Quantum angels

One of the reasons I get so impatient with woo-woos is that science is plenty cool enough without making shit up.

But because quantum physics is already weird even without any embellishment or misinterpretation, it's been particularly prone to being co-opted by woo-woos in their search for explanations supporting (choose one or more of the following):

• homeopathy
• psychic abilities
• astrology
• "natural healing"
• the soul
• "chakras" and "qi"
• auras

But you don't need to do any of this to make quantum physics cool, and I've got two good examples.  Let's start with an experiment regarding quantum entanglement -- the linking of two particles in a state describable by a single wave function.  While this might seem uninteresting at first, what it implies is that altering the spin state of particle A would instantaneously change the spin state of its entangled partner, particle B -- regardless of how far apart the two were.  It's almost as if the two were engaging in faster-than-light communication.  Most physicists, of course, do not believe this is what happens -- that it's more like separating a pair of gloves, each in its own sealed box, and sending one to Alpha Centauri.  Then you open the box that's still here on Earth, and find it contains the right-handed glove; at that point, you automatically know that the one on Alpha Centauri must contain the left-handed glove.  Information didn't travel anywhere; that knowledge is just a function of how the pairing works.

However, entanglement is still one of those things that isn't fully explained, even that way.  There's a further twist on this, and that's where things get even more interesting.  Most physicists couple the entanglement phenomenon with the idea of "local realism" -- that the two particles' spin must have been pointing in some direction prior to measurement, even if we didn't know what it was.  Thus, the two entangled particles might have "agreed" (to use an admittedly anthropomorphic term) on what the spin direction would be prior to being separated, simulating communication where there was none, and preserving Einstein's idea that the theories of relativity prohibit faster-than-light communication.

Right?

Scientists at Delft University of Technology in the Netherlands seem to have closed that loophole.  Using an extremely fast random number generator, they altered the spin state of one of two entangled particles separated by 1.3 kilometers, and measured the effect on its partner.  The distance makes it impossible for sub-light-speed communication between the two.  This tosses out the idea of local realism; if the experiment's results hold -- and they certainly seem to be doing so -- the particles were indeed communicating faster than light, something that isn't supposed to be possible.  Einstein was so repelled by this idea that he called it "spooky action at a distance."

To quote the press release:

With the help of ICFO’s quantum random number generators, the Delft experiment gives a nearly perfect disproof of Einstein's world-view, in which "nothing travels faster than light" and “God does not play dice.”  At least one of these statements must be wrong.  The laws that govern the Universe may indeed be a throw of the dice.

If this wasn't weird and cool enough, a second experiment performed right here at Cornell University supported one of the weirdest results of quantum theory -- that a system cannot change while you're watching it.

Graduate students Yogesh Patil and Srivatsan K. Chakram cooled about a billion atoms of rubidium to a fraction of a degree above absolute zero, and suspended them between lasers.  Under such conditions, the atoms formed an orderly crystal lattice.  But because of an effect called "quantum tunneling," even though the atoms were cold -- and thus nearly motionless -- they could shift positions in the lattice, leading to the result that any given atom could be anywhere in the lattice at any time.

Patel and Chakram found that you can stop this effect simply by observing the atoms.

This is the best experimental verification yet of what's been nicknamed the Quantum Zeno effect, after the Greek philosopher who said that motion was impossible because anyone moving from Point A to Point B would have to cross half the distance, then half the remaining distance, then half again, and so on ad infinitum -- and thus would never arrive.  Motion, Zeno said, must therefore be an illusion.

"This is the first observation of the Quantum Zeno effect by real space measurement of atomic motion," lab director Mukund Vengalattore said.  "Also, due to the high degree of control we've been able to demonstrate in our experiments, we can gradually 'tune' the manner in which we observe these atoms.  Using this tuning, we've also been able to demonstrate an effect called 'emergent classicality' in this quantum system."

Myself, I'm not reminded so much of Zeno as I am of another thing that doesn't move while you watch it.

See what I mean?  You don't need to add all sorts of woo-woo nonsense to this stuff to make it fascinating.  It's cool enough on its own.

Of course, the problem is, understanding it takes some serious effort.  Physics is cool, but it's not easy.  All of which supports a contention I've had for years; that woo-wooism is, at its heart, based in laziness.

Me, I'd rather work a little harder and understand reality as it is.  Even if it leaves me afraid to blink.

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Twists and turns

A recommendation to anyone who wants to completely revolutionize the scientific world: learn some damn science first.

It's why I get so completely fed up by people like Deepak Chopra, who blather on about "quantum frequencies" when I doubt he could give an accurate definition of either word.  Look, I get that physics is hard; I majored in physics, for fuck's sake.  Okay, I wasn't very good at it, but at least I came away with (1) a great deal of respect for the people who are smart enough to truly understand it, and (2) a determination not to pretend I'm an expert when I'm not.

But this isn't the perspective that a great many people have, to judge by the success of Chopra's books, which include -- I shit you not -- Quantum Healing and Quantum Body.

But I'm not here to rail about Deepak Chopra, who in any case has been something of a frequent flier here at Skeptophilia.  No, today's rant comes to you courtesy of a long-time loyal reader who asked me if I'd ever heard of "torsion field theory" and if so, what I thought about it.

My first thought was that any kind of field theory was going to involve mathematics on a level that would lose me after the first paragraph, so (Cf. my statement in paragraph two above) whatever opinion I had of it wouldn't be worth much.  But I'm nothing if not dedicated to my readers, so I said I'd look into it.

And... holy Moses.

Torsion field theory was born of some research (using the term loosely) in the 1980s by two Russian scientists (using that term loosely as well), Anatoly Akimov and Gennady Shipov.  The basic idea was that a particle's spin configuration causes it to give off "emanations" that allow for the transfer of information faster than the speed of light.

If you're thinking, "Wait... but... Einstein said...?", you're not the only one.  In 1991, physicist Yevgeny Aleksandrov exposed them as frauds, and called the grants they'd received from the Russian government to support their work "embezzlement."

