# What I Do

At the moment, I’m taking a quick break from reading some rather dense mathematical prose, and I spent yesterday plugging away at a draft of my research group’s next technical publication. This led me to reflect on a lesson that I think a lot of science education leaves out: Even in a technical article, you have to have a story to carry the progression through. “These are all the boffo weird roadside attractions we found while proving the theorems in our last paper” is honest, but not adequate.

Our research project is the reconstruction of the mathematical formalism of quantum theory from physical principles. We tease apart the theory, identify what is robustly strange about it — for many more quantum phenomena can be emulated with classical stochasticity than are often appreciated — and try to build a new representation that brings the most remarkable features of the physics to the forefront. In special relativity, we have Einstein’s postulates, and the dramatic tension between them: Inertial observers can come to agree upon the laws of physics, but they cannot agree upon a standard of rest. In thermodynamics, we have the Four Laws, which come with their own dramatic tension, in that energy is conserved while entropy is nondecreasing. Both of these theories are expressed in terms of what agents can and cannot do, yet they are more than “mere” engineering, because they apply to all agents. Or, to say it another way, it is to the benefit of any agent to pick up the theory and use it as a guide.

What, then, is the analogue for quantum theory? If the textbook presentation of quantum physics is like the formulae for the Lorentz transform, with all those square roots and whatnot, or the Maxwell relations in thermo, with all those intermingling partial derivatives that we invent hacks about determinants to remember, what is quantum theory’s version of Einstein’s postulates or the Four Laws?

That’s the grandiose version, anyway. The reason I got invited to speak at an American Mathematical Society meeting is that the geometric structures that arise in this work are vexingly fascinating. You want about Galois fields and Hilbert’s 12th problem? We’ve got ’em! How about sphere packing and unexpected octonions? We’ve got those, too! And the structure that leads down the latter path turns out, on top of that, to yield a new way of thinking about Mermin’s 3-qubit Bell inequality. It is all lovely, and it is all strange.

The SIC problem gives us the opportunity to travel all throughout mathematics, because, while the definition looks pretty small, the question is bigger on the inside.

# To Thems That Have

Occasionally, I think of burning my opportunities of advancing in the physics profession — or, more likely, just burning my bridges with Geek Culture(TM) — by writing a paper entitled, “Richard Feynman’s Greatest Mistake”.

I did start drafting an essay I call “To Thems That Have, Shall Be Given More”. There are a sizable number of examples where Feynman gets credit for an idea that somebody else discovered first. It’s the rich-get-richer of science.
Continue reading To Thems That Have

# 17 Equations that Clogged My Social-Media Timeline

An image burbled up in my social-media feed the other day, purporting to be a list of “17 Equations that Changed the World.” It’s actually been circulating for a while (since early 2014), and purports to summarize the book by that name written by Ian Stewart. This list is typo-ridden, historically inaccurate and generally indicative of a lousy knowledge-distribution process that lets us down at every stage, from background research to fact-checking to copy-editing.
Continue reading 17 Equations that Clogged My Social-Media Timeline

# Time Capsule

While looking through old physics books for alternate takes on my quals problems, I found a copy of Sir James Jeans’ Electricity and Magnetism (5th edition, 1925). It’s a fascinating time capsule of early views on relativity and what we know call the “old quantum theory,” that is, the attempt to understand atomic and molecular phenomena by adding some constraints to fundamentally classical physics. Jeans builds up Maxwellian electromagnetism starting from the assumption of the aether. Then, in chapter 20, which was added in the fourth edition (1919), he goes into special relativity, beginning with the Michelson–Morley experiment. Only after discussing many examples in detail does he, near the end of the chapter, say

If, then, we continue to believe in the existence of an ether we are compelled to believe not only that all electromagnetic phenomena are in a conspiracy to conceal from us the speed of our motion through the ether, but also that gravitational phenomena, which so far as is known have nothing to do with the ether, are parties to the same conspiracy. The simpler view seems to be that there is no ether. If we accept this view, there is no conspiracy of concealment for the simple reason that there is no longer anything to conceal.

# 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.

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

# 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.

# 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.
Continue reading What Can the LHC Tell Us?

# 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,

$$(i\displaystyle{\not} \partial – m)\psi = 0.$$

Recalling Feynman’s notation of slashed quantities,

$$\displaystyle{\not} a = \gamma^\mu a_\mu,$$

we can unpack this a little to

$$\left(i\gamma^\mu\partial_\mu – m\right) \psi = 0,$$

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

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

The Dirac Hamiltonian $$H_D$$ 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 $$\phi(x^1)$$. (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 $$m$$ term we had before.
Continue reading SUSY QM: The 1D Dirac Hamiltonian

# Intermezzo: The Dirac Equation

After you’ve been Pharyngulated a couple times, you develop a protective strategy to deal with the aftermath. “How,” you ask yourself, “can I get rid of the extra readers whom I’ve probably picked up?” The answer, for me at least, is clear:

Math!

RECAP

Science After Sunclipse has been presenting an introduction to supersymmetric quantum mechanics. This area of inquiry stemmed from attempts to understand the complicated implications of supersymmetry in a simpler setting than quantum field theory; just as supersymmetry began in string theory and developed into its own “thing,” so too has this offshoot become interesting in its own right. In a five-part series, we’ve seen how the ideas of “SUSY QM” can be applied to practical ends, such as understanding the quantum properties of the hydrogen atom. I have attempted to make these essays accessible to undergraduate physics students in their first or possibly second term of quantum theory. Having undergraduates solve the hydrogen atom in this fashion is rather unorthodox, but this is a safe kind of iconoclasm, as it was endorsed by three of my professors.

