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Category Archives: Quantum mechanics

Delayed Gratification

A post today by PZ Myers nicely expresses something which has been frustrating me about people who, in arguing over what can be a legitimate subject of “scientific” study, play the “untestable claim” card.

Their ideal is the experiment that, in one session, shoots down a claim cleanly and neatly. So let’s bring in dowsers who claim to be able to detect water flowing underground, set up control pipes and water-filled pipes, run them through their paces, and see if they meet reasonable statistical criteria. That’s science, it works, it effectively addresses an individual’s very specific claim, and I’m not saying that’s wrong; that’s a perfectly legitimate scientific experiment.

I’m saying that’s not the whole operating paradigm of all of science.

Plenty of scientific ideas are not immediately testable, or directly testable, or testable in isolation. For example: the planets in our solar system aren’t moving the way Newton’s laws say they should. Are Newton’s laws of gravity wrong, or are there other gravitational influences which satisfy the Newtonian equations but which we don’t know about? Once, it turned out to be the latter (the discovery of Neptune), and once, it turned out to be the former (the precession of Mercury’s orbit, which required Einstein’s general relativity to explain).

There are different mathematical formulations of the same subject which give the same predictions for the outcomes of experiments, but which suggest different new ideas for directions to explore. (E.g., Newtonian, Lagrangian and Hamiltonian mechanics; or density matrices and SIC-POVMs.) There are ideas which are proposed for good reason but hang around for decades awaiting a direct experimental test—perhaps one which could barely have been imagined when the idea first came up. Take directed percolation: a simple conceptual model for fluid flow through a randomized porous medium. It was first proposed in 1957. The mathematics necessary to treat it cleverly was invented (or, rather, adapted from a different area of physics) in the 1970s…and then forgotten…and then rediscovered by somebody else…connections with other subjects were made… Experiments were carried out on systems which almost behaved like the idealization, but always turned out to differ in some way… until 2007, when the behaviour was finally caught in the wild. And the experiment which finally observed a directed-percolation-class phase transition with quantitative exactness used a liquid crystal substance which wasn’t synthesized until 1969.

You don’t need to go dashing off to quantum gravity to find examples of ideas which are hard to test in the laboratory, or where mathematics long preceded experiment. (And if you do, don’t forget the other applications being developed for the mathematics invented in that search.) Just think very hard about the water dripping through coffee grounds to make your breakfast.

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Of Two Time Indices

In the appendix to a paper I am currently co-authoring, I recently wrote the following within a parenthetical excursus:

When talking of dynamical systems, our probability assignments really carry two time indices: one for the time our betting odds are chosen, and the other for the time the bet concerns.

A parenthesis in an appendix is already a pretty superfluous thing. Treating this as the jumping-off point for further discussion merits the degree of obscurity which only a lengthy post on a low-traffic blog can afford.

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Bohr’s Horseshoe

Now and then, one hears physicist stories of uncertain origin. Take the case of Niels Bohr and his horseshoe. A short version goes like the following:

It is a bit like the story of Niels Bohr’s horseshoe. Upon seeing it hanging over a doorway someone said, “But Niels, I thought you didn’t believe horseshoes could bring good luck.” Bohr replied, “They say it works even if you don’t believe.” [source]

I find it interesting that nobody seems to know where this story comes from. The place where I first read it was a jokebook: Asimov’s Treasury of Humor (1971), which happens to be three years older than the earliest appearance Wikiquote knows about. In this book, Isaac Asimov tells a lot of jokes and offers advice on how to deliver them. The Bohr horseshoe, told at slightly greater length, is joke #80. Asimov’s commentary points out a difficulty with telling it:

To a general audience, even one that is highly educated in the humanities, Bohr must be defined — and yet he was one of the greatest physicists of all time and died no longer ago than 1962. But defining Bohr isn’t that easy; if it isn’t done carefully, it will sound condescending, and even the suspicion of condescension will cool the laugh drastically.

Note the light dusting of C. P. Snow. Asimov proposes the following solution.

If you despair of getting the joke across by using Bohr, use Einstein. Everyone has heard of Einstein and anything can be attributed to him. Nevertheless, if you think you can get away with using Bohr, then by all means do so, for all things being equal, the joke will then sound more literate and more authentic. Unlike Einstein, Bohr hasn’t been overused.