Anyone who's saying, "Okay, well, that was that, then," obviously doesn't understand how persistent the purveyors of pseudoscience can be.  Akimov and Shipov portrayed Aleksandrov's attacks as coming from a hidebound scientific establishment that couldn't handle being challenged -- and also wanted to keep all the grant money for itself.  (Similar to all of the alt-med proponents complaining about being suppressed by "Big Pharma.")  They fought back -- and won, receiving grants from the Russian government throughout the 1990s, and ultimately founding "The International Institute for Theoretical and Applied Physics" to continue doing their thing.  (Thus showing that having a fancy-sounding name for your "institute" doesn't mean that you're doing actual science.)

Not a single thing they did -- not one -- ever generated a paper in a peer-reviewed physics journal.  Despite this, "torsion field theory" is still being talked about as a "revolution in physics" (and its proponents still claim the physics community is suppressing it), and it has been used to explain -- once again, I feel obliged to mention that I am not making this up -- such phenomena as telepathy, clairvoyance, telekinesis, and levitation.  It's said to be the basis of homeopathic "remedies," perpetual motion machines, stargates, and UFO propulsion systems.

Did you notice a commonality between every one of the things I just listed?

Yeah, me too.

Here's the problem.  This is not how science works.  Proposing a "theory" that flies in the face of not one, but two of the most thoroughly tested models in physics (the theories of relativity and quantum mechanics), based upon exactly zero evidence, and then using that "theory" to explain a bunch of phenomena that to the best of our current knowledge, don't exist, isn't science.  It's self-delusion at best, and outright fraud at worst.  And it doesn't improve things when you name it by swiping some actual terms from physics (torsion means a twisting force; a field is a distribution of values of a quantity in space).

A diagram of the torsion tensor. Like, you know, actual science [Image licensed under the Creative Commons Circle development with torsion, CC BY-SA 4.0]

No, science isn't perfect.  But it does have one enormously important thing going for it -- it self-corrects.  And scientists, far from being the sticks-in-the-mud the pseudoscientific community would like you to believe, are always on the lookout for the places it's not working, because identifying and correcting those places is how careers are made.  If there really was some mysterious twisty-turny field generated by quantum spin that could generate faster-than-light information transfer, the physicists would be clambering over each other to get their papers published first.

That'd be Nobel Prize material, right there.

So many thanks to the reader who suggested I research "torsion field theory."  I now have many dents in my forehead from all the faceplants I did.  If you find any other revolutionary developments in physics that for some reason no actual physicists are working on, though, I'd rather not know about them.

Maybe you should just send them directly to Deepak Chopra.

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The forbidden light

In the early nineteenth century, two scientists -- Joseph von Fraunhofer and Charles Wheatstone -- independently observed something strange; if you heated up samples of various elements, they emitted a light spectrum that contained strong peaks at certain frequencies, showing up as bright lines instead of a continuous rainbow of colors.

It quickly became obvious that this property could be used to identify the presence of different elements in mixed samples.  In fact, helium was discovered when French astronomer Georges Rayet found emission lines in the solar spectrum that didn't correspond to any other known element, making it the only element in the periodic table first detected somewhere other than on Earth.  (The name helium comes from the Greek Ἥλιος, meaning the Sun.)

Figuring out why this phenomenon occurred, though, took almost a hundred years.  The explanation, due in large part to the work of Danish physicist Niels Bohr, has to do with the fact that the electron shells in atoms are quantized -- there are only certain allowed energy levels, so an atom has to absorb a particular frequency of light in order for one of its electrons to jump to the next level (or, conversely, to drop to a lower level, the atom has to emit a photon of a particular frequency).  This simultaneously explained the specificity of emission spectra and the odd phenomenon of absorption spectra, where broad-spectrum light passing through transparent substances shows dark lines where certain frequencies are absorbed, effectively subtracting them from the beam.

So each element has its own distinctive "fingerprint" of spectral lines, which is how researchers here on Earth can determine the chemical composition of distant stars, and even the constituents of the atmospheres of exoplanets.

The emission spectrum of iron [Image is in the Public Domain]

However -- as usual -- even this rather complex model has some unexpected twists.

Very rarely, the electrons in atoms will undergo forbidden transitions, resulting in light being emitted that should not be possible from the element in question.  (A simple analogy is if you were climbing a staircase, and somehow were able to go up by one-and-three-quarters steps.)  These transitions are highly unstable (just as your attempted ascent would be), and the electron almost instantaneously collapses back into one of the allowed energy states, but when it does so the atom emits a frequency of light you wouldn't expect.  So these aren't so much forbidden as they are extremely improbable; in ordinary situations, their contribution to the light spectrum is vanishingly small.

But in very high energy conditions, where the electrons are bouncing all over the place millions of times per second, you begin to see a significant contribution from forbidden transitions.

The reason this comes up is because of a study of a Seyfert galaxy named MCG 01-24-014.  Seyfert galaxies, named after American astronomer Carl Keenan Seyfert who studied them extensively, look superficially like ordinary spiral galaxies, but have an active galactic nucleus.  This latter name is a massive understatement, mostly because astronomers shy away from calling something "Holy Shit This Thing Is Super Bright, No Really You Have No Idea How Bright It Is."  The center bit of a Seyfert galaxy has a luminosity equal to the luminosity of all the stars of the Milky Way put together, and is thought to be the result of large quantities of material falling rapidly into a supermassive black hole.  Most of the light emitted is outside of the visible spectrum -- thus their ordinary appearance through a telescope -- but when viewed in other frequency ranges, it becomes obvious how weird they are.  

The Circinus Galaxy, one of the best-studied Seyfert galaxies [Image is in the Public Domain courtesy of NASA/JPL]

And MCG 01-24-014 is really peculiar -- emitting far more light from forbidden transitions than even an average Seyfert galaxy would.  So whatever is powering its galactic core is running full-throttle.

The forbidden light of Seyfert galaxies provides us with yet another example of "you think you understand, then nature throws you a curve ball."

Sometimes you hear the criticism levied at scientists that all the technical details somehow take away from the wonder of simply looking up and delighting at the beauty of the night sky.  I can't speak for anyone else, but for me, the exact opposite is true.  I can still go outside on a clear winter's night and look up at my favorite naked-eye astronomical object -- the Pleiades -- and fully appreciate how lovely it is, but my enjoyment is increased further by knowing that it's a cluster of recently-formed hot blue supergiant stars inside the wispy strands of a reflection nebula.  