The posts in this series to date are as follows:

Having solved the “Coulomb problem,” we have attained a plateau and can move in several directions. The solution technique of shape-invariant partner potentials is broadly applicable; virtually all potentials for which introductory quantum classes solve the Schrödinger Equation can be brought into this framework. We can also move into new conceptual territory, connecting these ideas from quantum physics to statistical mechanics, for example, or moving from the non-relativistic regime we’ve studied so far into the territory of relativity. Today, we’ll take the latter route.

We’re going to step aside for a brief interlude on the Dirac Equation. Using some intuition about special relativity, we’re going to betray our Vulcan heritage and take a guess — an inspired guess, as it happens — one sufficiently inspired that I strongly doubt I could make it myself. Fortunately, Dirac made it for us. After reliving this great moment in TwenCen physics, we’ll be in an excellent position to explore another aspect of SUSY QM.

REFRESHER ON RELATIVITY

Let’s ground ourselves with the basic principles of special relativity. (Recently, Skulls in the Stars covered the history of the subject.) First, we have that the laws of physics will appear the same in all inertial frames: if Joe and Moe are floating past each other in deep space, Joe can do experiments with springs and whirligigs and beams of light to deduce physical laws, and Moe — who Joe thinks is moving past with constant velocity — will deduce the same physical laws. Thus, neither Joe nor Moe can determine who is “really moving” and who is “really standing still.”

Second, all observers will measure the same speed of light. In terms of a space-time diagram, where time is conventionally drawn as the vertical axis and space as the horizontal, Joe and Moe will both represent the progress of a light flash as a diagonal line with the same slope. (This video has some spiffy CG renditions of the concept.) To make life easy on ourselves, we say that this line has a slope of 1, and is thus drawn at a 45-degree angle from the horizontal. This means we’re measuring distance and time in the same units, a meter of time being how long it takes light to travel one meter.
Continue reading Intermezzo: The Dirac Equation

# Yet Another Relativity Denier

Exercise: find the mistake in this attempt to challenge Einstein. Hint: if an observer in one Lorentz frame measures the position of a particle to be changing as $$x = ct$$, then that particle is traveling at the speed of light, and all observers in other Lorentz frames will agree.

Bonus point: explain the difference between the speed of light in a vacuum and the speed of light as measured when light is passing through matter.

(Thanks to the reader who noticed the “relativity challenge” Google ad in my sidebar. You know, it’s not quite cricket for me to plead that the Gentle Reader click on those links, but I can’t help it if other people appreciate the irony of pseudoscience making micropayments to science.)

# Happenings

The Internet is making me feisty and argumentative (exhibit A). In my current mood, I’d be apt to fill this space with spite; fortunately, other people blag so I don’t have to.

First in a random ordering, Ben Allen asks, “How complex is a human?” Entertaining arithmetic ensues in the comments. Next, our friend gg kicks off a series of posts on Einstein’s relativity. Oddly enough, relativity has also showed up in Steven Novella’s latest debunking of Michael Egnor.

Finally, Russell Blackford has saved me the trouble of blagging about an odd story concerning religion and nanotechnology.

I’m currently supposed to be writing two different journal articles at my day job, but I’ll see if I can eke out the time to continue my supersymmetry series. There’s no better way to get myself a little peace and quiet than to post lots of equations!

# Teh Burning Stupid: Relativity Edition

To put the “moral” at the beginning, let’s summarize. If you want to raise my blood pressure, one good way to do it is to write a completely wrong, back-to-front absurd tirade against all of twentieth-century physics. Anyone can slip a few errors into an essay, or even a few “fundamental” errors, but if you want the brass ring, you need at the very least to misrepresent the special theory of relativity, the general theory of relativity, quantum mechanics, the use of mathematics in physics and the scientific method. Bonus points if you confuse general relativity with quantum physics; a woop-woop-woop special prize for taking a non-true assertion and calling it a “fundamental premise” of quantum mechanics; and an extra cherry on top if you take three famous observations which support general relativity, lie about two of them and forget the third.

The story so far:

I might write an actual, non-linkfesty post about this. . . but then again, other corners of the Network are calling to me and reminding me of overdue obligations, so I might leave it to my colleagues.

UPDATE (6 December): gg now has Part 2 out on the blogonets.

UPDATE (16:47 o’clock): Tyler DiPietro dons the asbestos and joins the fun, followed quickly by Mark Chu-Carroll.

UPDATE (9 December): Flavin of the St. Louis Skeptical Society offers an essay.

# Autodynamics

Ah, some light Friday fare!

By now, everybody has probably heard about the forthcoming crackpot “documentary” from David de Hilster, Einstein Wrong – The Miracle Year. Currently looking for financial backing, de Hilster hopes to release this flick in 2008, doing for relativity what What the Bleep Do We Know (2004) did for quantum physics: namely, let the fractured ceramics have free play.

As it turns out, David de Hilster is one of the Network’s classic relativity cranks. He’s been pushing his pet theory, “Autodynamics,” since at least the early 1990s (on the sci.physics Usenet group). As it also turns out, Autodynamics has plenty of problems. For example, it chucks out the Lorentz transformations, thereby making itself inconsistent with the Maxwell equations, which form our basic understanding of electricity and magnetism, without which the technological support system of modern society couldn’t exist.

What’s more, they don’t like that nasty ol’ equation

$$E = mc^2.$$

The Autodynamicist revulsion at this horrible formula has led them to propose — no, I’m not making this up — that $$E$$ should equal $$mc^3$$ instead.