009, putting it all on a firmer foundation — and writing up the lecture notes — in 2010, and refined over the last two years. This year’s teaching evaluations should come through soon, so I’ll have some feedback, and there are still about six weeks until the students’ understanding — and my explanations — are tested in the exam.
Next year, I’ve got the frankly daunting responsibility of teaching second-year quantum mechanics: 30 lectures, lots of problem sheets, in-class problems to work through, and of course the mindbending weirdness of the subject itself. I’d love to teach them Dirac’s very useful notation which unifies the physical concept of quantum states with the mathematical ideas of vectors, matrices and operators — and which is used by all actual practitioners from advanced undergraduates through working physicists. But I’m told that students find this an extra challenge rather than a simplification. Comments from teachers and students of quantum mechanics are welcome.
OK, this is going to be a very long post. About something I don’t pretend to be expert in. But it is science, at least.
A couple of weeks ago, Radio 4’s highbrow “In Our Time” tackled the so-called “Measurement Problem”. That is: quantum mechanics predicts probabilities, not definite outcomes. And yet we see a definite world. Whenever we look, a particle is in a particular place. A cat is either alive or dead, in Schrodinger’s infamous example. So, lots to explain in just setting up the problem, and even more in the various attempts so far to solve it (none quite satisfactory). This is especially difficult because the measurement problem is, I think, unique in physics: quantum mechanics appears to be completely true and experimentally verified, without contradiction so far. And yet it seems incomplete: the “problem” arises because the equations of quantum mechanics only provide a recipe for the calculations of probabilities, but doesn’t seem to explain what’s going on underneath. For that, we need to add a layer of interpretation on top. Melvyn Bragg had three physicists down to the BBC studios, each with his own idea of what that layer might look like.
Unfortunately, the broadcast seemed to me a bit of a shambles: the first long explanation by Basil Hiley of Birkbeck of quantum mechanics used the terms “wavefunction” and “linear superposition” without even an attempt at a definition. Things got a bit better as Bragg tried to tease things out, but I can’t imagine the non-physicists that were left listening got much out of it. Hiley himself worked with David Bohm on one possible solution to the measurement problem, the so-called “Pilot Wave Theory” (another term which was used a few times without definition) in which quantum mechanics is actually a deterministic theory — the probabilities come about because there is information to which we do not — and in principle cannot — have access to about the locations and trajectories of particles.
Roger Penrose proved to be remarkably positivist in his outlook: he didn’t like the other interpretations on offer simply because they make no predictions beyond standard quantum mechanics and are therefore untestable. (Others see this as a selling point for these interpretations, however — there is no contradiction with experiment!) To the extent I understand his position, Penrose himself prefers the idea that quantum mechanics is actually incomplete, and that when it is finally reconciled with General Relativity (in a Theory of Everything or otherwise), we will find that it actually does make specific, testable predictions.
There was a long discussion by Simon Saunders of that sexiest of interpretations of quantum mechanics, the Many Worlds Interpretation. The latest incarnation of Many-Worlds theory is centered around workers in or near Oxford: Saunders himself, David Wallace and most famously David Deutsch. The Many-Worlds interpretation (also known as the Everett Interpretation after its initial proponent) attempts to solve the problem by saying that there is nothing special about measurement at all — the simple equations of quantum mechanics always obtain. In order for this to occur, then all possible outcomes of any experiment must be actualized: that is, their must be a world for each outcome. But we’re not just talking about outcomes of science experiments here, but rather any time that quantum mechanics could have predicted something other than what (seemingly) actually happened. Which is all the time, to all of the particles in the Universe, everywhere. This is, to say the least, “ontologically extravagant”. Moreover, it has always been plagued by at least one fundamental problem: what, exactly, is the status of probability in the many-worlds view? When more than one quantum-mechanical possibility presents itself, each splits into its own world, with a probability related to the aforementioned wavefunction. But what beyond this does it mean for one branch to have a higher probability? The Oxonian many-worlders have tried to use decision theory to reconcile this with the prescriptions of quantum mechanics: from very minimal requirements of rationality alone, can we derive the probability rule? They claim to have done so, and they further claim that their proof only makes sense in the Many-Worlds picture. This is, roughly, because only in the Everett picture is their no “fact of the matter” at all about what actually happens in a quantum outcome — in all other interpretations the very existence of a single actual outcome is enough to scupper the proof. (I’m not so sure I buy this — surely we are allowed to base rational decisions on only the information at hand, as opposed to all of the information potentially available?)
At bottom, these interpretations of quantum mechanics (aka solutions to the measurement problem) are trying to come to grips with the fact that quantum mechanics seems to be fundamentally about probability, rather than the way things actually are. And, as I’ve discussed elsewhere, time and time again, probability is about our states of knowledge, not the world. But we are justly uncomfortable with 70s-style “Tao-of-Physics” ideas that make silly links between consciousness and the world at large.