Understanding and appreciation shouldn't be inversely proportional.  The more I know, the more I wonder at the beauty, complexity, and strangeness of this universe in which we live.  The only frustrating part about it all is the limitation of my mind in comprehending it all.

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Quantum pigeons

I have a fascination for quantum physics.  Not that I can say I understand it that well; but no less than Nobel laureate and generally brilliant guy Richard Feynman said (in his lecture "The Character of Physical Law"), "If you think you understand quantum mechanics, you don't understand quantum mechanics," so I figure I have a pretty good excuse for my lack of deep comprehension.  I have a decent, if superficial, grasp of such loopy ideas as quantum indeterminacy, superposition, entanglement, and so on, but that's about the best I can do.  At least I understand enough to find the following joke absolutely hilarious:

Heisenberg and Schrödinger were out for a drive one day, and they got pulled over by a cop.  The cop says to Heisenberg, who was driving, "Hey, buddy, do you know how fast you were going?"

 
Heisenberg says, "No, but I know exactly where I am."

 
The cop says, "You were doing 85 miles per hour!"

 
Heisenberg throws his hands in the air and responds, "Great!  Now I'm lost."

 
The cop scowls at him.  "All right, pal, if you're going to be a smartass, I'm going to search your car."  So he opens the trunk, and there's a dead cat inside it.  He says, "Did you know there's a dead cat in your trunk?"

 
Schrödinger says, "Well, there is now."

Thanks, you're a great audience. I'll be here all week.

In any case, the topic comes up because of a paper in Proceedings of the National Academy of Sciences called, "Experimental Demonstration of the Quantum Pigeonhole Paradox," by a team of physicists at China's University of Science and Technology, which was enough to make my brain explode.  Here's the gist of it, although be forewarned that if you ask me for further explanation, you're very likely to be out of luck.

There's something called the pigeonhole principle in number theory, that seems kind of self-evident to me but apparently is highly profound to number theorists and other people who delve into things like sets, one-to-one correspondences, and mapping. It goes like this: if you try to put three pigeons into two pigeonholes, one of the pigeonholes must be shared by two pigeons.

See, I told you it was self-evident.  Maybe you have to be a number theorist before you find these kind of things remarkable.

[Image licensed under the Creative Commons Razvan Socol, Rock Pigeon (Columba livia) in Iași, CC BY-SA 3.0]

In any case, what the research showed is that on the quantum level, the pigeonhole principle doesn't hold true.  In the experiment, photons take the place of pigeons, and polarization states (either horizontal or vertical) take the place of the pigeonholes.  And when you do this, you find...

... that when you compare the polarization states of the three photons, no two of them are alike.

Hey, don't yell at me.  I didn't discover this stuff, I'm just telling you about it.

"The quantum pigeonhole effect challenges our basic understanding….   So a clear experimental verification is highly needed," study co-authors Chao-Yang Lu and Jian-Wei Pan wrote in an e-mail.  "The quantum pigeonhole may have potential applications to find more complex and fundamental quantum effects."

It's not that I distrust them or am questioning their results (I'm hardly qualified to do so), but I feel like what they're claiming makes about as much sense as saying that 2 + 2 = 5 for large values of 2.  Every time I'm within hailing distance of getting it, my brain goes, "Nope.  If the first two photons are, respectively, horizontally polarized and vertically polarized, the third has to be either horizontal or vertical."

But apparently that's not true. Emily Conover, writing for Science News,writes:

The mind-bending behavior is the result of a combination of already strange quantum effects.  The photons begin the experiment in an odd kind of limbo called a superposition, meaning they are polarized both horizontally and vertically at the same time.  When two photons’ polarizations are compared, the measurement induces ethereal links between the particles, known as quantum entanglement.  These counterintuitive properties allow the particles to do unthinkable things.

Which helps.  I guess.  Me, I'm still kind of baffled, which is okay.  I love it that science is capable of showing us wonders, things that stretch our minds, cause us to question our understanding of the universe.  How boring it would be if every new scientific discovery led us to say, "Meh.  Confirms what I already thought."

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Reality vs. allegory

Today's topic came to me a couple of days ago while I was watching a new video by one of my favorite YouTubers, Sabine Hossenfelder.

Sabine's channel is called Science Without the Gobbledygook, and is well worth subscribing to.  She's gotten a reputation for calling out people (including her colleagues) for misleading explanations of scientific research aimed at laypeople.  Her contention -- laid out explicitly in the specific video I linked -- is that if you take the actual model of quantum mechanics (which is entirely mathematical) and try to put it into ordinary language, you will always miss the mark, because we don't have unambiguous words to express the reality of the mathematics.  The effect this has is to create in the minds of non-scientists the impression that the science is saying something that it most definitely is not.

It reminded me of when I was about twenty, and I stumbled upon the book The Dancing Wu-Li Masters by Gary Zukav.  This book provides a non-mathematical introduction to the concepts of quantum mechanics, which is good, I suppose; but then it attempts to tie it to Eastern mysticism, which is troubling to anyone who actually understands the science.

But as a twenty-year-old -- even a twenty-year-old physics major -- I was captivated.  I went from there to Fritjof Capra's The Tao of Physics, which pushes further into the alleged link between modern physics and the wisdom of the ancients.  In an editorial review of the book, we read:

First published in 1975, The Tao of Physics rode the wave of fascination in exotic East Asian philosophies.  Decades later, it still stands up to scrutiny, explicating not only Eastern philosophies but also how modern physics forces us into conceptions that have remarkable parallels...  (T)he big picture is enough to see the value in them of experiential knowledge, the limits of objectivity, the absence of foundational matter, the interrelation of all things and events, and the fact that process is primary, not things. Capra finds the same notions in modern physics.

In part, I'm sure my positive reaction to these books was because I was in the middle of actually taking a class in quantum mechanics, and it was, to put not too fine a point on it, really fucking hard.  I had thought of myself all along as quick at math, but the math required for this class was brain-bendingly difficult.  It was a relief to escape into the less rigorous world of Capra and Zukav.