I find this, except for the last sentence, strangely appropriate in the context of quantum-foundations arguments.

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Know Thy Audience?

D. W. Logan et al. have an editorial in PLoS Computational Biology giving advice for scientists who want to become active Wikipedia contributors. I was one, for a couple years (cue the “I got better”); judging from my personal experience, most of their advice is pretty good, save for item four:

Wikipedia is not primarily aimed at experts; therefore, the level of technical detail in its articles must be balanced against the ability of non-experts to understand those details. When contributing scientific content, imagine you have been tasked with writing a comprehensive scientific review for a high school audience. It can be surprisingly challenging explaining complex ideas in an accessible, jargon-free manner. But it is worth the perseverance. You will reap the benefits when it comes to writing your next manuscript or teaching an undergraduate class.

Come again?

Whether Wikipedia as a whole is “primarily aimed at experts” or not is irrelevant for the scientist wishing to edit the article on a particular technical subject. Plenty of articles — e.g., Kerr/CFT correspondence or Zamolodchikov c-theorem — have vanishingly little relevance to a “high school audience.” Even advanced-placement high-school physics doesn’t introduce quantum field theory, let alone renormalization-group methods, centrally extended Virasoro algebras and the current frontiers of gauge/gravity duality research. Popularizing these topics may be possible, although even the basic ideas like critical points and universality have been surprisingly poorly served in that department so far. While it’s pretty darn evident for these examples, the same problem holds true more generally. If you do try to set about that task, the sheer amount of new invention necessary — the cooking-up of new analogies and metaphors, the construction of new simplifications and toy examples, etc. — will run you slap-bang into Wikipedia’s No Original Research policy.

Even reducing a topic from the graduate to the undergraduate level can be a highly nontrivial task. (I was a beta-tester for Zwiebach’s First Course in String Theory, so I would know.) And, writing for undergrads who already have Maxwell and Schrödinger Equations under their belts is not at all the same as writing for high-school juniors (or for your poor, long-suffering parents who’ve long since given up asking what you learned in school today). Why not try that sort of thing out on another platform first, like a personal blog, and then port it over to Wikipedia after receiving feedback? Citing your own work in the third person, or better yet recruiting other editors to help you adapt your content, is much more in accord with the letter and with the spirit of Wikipedia policy, than is inventing de novo great globs of pop science.

Popularization is hard. When you make a serious effort at it, let yourself get some credit.

Know Thy Audience, indeed: sometimes, your reader won’t be a high-school sophomore looking for homework help, but is much more likely to be a fellow researcher checking to see where the minus signs go in a particular equation, or a graduate student looking to catch up on the historical highlights of their lab group’s research topic. Vulgarized vagueness helps the latter readers not at all, and gives the former only a gentle illusion of learning. Precalculus students would benefit more if we professional science people worked on making articles like Trigonometric functions truly excellent than if we puttered around making up borderline Original Research about our own abstruse pet projects.

ARTICLE COMMENTED UPON

  • Logan DW, Sandal M, Gardner PP, Manske M, Bateman A, 2010 Ten Simple Rules for Editing Wikipedia. PLoS Comput Biol 6(9): e1000941. doi:10.1371/journal.pcbi.1000941
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Textbook Cardboard and Physicist’s History

By the way, what I have just outlined is what I call a “physicist’s history of physics,” which is never correct. What I am telling you is a sort of conventionalized myth-story that the physicists tell to their students, and those students tell to their students, and is not necessarily related to the actual historical development, which I do not really know!

Richard Feynman

Back when Brian Switek was a college student, he took on the unenviable task of pointing out when his professors were indulging in “scientist’s history of science”: attributing discoveries to the wrong person, oversimplifying the development of an idea, retelling anecdotes which are more amusing than true, and generally chewing on the textbook cardboard. The typical response? “That’s interesting, but I’m still right.”

Now, he’s a palaeontology person, and I’m a physics boffin, so you’d think I could get away with pretending that we don’t have that problem in this Department, but I started this note by quoting Feynman’s QED: The Strange Theory of Light and Matter (1986), so that’s not really a pretence worth keeping up. When it comes to formal education, I only have systematic experience with one field; oh, I took classes in pure mathematics and neuroscience and environmental politics and literature and film studies, but I won’t presume to speak in depth about how those subjects are taught.