But there is an interpretation that takes subjective probability seriously without resorting to the extravagance of many (very, very many) worlds. Chris Fuchs, along with his collaborators Carlton Caves and Ruediger Schack have pursued this idea with some success. Whereas the many-worlds interpretation requires a universe that seems far too full for me, the Bayesian interpretation is somewhat underdetermined: there is a level of being that is, literally unspeakable: there is no information to be had about the quantum realm beyond our experimental results. This is, as Fuchs points out, a very strong restriction on how we can assign probabilities to events in the world. But I admit some dissatisfaction at the explanatory power of the underlying physics at this point (discussed in some technical detail in a review by yet another Oxford philosopher of science, Christopher Timpson).
In both the Bayesian and Many Worlds interpretations (at least in the modern versions of the latter), probability is supposed to be completely subjective, as it should be. But something still seems to be missing: probability assignments are, in fact, testable, using techniques such as Bayesian model selection. What does it mean, in the purely subjective interpretation, to be correct, or at least more correct? Sometimes, this is couched as David Lewis’ “principal principle” (it’s very hard to find a good distillation of this on the web, but here’s a try): there is something out there called “objective chance” and our subjective probabilities are meant to track it (I am not sure this is coherent, and even Lewis himself usually gave the example of a coin toss, in which there is nothing objective at all about the chance involved: if you know the initial conditions of the coin and the way it is flipped and caught, you can predict the outcome with certainty.) But something at least vaguely objective seems to be going on in quantum mechanics: more probable outcomes happen more often, at least for the probability assignments that physicists make given what we know about our experiments. This isn’t quite “objective chance” perhaps, but it’s not clear that there isn’t another layer of physics still to be understood.
This morning I found what is undoubtedly one of the weirdest papers ever to appear on the arXiv, “Ettore Majorana: quantum mechanics of destiny”, by O. B. Zaslavskii. On the one hand, it’s a short retelling of the life of Ettore Majorana, a major figure in the development of mid-20th-Century particle physics. On the other, it’s a weird structural/semiotic analysis of Majorana’s life in the context of his work on quantum mechanics. That is, not an analysis of his work, but an analysis of his life as if it were a quantum-mechanical system!
Majorana is remembered nowadays for his work on the fundamental properties of particles, in particular neutrinos and the equations that can describe them. If neutrinos are their own antiparticles, they are called Majorana neutrinos (otherwise they are Dirac neutrinos, after the British physicist who wrote down another possible set of equations that describe particles like electrons which are different from their antiparticles). But he is also known for having disappeared in the late 1930s under so-called “mysterious circumstances”.
The paper makes the claim that Majorana’s disappearance was an example of his applying the logic of quantum mechanics to his own life (and death) — superposition, probability, uncertainty. If artists live their lives as works of art, why shouldn’t scientists live theirs as if they embodied their scientific ideas? Or at least, why can’t the historian use quantum mechanics as an interpretive structure for understanding the past? (Like, say, Freudian and Marxist literary criticism, or, more recently, the application of Darwinian evolution to literary theory — although “it would be pointless and, indeed, comical to base literary criticism on quantum mechanics, string theory, or general relativity” according to this article on Darwinian criticism.)
Well, I am all for breaking down the barriers between the two cultures…
For some light holiday reading, check out this slightly mistitled article from the NY Times on the still-unsolved mysteries of Quantum Mechanics. It’s always good PR to have Einstein’s name in the title, but really it’s about a theory -- Quantum Mechanics -- that Einstein didn’t like much. That article spawned an excellent post from Sean Carroll attempting to tease out the Quantum-Mechanical idea that the world is, at bottom, probabilistic. 97 thought-provoking comments and counting, including my own reaction to Sean’s statement that “what we can observe is only a small fraction of what really exists” and pointing to the quantum-mechanical wavefunction as “what really exists”: I don’t think this can be correct, since the wavefunction depends on what we know, not on what is out in the world. The wavefunction is certainly a useful (to again use Sean’s words) construct to enable us to do calculations; but that doesn’t make it real.
Today, Sean has posted this followup, talking about the many-worlds interpretation of Quantum Mechanics (which I admit seems ridiculous to me, although it has many eminent adherents): whenever a quantum-mechanical “decision” happens, the world splits in two, one with each of the possibilities. One world where the canonical Schrodinger’s cat is dead, one where it is, happily, alive.
Now, the problem with all of these discussions is that most of the possible interpretations of Quantum Mechanics are distinctions without difference: they make identical predictions which the fit the well-known and well-tested rules. (Not quite like Intelligent Design vs Evolution: ID makes the same predictions as Evolution by throwing away any predictions that don’t fit the facts a posteriori, whereas the different interpretations of Quantum Mechanics are a priori identical, alas.) Chris Fuchs likens the situation to physics before Einstein came up with Special Relativity -- one hundred years ago. All of the facts were in place -- Lorentz contractions, Maxwell’s equations of electromagnetism, the Michelson-Morley experiment showing that light didn’t seem to propagate through an ether -- but it took Einstein’s genius to notice that this could be explained by two simple principles: the speed of light is constant, and physics is the same in all inertial frames. So, it all comes back to Einstein in the end, especially while it’s still 2005.