As a basis for comparison, read a quote from the Wikipedia article on quantum electrodynamics, chosen because it was one of the easier ones to understand:

(B)eing closed loops, (they) imply the presence of diverging integrals having no mathematical meaning.  To overcome this difficulty, a technique called renormalization has been devised, producing finite results in very close agreement with experiments.  It is important to note that a criterion for theory being meaningful after renormalization is that the number of diverging diagrams is finite.  In this case the theory is said to be renormalizable.  The reason for this is that to get observables renormalized one needs a finite number of constants to maintain the predictive value of the theory untouched.  This is exactly the case of quantum electrodynamics displaying just three diverging diagrams.  This procedure gives observables in very close agreement with experiment as seen, e.g. for electron gyromagnetic ratio.

Compare that to Capra's take on things, in a quote from The Tao of Physics:

Modern physics has thus revealed that every subatomic particle not only performs an energy dance, but also is an energy dance; a pulsating process of creation and destruction.  The dance of Shiva is the dancing universe, the ceaseless flow of energy going through an infinite variety of patterns that melt into one another.  For the modern physicists, then Shiva’s dance is the dance of subatomic matter.  As in Hindu mythology, it is a continual dance of creation and destruction involving the whole cosmos; the basis of all existence and of all natural phenomenon.  Hundreds of years ago, Indian artists created visual images of dancing Shivas in a beautiful series of bronzes.  In our times, physicists have used the most advanced technology to portray the patterns of the cosmic dance.

[Image licensed under the Creative Commons Arpad Horvath, CERN shiva, CC BY-SA 3.0]

It all sounds nice, doesn't it?  No need for hard words like "renormalization" and "gyromagnetic ratio," no abstruse mathematics.  All you have to do is imagine particles dancing, waving around their four little quantum arms, just like Shiva.

The problem here, though, isn't just laziness; and I've commented on the laziness inherent in the woo-woo mindset often enough that I don't need to write about it further.  But there's a second issue, one often overlooked by laypeople, and that is "mistaking analogy for reality."

Okay, I'll go so far as to say that the verbal descriptions of quantum mechanics sound like some of the "everything that happens influences everyone, all the time" stuff from Buddhism and Hinduism -- the interconnectedness of all, a concept that is explained in the beautiful allegory of "Indra's Net" (the version quoted here comes from Douglas Hofstadter's Gödel, Escher, Bach: An Eternal Golden Braid):

Far away in the heavenly abode of the great god Indra, there is a wonderful net which has been hung by some cunning artificer in such a manner that it stretches out infinitely in all directions.  In accordance with the extravagant tastes of deities, the artificer has hung a single glittering jewel in each "eye" of the net, and since the net itself is infinite in dimension, the jewels are infinite in number.  There hang the jewels, glittering like stars in the first magnitude, a wonderful sight to behold.  If we now arbitrarily select one of these jewels for inspection and look closely at it, we will discover that in its polished surface there are reflected all the other jewels in the net, infinite in number.  Not only that, but each of the jewels reflected in this one jewel is also reflecting all the other jewels, so that there is an infinite reflecting process occurring.

But does this mean what some have claimed, that the Hindus discovered the underlying tenets of quantum mechanics millennia ago?

Hardly.  Just because two ideas have some superficial similarities doesn't mean that they are, at their basis, saying the same thing.  You could say that Hinduism has some parallels to quantum mechanics, parallels that I would argue are accidental, and not really all that persuasive when you dig into them more deeply.  Those parallels don't mean that Hinduism as a whole is true, nor that the mystics who devised it somehow knew about submicroscopic physics.

In a way, we science teachers are at fault for this, because so many of us teach by analogy.  I did it all the time: antibodies are like cellular trash tags; enzyme/substrate interactions are like keys and locks; the Krebs cycle is like a merry-go-round where two kids get on at each turn and two kids get off.  But hopefully, our analogies are transparent enough that no one comes away with the impression that they are describing what is really happening.  For example, I never saw a student begin an essay on the Krebs cycle by talking about literal microscopic merry-go-rounds and children.

The line gets blurred, though, when the reality is so odd, and the actual description of it (i.e. the mathematics) so abstruse, that most non-scientists can't really wrap their brains around it.  As Sabine Hossenfelder points out, we might not even have the language to express in words what quantum mechanics is saying mathematically.  Then there is a real danger of substituting a metaphor for the truth.  It's not helped by persuasive, charismatic writers like Capra and Zukav, nor by the efforts of True Believers to cast the science as supporting their religious ideas because it helps to prop up their own worldview (you can read an especially egregious example of this here).

After a time in my twenties when I was seduced by pretty allegories, I finally came to the conclusion that the reality was better -- and, in its own way, breathtakingly beautiful.  Take the time to learn what the science actually says, or at least listen to straight-shooting science vloggers like Sabine Hossenfelder and  Derek Muller (of the amazing YouTube channel Veritasium).  I think you'll find what you'll learn is a damnsight more interesting and elegant than Shiva and Indra and the rest of 'em.  And best of all: it's actually true.

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Sending pucks to Bolivia

Over the last few days I've been reading physicist Sean Carroll's wonderful book Something Deeply Hidden, which is about quantum physics, and although a lot of it (so far) is at least familiar to me in passing, he has a way of explaining things that is both direct and simultaneously completely mind-blowing.

I'm thinking especially of the bit I read last night, about the fact that even the physicists are unsure what quantum mechanics is really describing.  It's not that it doesn't work; the model has been tested every different way you can think of (and probably ones neither one of us would have thought of), and it's passed every test, often to levels of precision other realms of physics can only dream of.  The equations work; there's no doubt about that.  But what is it, exactly, that they're describing?

Here's the analogy he uses.  Suppose there was some physicist who was able to program a computer with all of Newton's laws of motion and the other equations of macroscopic physics that have been developed since Newton's time.  So if you wanted to know anything about the position, velocity, momentum, or energy of an object, all you have to do is input the starting conditions, and the computer will spit out the final state after any given amount of time elapsed.

A simple example: a cannon fires a cannonball with an initial velocity of 150 m/s at an incline of 45 degrees.  The (constant) acceleration due to gravity is -9.8 m/s^2 (the negative sign is because the acceleration vector points downward).  Ignoring air resistance, what is the highest point in its trajectory?