So, with all those caveats stated, I can at least sketch what I suspect to be a contributing factor (which other sciences might encounter to a lesser extent or in a different way).

Suppose I want to teach a classful of college sophomores the fundamentals of quantum mechanics. There’s a standard “physicist’s history” which goes along with this, which touches on a familiar litany of famous names: Max Planck, Albert Einstein, Niels Bohr, Louis de Broglie, Werner Heisenberg, Ernst Schrödinger. We like to go back to the early days and follow the development forward, because the science was simpler when it got started, right?

The problem is that all of these men were highly trained, professional physicists who were thoroughly conversant with the knowledge of their time — well, naturally! But this means that any one of them knew more classical physics than a modern college sophomore. They would have known Hamiltonian and Lagrangian mechanics, for example, in addition to techniques of statistical physics (calculating entropy and such). Unless you know what they knew, you can’t really follow their thought processes, and we don’t teach big chunks of what they knew until after we’ve tried to teach what they figured out! For example, if you don’t know thermodynamics and statistical mechanics pretty well, you won’t be able to follow why Max Planck proposed the blackbody radiation law he did, which was a key step in the development of quantum theory.

Consequently, any “historical” treatment at the introductory level will probably end up “conventionalized.” One has to step extremely carefully! Strip the history down to the point that students just starting to learn the science can follow it, and you might not be portraying the way the people actually did their work. That’s not so bad, as far as learning the facts and formulæ is concerned, but you open yourself up to all sorts of troubles when you get to talking about the process of science. Are we doing physics differently than folks did N or 2N years ago? If we are, or if we aren’t, is that a problem? Well, we sure aren’t doing it like they did in chapter 1 of this textbook here. . . .

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Calloo! Callay!

My copy of Quantum Mechanics and Path Integrals by Feynman and Hibbs just arrived! If, say, David Griffiths’ textbook epitomizes the ordinary “vernacular” treatment of quantum mechanics, QMaPI is a classic unorthodox approach. Intended for students who already have a bit of background in the subject, it builds up the Lagrangian alternative to the Hamiltonian method, a highly useful idea when one goes on to study field theory, string theory or advanced statistical physics.

For years, this book was only available in beat-up old library copies and illegal DJVU files from Lithuania, but now, Dover has brought forth a new edition. I’m not certain on this, but it appears as if the book was so heavily pre-ordered that Amazon.com sold out of it the day it became available for purchase.

EDIT TO ADD: One erratum — on p. 364, Thorber should be Thornber.

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Monday BPSDB: Null Physics

BPSDBA fellow named Terry Witt has been advertising his self-published book, Our Undiscovered Universe, in places like Discover magazine and Scientific American. Unfortunately, the ad pages aren’t exactly peer-reviewed, or even cross-checked with a nearby grad student; being businesses, magazines naturally care about revenue. Upon examination, Our Undiscovered Universe turns out to be brimming over with crank physics and general nonsense. Ben Monreal, who was one of the intimidatingly smart people in the lab where I did my undergrad thesis, has weighed Witt’s “Null Physics” and found it wanting; his review of Our Undiscovered Universe is quite a good read.

Witt’s book starts with pseudomathematics before moving on to pseudophysics. As Ben explains,

Chapter 1 is where Witt lays out a series of “proofs” derived from what he calls the “Null Axiom”. That axiom is: “Existence sums to nonexistence” (pg. 28)—something that Witt calls self-evident after a page of invalid set theory. The central mistake, if I had to identify one, is the claim that “X does not exist” is the same as “everything except X exists”. This is utter baloney, whether in formal logic or in set theory or in daily experience.

Actually, as the book unfolds, Witt doesn’t appear to use this dead-in-the-water non-axiom for anything. He does, however, pile on more pseudomathematics:

Chapter 3 contains such gems as Theorem 3.1: “The Existence of Any Half of the Universe is Equal to the Nonexistence of the Other Half” (pg. 66) and Theorem 3.9: “The Time Required for Light to Traverse the Universe is Eternity, infinity/c” (pg. 72). I am not making this up. Witt throws around “infinity” as though it were an ordinary real number; he multiplies and divides by it, etc., with normal algebraic cancellation. This is complete nonsense; there are two centuries of mathematical thought figuring out the mathematical properties of infinity, and Witt’s approach is valid in exactly none of them. (Witt later explained on his online forum—currently disabled—that he’s reinvented all of the mathematics associated with “infinity”. His reasoning, if that’s what you call it, was that his new definition jibed with a grand idea about math being dependent on nature; it was an argument from incredulity.)