And the computer spits out 574.4 meters.

Now, anyone who took high school physics could figure this out with a few calculations.  But the point Carroll makes is this: could someone input numbers like that into the software, and get an output number, without having any clue what the model is actually doing?

The answer, of course, is yes.  You might even know what the different variables mean, and know that your answer is "maximum height of the cannonball," and that when you check, the answer is right.  But as far as knowing why it works, or even what's happening in the system that makes it work, you wouldn't have any idea.

That's the situation we're in with quantum physics.

And of course, quantum physics is a hell of a lot less intuitive than Newtonian mechanics.  I think the piece if it that always boggles me the most is the probabilistic nature of matter and energy on the submicroscopic level.  

Let me give you an example, analogous to the cannonball problem.  Given a certain set of conditions, what is the position of an electron?

The answer -- which, to reiterate, has been confirmed experimentally countless times -- is that prior to observation, the electron kind of isn't anywhere in particular.  Or it's kind everywhere at once, which amounts to the same thing.  Electrons -- and all other forms of matter and energy -- are actually fields of probabilities.  You can calculate those probabilities to as many decimal places as you like, and it gives phenomenally accurate predictions.  (In fact, the equations describing those probabilities have a load of real-world applications, including semiconductors, microchips, and lasers.)  But even so, there's no doubt that it's weird.  Let's say you repeatedly measure electron positions hundreds or thousands of times, and plot those points on a graph.  The results conform perfectly to Schrödinger's wave equation, the founding principle of quantum physics.  But each individual measurement is completely uncertain.  Prior to measurement, the electron really is just a smeared-out field of probabilities; after measurement, it's localized to one specific place.

Now, let me point out something that this isn't saying.  Quantum physics is not claiming that the electron actually is in a specific location, and we simply don't have enough information to know where.  This is not an issue of ignorance.  This was shown without any question by the famous double-slit experiment, where photons are shot through a pair of closely-spaced slits, and what you see at the detector on the other side is an interference pattern, as if the photons are acting like waves -- basically, going through both slits at the same time.  You can even shoot one photon at a time through the slits, and the detector (once again after many photons are launched through), still shows an interference pattern.  Now, change one thing: add another detector at each slit, so you know for sure which slit each photon went through.  When you do that, the interference pattern disappears.  The photons, apparently, aren't little packets of energy; they're spread-out fields of probabilities, and when they're moving they take all possible paths to get from point A to point B simultaneously.  If you don't observe its path, what you measure is the sum of all the possible paths the photon could have taken; only if you observe which slit it went through do you force it to take a single path.

It's as if when Wayne Gretzky winds up for a slap shot, the puck travels from his stick to the net taking every possible path, including getting there via Bolivia, unless you're following it with a high-speed camera -- if you do that, the puck only takes a single path.

[Image licensed under the Creative Commons Alexandre Gondran, 100 trajectories guided by the wave function, CC BY-SA 4.0]

If you're saying, "what the hell?" -- well, so do we all.  The most common interpretation of this -- called the Copenhagen interpretation, after the place it was dreamed up -- is that observing the electron "collapses the wave function," meaning that it forces the electron to condense into a single place described by a single path.  But this opens up all sorts of troublesome questions.  Why does observation have that effect?  What counts as an observer?  Does it have to be a sentient being?  If a photon lands on the retina of a cat, does its wave function collapse?  What if the photon is absorbed by a rock?  Most importantly -- what is actually happening that makes the wave function collapse in the first place?

To add to the mystery, there's also the Heisenberg uncertainty principle, which states that for certain pairs of variables -- most famously, position and velocity -- you can't know both of them to high precision at the same time.  The more you know about a particle's position, the less you can know even theoretically about its velocity.  Or, more accurately, if you pinpoint a particle's position, its velocity can only be described as a wide field of probabilities.  And vice versa.

I think the passage in Carroll's book that made me the most astonished was the following summation of all this:

Classical [Newtonian] mechanics offers a clear and unambiguous relationship between what we see and what the theory describes.  Quantum mechanics, for all its successes, offers no such thing.  The enigma at the heart of quantum reality can be summed up in a simple motto: what we see when we look at the world seems to be fundamentally different from what actually is.

So.  Yeah.  You can see why I was kind of wide-eyed, and I'm not even a quarter of the way through the book yet.  

Anyhow, maybe we should lighten things up by ending with my favorite joke.

Schrödinger and Heisenberg are out for a drive, with Heisenberg at the wheel.  After a while, they get pulled over by a cop.

The cop says to Heisenberg, "Do you have any idea how fast you were going?"

Heisenberg replies, "No, but I know exactly where I am."

The cop says, "You were going 85 miles an hour!"

Heisenberg throws his hands up and the air and says, "Great!  Now I'm lost!"

The cop by this time is getting pissed off, and says, "Fine, if you're going to be a smartass, I'm gonna search your car."  So he opens the trunk, and in the trunk is a dead cat.

The cop says, "Did you know there's a dead cat in your trunk?"

Schrödinger says, "Well, there is now."

Thanks.  You've been a great audience.  I'll be here all week.

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Return to sender

One of the hardest things about understanding quantum physics is that it is so fundamentally different from the way things act on the macroscopic level.

Even a layperson's grasp of the subject -- leaving aside all the abstruse mathematics -- requires one to jettison every expectation that the everyday objects we see and interact with will behave in the same fashion as the "objects" (as it were) on the subatomic level.  I put the word "objects" in quotes advisedly; the word "particle" brings to mind a hard, discrete little lump of matter, and that's still how they're drawn in science books:

[Image licensed under the Creative Commons Richie Bendall, Atomic structure of Lithium-7, CC BY-SA 4.0]

The reality is far weirder, and far harder to picture; particles, all the way down to photons of light, aren't little miniature bullets zinging around, they're actually smeared-out fields of probabilities.  The reassuringly solid matter we, and everything else, are made of turns out to be (at its basis) composed of something that is ephemeral, not even existing at one particular location in any real sense.

But it bears mention that however bizarre this is, it is not just a wild guess.  The predictions of quantum mechanics have been tested every which way from Sunday, and each time, the results have been spot-on.  So it may be unsettling, it certainly is counter-intuitive, but if we buy the methods of science at all, we have to conclude that whether we like it or not, this is what reality is.

Take, for example, the quantum boomerang effect, which I only found out about a couple of days ago because of some research out of the University of California - Santa Barbara.  The idea here, so far as I understand it -- and I will once again throw in the caveat that I'm not much better than a layperson myself, so bear with me -- has to do with what occurs when electrons in a substance are given a repeated kick of energy.

Picture, for example, something that spins freely on an axle, like a fan.  Imagine taking a fan (Nota bene: Mr. Safety says unplug it first!), and giving a regularly-timed tap on the blades with one finger.  The fan would absorb the energy, overcoming any resistance in the axle due to friction, and the blades would begin to turn; if you timed it right, you could get it spinning at a decent clip.

So far, nothing odd.  Now, imagine an analogous situation on the subatomic level.  Suppose you had a substance with atoms arranged in a lattice, but there are some defects in the lattice -- impurities, gaps, and so on.  In a metallic lattice, electrons are fairly free to move (this is why metals make good conductors); but the defects inhibit electron transfer, just as friction was working against you in turning the rotor blades.  Here, though, something completely different happens when you disturb the system.  If you give the lattice regular pulses of energy, the electrons are jolted out of their position, but they don't keep moving -- they immediately turn around and settle back down in their original positions.

Thus the nickname "the boomerang effect."

"It's really a fundamentally quantum mechanical effect," said physicist David Weld, who co-authored the paper, in an interview with Science Daily.  "There's no classical explanation for this phenomenon...  In a classical system, a rotor kicked in this way would respond by constantly absorbing energy from the kicks.  Take a quantum version of the same thing, and what you see is that it starts gaining energy at short times, but at some point it just stops and it never absorbs any more energy.  It becomes what's called a dynamically localized state."

The explanation, Weld says, lies in the dual particle-wave nature of subatomic particles.  Because matter on the smallest scales has both particle-like and wave-like properties, it's going to exhibit some weird properties as compared to the solid stuff we see around us.  "That chunk of stuff that you're pushing away is not only a particle, but it's also a wave, and that's a central concept of quantum mechanics," Weld said.  "Because of that wave-like nature, it's subject to interference, and that interference in this system turns out to stabilize a return and dwelling at the origin."

So we can add that to our list of weird and counterintuitive behavior on the quantum level.  The universe is a strange, compelling, beautiful place, and the more you study it, the stranger it gets.  Me, I kind of like that.  I don't mind that things aren't as they seem.  How boring things would be if our "common sense" got it right every single time.

Even if I don't fully understand it -- even if I never fully understand it -- I'd much prefer that the cosmos never loses its ability to astonish us.

**************************************

Even spookier action

Once again, I've had my mind blown by a set of experiments about the behavior of subatomic particles that teeters on the edge of what my layman's brain can understand.  So I'm gonna tell you about it as best I can, and I would ask that any physics types in the studio audience let me know about any errors I make so I can correct 'em.

You're undoubtedly aware of the quote by Einstein having to do with "spooky action at a distance," which is how he viewed the bizarre and counterintuitive features of the physics of the very small such as quantum superposition and entanglement.  Both of these phenomena, though, have been explained by the model that particles aren't the little pinpoint masses we picture them as, but spread-out fields of probabilities that can interact even when they're not near each other.

But that still leaves intact the conventional view, certainly the common-sense one, that one object can't affect another unless the field generated by one of them intersects the field generated by the other, whether that field be gravity, electromagnetism, or either of the two less-familiar nuclear forces (strong and weak).  Not as obvious is that this influence is generally transmitted by some sort of carrier particle being exchanged between the two -- although the carrier particle that transmits the gravitational force has yet to be discovered experimentally.

This is one of the main reasons that unscientific superstitions like astrology can't be true; it's positing that your personality and life's path are affected by the position of the Sun or one of the planets relative to a bunch of stars that only appear to be near each other when viewed from our perspective.  Most of those stars are tens to hundreds of light years away, so any influence they might have on you via the four fundamental forces is about as close to zero as you could possibly get, because all four of them dramatically decrease in intensity the farther away you get.  (As Carl Sagan quipped, at the moment of your birth, the obstetrician who delivered you was exerting a greater gravitational pull on you than Jupiter was.)

So the bottom line appears to be: no interaction between the fields generated by two objects, no way can they influence each other in any fashion.

But.

In 1959, two physicists, Yakir Aharonov and David Bohm, published a paper on what has come to be known as the Aharonov-Bohm effect.  This paper concluded that under certain conditions, an electrically-charged particle can be affected by an electromagnetic field -- even when the particle itself is shielded in such a way that both the electric field and magnetic field it experiences is exactly equal to zero, and the particle's wave function is blocked from the region that is experiencing the field.

So that leaves us with one of two equally distasteful conclusions.  Either the measured electric and magnetic fields in a region don't tell us all we need to know to understand the electromagnetic potential a particle is experiencing, or we have to throw away the principle of locality -- that an object can only be influenced by the conditions in its local environment.

(Nota bene: in physics, "local" has a rigorous definition; two phenomena are local relative to each other if the amount of time a cause from one can precede an effect on the other is equal to or greater than the amount of time it would take light to travel from the position of the cause to the position of the effect.  This is the basis of the reluctance of physicists to believe in any kind of superluminal information transfer.)

What's more troubling still is that this isn't just some theoretical meandering; the Aharonov-Bohm effect has been demonstrated experimentally.  So as bafflingly weird as it sounds, it apparently is a built-in feature of quantum physics, as if we needed anything else to make it even crazier.

But maybe this is just some weirdness of electromagnetism, right?  Well, that might have been believable...

... until now.

In a paper three days ago in Science, five physicists at Stanford University -- Chris Overstreet, Peter Asenbaum, Joseph Curti, Minjeong Kim, and Mark Kasevich -- have demonstrated that the same thing works for gravitational interactions.