When Witt does finally get around to physics, five chapters into the book, he doesn’t do any better.
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Quantum Woo, Part N

BPSDBTime for a little BPSDB! The redoubtable Ben Goldacre has the dirt on Bill Nelson’s “QXCI machine,” a device for “bioenergetic health auditing,” a medical procedure well-known among specialists as an essential step in the surgical removal of cash from wallets. Best of all, though, is what QXCI stands for: Quantum Xrroid Consciousness Interface. Now, quantum physics has jack to do with consciousness, but more importantly, “quantum xrroid” just sounds. . . painful. Like a blood boil growing inside your X, if you know what I mean.

Maybe a “quantum xrroid” means that your X is in a superposition of inflamed and not inflamed and only settles on one or the other option when your doctor examines it.

(Incidentally, I met the redoubtable Ben Goldacre in Vegas a few weeks ago — and thereby would hang a tale, if he weren’t still hoarding the photo evidence.)

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Quantum One

Michael Nielsen’s recent essay “Why the world needs quantum mechanics,” about the quintessential weirdness of quantum phenomena, provoked Dave Bacon to ask if there’s a better way to teach introductory courses in quantum physics. This question strikes a chord with me, since my first semester of college quantum — the class known as “8.04″ — was rather remarkably dreadful.

It began with some fluff about early models of the atom, leaving out most of the ideas actually proposed in favor of a “textbook cardboard” version of the discoveries made in early TwenCen. If we can’t teach history well, why teach it at all? We’re certainly not promoting a genuine understanding of how science works if we only present a caricature of it. I doubt one could even instil an appreciation for the problems which Bohr, Heisenberg, Schrödinger and company solved in the years leading up to 1927: sophomore physics students can’t follow in their footsteps, because sophomore physics students don’t know as much classical physics as professional physicists of the 1920s did. To understand their starting point and the steps they took requires, oddly enough, subject material which even MIT undergrads don’t learn until later.
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Decoherent Dewey

Click to embiggen slightly:

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Quantum Mechanics in Your Face

Via Imaginary Potential comes Sidney Coleman’s lecture on how quantum mechanics differs from classical and what that whole “collapsing the wave function” business is all about. The lecture is geared to those who have a working familiarity with first-term quantum physics: the Schrödinger Equation, spin operators and such.

The video quality is not always quite good enough to capture what’s written on the transparencies, but the audio makes up for it.

EDIT TO ADD: I don’t actually agree with the final thesis of Coleman’s lecture (I’ve gone too far in my reading of Appleby, Barnum, Caves, Fuchs, Kent, Leifer, Peres, Schack, Spekkens, Unruh, Zeilinger and so on to make that retreat). However, I would say that (a) the GHZ story is easier to remember than the Bell story, and (b) “vernacular” quantum mechanics is a good term to have on hand, as the mishmash we get from several generations of skipping-past-the-weird-bits shouldn’t necessarily be called a “school of thought” in its own right.

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Quote of the Day

Today’s prize for best use of scientific terminology for humorous effect goes to Jeff Medkeff:

I did notice in the course of my reading that the density of baloney in this story is very high — maybe even degenerate.

Well, I laughed.

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What Can the LHC Tell Us?

What can the LHC tell us, and how long will we have to wait to find out?

Over at Symmetry Breaking, David Harris has a timeline for when the amount of data gathered at the LHC will be large enough to detect particular exciting bits of physics which we expect might be lurking in wait, at high-energy realms we can’t currently reach. (The figures come from Abe Seiden’s presentation at the April 2008 meeting of the American Physical Society.) Assuming the superconducting cables — all 7000 kilometers of them! — get chilled down to their operating temperatures by mid-June and particles start whirling around the ring on schedule after that, then we could hope to spot the Higgs boson as early as 2009.
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An Unusual Occurrence

So there I was, quietly standing in Lobby 10, queuing to buy myself and a few friends advance tickets to Neil Gaiman’s forthcoming speech at MIT, when a strange odor proturbed onto my awareness. “That’s odd,” thought I, “it smells like backstage at my high school’s auditorium. [snif snif] Or the bathroom at Quiz Bowl state finals. . . And it’s not even 4:20. Something very unusual is going on, here on this university campus.”