This is bizarre for a variety of reasons.  First, the Aharonov-Bohm effect is just bizarre, in and of itself.  Second, as I mentioned earlier, we don't even have experimental proof that gravity has a carrier particle, or if perhaps it is just a description of the curvature of space -- i.e., if gravity is a completely different animal from the other three fundamental forces.  Third, and weirdest, the equations governing gravity don't mesh with the equations governing the other three forces, and every effort to coalesce them and create a "Grand Unified Theory" has met with failure.  Combining the gravitational field equations with the ones in the quantum realm generates infinities -- and you know what that does.  

"Every time I look at this experiment, I’m like, 'It’s amazing that nature is that way,'" said study co-author Mark Kasevich, in an interview with Science News.

"Amazing" isn't how I would have put it.  In Kasevich's situation, I think what I'd have said would have been more like, "Holy shit, what the hell is going on here?"  But I'm kind of unsubtle that way.

So what it seems to indicate to me is that we're missing something pretty fundamental about how forces work, and that this is an indication that there's a serious gap in the theoretical underpinning of physics.

(Nota bene #2: I still think astrology is bullshit, though.)

It's tempting for us laypeople to just throw our hands up in despair and say, "Okay, this stuff is so weird it can't be true."  The problem is, if you buy into the methods of science -- which I hope all of us do -- that's the one response you can't have.  The experimental evidence is what it is, whether you like (or understand) it or not, and if it contradicts your favorite model of how things work, you have to chuck the model, not the evidence.  Or, as Neil deGrasse Tyson more eloquently and succinctly put it, "The wonderful thing about science is that it works whether or not you believe in it."

So it looks like we're stuck with this even-spookier-action-at-a-distance, as counterintuitive as it sounds.  Objects can interact with each other gravitationally even when the gravitational field produced by object #1 is exactly zero where object #2 is currently sitting.  And this is about the limit of what I can explain, so if you ask me to clarify further, I'm afraid my response will be a puzzled head-tilt much like what my dog gives me when I tell him something he just can't comprehend, like why I don't want to go outside and play ball with him when it's subzero temperatures and snowing.

But I'll end on a more academic note, with a quote by the famous biologist J. B. S. Haldane, that I've used before in posts about quantum physics: "The universe is not only queerer than we imagine, it is queerer than we can imagine."

*************************************

Since reading the classic book by Desmond Morris, The Naked Ape, when I was a freshman in college, I've been fascinated by the idea of looking at human behavior as if we were just another animal -- anthropology, as it were, through the eyes of an alien species.  When you do that, a lot of our sense of specialness and separateness simply evaporates.

The latest in this effort to analyze our behavior from an outside perspective is Pascal Boyer's Human Cultures Through the Scientific Lens: Essays in Evolutionary Cognitive Anthropology.  Why do we engage in rituals?  Why is religion nearly universal to all human cultures -- as is sports?  Where did the concept of a taboo come from, and why is it so often attached to something that -- if you think about it -- is just plain weird?

Boyer's essays challenge us to consider ourselves dispassionately, and really think about what we do.  It's a provocative, fascinating, controversial, and challenging book, and if you're curious about the phenomenon of culture, you should put it on your reading list.

[Note: if you purchase this book using the image/link below, part of the proceeds goes to support Skeptophilia!]

A smile without a cat

Every time I hear some new discovery in quantum physics, I think, "Okay, it can't get any weirder than this."

Each time, I turn out to be wrong.

A few of the concepts I thought had blown my mind as much as possible:

• Quantum superposition -- a particle being in two states at once until you observe it, at which point it apparently decides on one of them (the "collapse of the wave function")
• The double-slit experiment -- if you pass light through a closely-spaced pair of slits, it creates a distinct interference pattern -- an alternating series of parallel bright and dark bands.  The same interference pattern occurs if you shoot the photons through one of the slits, one photon at a time.  If you close the other slit, the pattern disappears.  It's as if the photons passing through the left-hand slit "know" if the right-hand slit is open or closed.
• Quantum entanglement -- two particles that somehow are "in communication," in the sense that altering one of them instantaneously alters the other, even if it would require superluminal information transfer to do so (what Einstein called "spooky action-at-a-distance")
• The pigeonhole paradox -- you'd think that if you passed three photons through polarizing filters that align their vibration plane either horizontally or vertically, there'd be two of them polarized the same way, right?  It's a fundamental idea from set theory; if you have three gloves, it has to be the case that either two are right-handed or two are left-handed.  Not so with photons.  Experiments showed that you can polarize three photons in such a way that no two of them match.

Bizarre, counterintuitive stuff, right there.  But wait till you hear the latest:  three physicists, Yakim Aharonov of Tel Aviv University, Sandu Popescu of the University of Bristol, and Eliahu Cohen of Bar Ilan University, have demonstrated something they're calling a quantum Cheshire Cat.  Apparently under the right conditions, a particle's properties can somehow come unhooked from the particle itself and move independently of it -- a bit like Lewis Carroll's cat disappearing but leaving behind its disembodied grin.

The Cheshire Cat from John Tenniel's illustrations for Alice in Wonderland (1865) [Image is in the Public Domain]

I'll try to explain how it works, but be aware that I'm dancing right along the edge of what I'm able to understand, so if you ask for clarification I'll probably say, "Damned if I know."  But here goes.

Imagine a box containing a particle with a spin of 1/2.  (Put more simply, this means that if you measure the particle's spin along any of the three axes (x, y, and z), you'll find it in an either-or situation -- right or left, up or down, forward or backward.)  The box has a partition down the middle that is fashioned to have a small, but non-zero, probability of the particle passing through.  At the other end of the box is a second partition -- if the particle is spin-up, it passes through; if not, it doesn't and is reflected back into the box.

With me so far?  'Cuz this is where it gets weird.

In quantum terms, the fact that there's a small but non-zero chance of the particle leaking through means that part of it does leak through; this is a feature of quantum superposition, which boils down to particles being in two places at once (or, more accurately, their positions being fields of probabilities rather than one specific location).  If the part that leaks through is spin-up, it passes through the right-hand partition and out of the box; otherwise it reflects back and interacts with the original particle, causing its spin to flip.

The researchers found that this flip occurs even if measurements show that the particle never left the left-hand side of the box.

So it's like the spin of the particle becomes unhooked from the particle itself, and is free to wander about -- then can come back and alter the original particle.  See why they call it a quantum Cheshire Cat?  Like Carroll's cat's smile, the properties of the particle can somehow come loose.