I became aware of a, well, perhaps a presence would be the best way to describe it: the sort of feeling which people report when their temporal and parietal lobes are stimulated by magnetic fields. Something tall and imposing was standing. . . just. . . over. . . my. . . right. . . shoulder! But when I turned to see, I saw nothing there.

Feeling a little perturbed, I bought my tickets and tried to shrug it off. Not wanting to deal with the wet and yucky weather currently sticking down upon Cambridge, I descended the nearest staircase and began to work my way eastward through MIT’s tunnel system, progressing through the “zeroth floors” of the classroom and laboratory buildings, heading for Kendall Square and the T station. Putting my unusual experience in the ticket queue out of my mind, I returned to contemplating the junction of physics and neuroscience:

“So, based on the power-law behavior of cortical avalanches, we’d guess that the cortex is positioned at a phase transition, a critical point between, well, let’s call them quiescence and epileptic madness. This would allow the cortex to sustain a wide variety of functional patterns. . . but at a critical point, the Wilson-Cowan equations should yield a conformal field theory in two spatial dimensions. . . .

“But if you reinterpret the classical partition function as a quantum path integral, a field theory in 2D becomes a quantum field theory in one spatial and one temporal dimension. And the central charge of the quantum conformal field theory is equal to the normalized entropy density. . . so we should be able to apply gauge/gravity duality and model the cortex as a black hole in anti-de Sitter spacetime —”

Suddenly, a tentacle wrapped around my chest, and constricted, and pulled, and lifted — not up, but in a direction I had never moved before. Like a square knocked out of Flatland, I had been displaced.
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Categorical Information Theory

Ben Allen is now on the arXivotubes, with a category-theoretic arithmetic of information.

The concept of information has found application across the sciences. However, the conventional measures of information are not appropriate for all situations, and a more general mathematical concept is needed. In this work we give axioms that characterize the arithmetic of information, i.e. the way that pieces of information combine with each other. These axioms allow for a general notion of information functions, which quantify the information transmitted by a communication system. In our formalism, communication systems are represented as category-theoretic morphisms between information sources and destinations. Our framework encompasses discrete, continuous, and quantum information measures, as well as familiar mathematical functions that are not usually associated with information. We discuss these examples and prove basic results on the general behavior of information.

It looks like a discussion about this is starting over at the n-Category Café. If I didn’t have to spend today cutting down a 12-page paper to eight pages for an overpriced book of conference proceedings which nobody will read, I’d totally be writing more about it!

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SUSY QM: The 1D Dirac Hamiltonian

Whew! We spent a considerable amount of wordage developing the Dirac Equation. Now, it’s time to tie this development back to the supersymmetry material we studied earlier in the non-relativistic context. The result will be a surprising mapping between relativistic and non-relativistic quantum mechanics. Today, we’ll just get the gist of it, and to get started, we’ll begin with the final equation we had before,

[tex](i\displaystyle{\not} \partial – m)\psi = 0.[/tex]

Recalling Feynman’s notation of slashed quantities,

[tex]\displaystyle{\not} a = \gamma^\mu a_\mu,[/tex]

we can unpack this a little to

[tex]\left(i\gamma^\mu\partial_\mu – m\right) \psi = 0,[/tex]

which we can elaborate to include an electromagnetic field as follows:

[tex]{\left[i\gamma^\mu(\partial_\mu + iA_\mu) - m\right] \psi = 0.[/tex]

The Dirac Hamiltonian [tex]H_D[/tex] has a rich SUSY structure, of which we can catch a glimpse even having pared the problem down to its barest essentials. To take the simplest possible case, consider a Dirac particle living in one spatial dimension, on which there also lives a scalar potential [tex]\phi(x^1)[/tex]. (We could call this a “1+1-dimensional” system, to remind ourselves of the difference between time and space.) The SUSY structure can be seen most clearly when we look at the limit of a massless particle; this eliminates the [tex]m[/tex] term we had before.
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