Whatever a "loose property" actually means.

The researchers have suggested that this bizarre phenomenon might allow counterfactual communication -- communication between two observers without any particle or energy being transferred between them.  In the setup I described, the observer left of the box would know if the observer on the right had turned the spin-dependent barrier on or off by watching to see if the particle in the left half of the box had altered its spin.  More spooky action-at-a-distance, that.

What I have to keep reminding myself is that none of this is some kind of abstract idea or speculation of what could be; these findings have been experimentally verified over and over.  Partly because it's so odd and counterintuitive, the theories of quantum physics have been put through rigorous tests, and each time they've passed with flying colors.  As crazy as it sounds, this is what reality is, despite how hard it is to wrap our minds around it.

"What is the most important for us is not a potential application – though that is definitely something to look for – but what it teaches us about nature," said study co-author Sandu Popescu.  "Quantum mechanics is very strange, and almost a hundred years after its discovery it continues to puzzle us.  We believe that unveiling even more puzzling phenomena and looking deeper into them is the way to finally understand it."

Indeed.  I keep coming back to the fact that everything you look at -- all the ordinary stuff we interact with on a daily basis -- is made of particles and energy that defy our common sense at every turn.  As the eminent biologist J. B. S. Haldane famously put it, "The universe is not only queerer than we imagine -- it is queerer than we can imagine."

**********************************

Some of the most enduring mysteries of linguistics (and archaeology) are written languages for which we have no dictionary -- no knowledge of the symbol-to-phoneme (or symbol-to-syllable, or symbol-to-concept) correspondences.

One of the most famous cases where that seemingly intractable problem was solved was the near-miraculous decipherment of the Linear B script of Crete by Alice Kober and Michael Ventris, but it bears keeping in mind that this wasn't the first time this kind of thing was accomplished.  In the early years of the nineteenth century, this was the situation with the Egyptian hieroglyphics -- until the code was cracked using the famous Rosetta Stone, by the dual efforts of Thomas Young of England and Jean-François Champollion of France.

This herculean, but ultimately successful, task is the subject of the fascinating book The Writing of the Gods: The Race to Decode the Rosetta Stone, by Edward Dolnick.  Dolnick doesn't just focus on the linguistic details, but tells the engrossing story of the rivalry between Young and Champollion, ending with Champollion beating Young to the solution -- and then dying of a stroke at the age of 41.  It's a story not only of a puzzle, but of two powerful and passionate personalities.  If you're an aficionado of languages, history, or Egypt, you definitely need to put this one on your to-read list.

[Note: if you purchase this book using the image/link below, part of the proceeds goes to support Skeptophilia!]

Wibbly-wobbly...

Have I told you my favorite joke?

Heisenberg and Schrödinger are out for a drive, and a cop pulls them over.

The cop says to Heisenberg, who was driving, "Hey, buddy, do you know how fast you were going?"

Heisenberg says, "No, but I know exactly where I am."

The cop says, "You were doing 70 miles per hour!"

Heisenberg throws his hands up in annoyance and says, "Great!  Now I'm lost."

The cop scowls and says, "Okay, if you're going to be a wiseguy, I'm gonna search your car."  So he opens the trunk, and there's a dead cat inside.

The cop says, "Did you know there's a dead cat in your trunk?"

Schrödinger says, "Well, there is now."

*brief pause so you can all stop chortling*

The indeterminate nature of reality at the smallest scales always tends to make people shake their head in wonderment at how completely weird the universe is, if they don't simply disbelieve it entirely.  The Uncertainty Principle, peculiar as it sounds, is a fact.  It isn't a limitation of our measurement technique, as if you were trying to find the size of something small and had a poorly-marked ruler, so you could get a more accurate number if you found a better one.  This is something fundamental and built-in about reality.  There are pairs of measurements for which precision is mutually exclusive, such as velocity and position -- the more accurate your information is about one of them, the less you can even theoretically know about the other.

Likewise, the collapse of the wave function, which gave rise to the story of the famous (but ill-fated) cat, is an equally counterintuitive part of how reality is put together.  Outcomes of purely physical questions -- such as where a particular electron is at a given time -- are probabilities, and only become certainties when you measure them.  Again, this isn't a problem with measurement; it's not that the electron really is in a specific location, and you just don't know for sure where until you look.  Before you measure it, the electron's reality is that it's a spread-out field of probabilities.  Something about interacting with it using a measuring device makes that field of probabilities collapse into a specific location -- and no one knows exactly why.

But if you want your mind blown further -- last week in a paper in Physical Review Letters we found out how long it takes.

It turns out the wave function collapse isn't instantaneous.  In "Tracking the Dynamics of an Ideal Quantum Measurement," by a team led by Fabian Pokorny of Stockholm University, the researchers describe a set of experiments involving "nudging" a strontium atom with a laser to induce the electrons to switch orbits (i.e. making them assume a particular energy, which is one of those quantum-indeterminate things like position).  The fidelity of the measurement goes down to the millionths of a second, so the scientists were able to keep track of what happened in fantastically short time intervals.

And the more they homed in on what the electron was doing, the fuzzier things got.  The theory is that as you get down on those scales, time itself becomes blurred -- so the shorter the time interval, the less certain you are about when exactly something happened.

"People assume that time is a strict progression from cause to effect, but actually, from a non-linear non-subjective viewpoint, it's more of a big ball of wibbly-wobbly timey-wimey... stuff." -- The Tenth Doctor, "Blink"

I don't know about you, but I thought I had kinda sorta wrapped my brain around the quantum indeterminacy of position thing, but this just blew my mind all over again.  Even time is fuzzy?  I shouldn't be surprised; for something so damn familiar, time itself is really poorly understood.  With all of the spatial dimensions, you can move any direction you want; why is time one-way?  It's been explained using the Second Law of Thermodynamics, looking at ordered states and disordered states -- the explanation goes something like this:

Start with an ordered state, such as a hundred pennies all heads-up.  Give them a quick shake.  A few will flip, but not many.  Now you might have 83 heads and 17 tails.  There are a great many possible ways you could have 83 heads and 17 tails as long as you don't care which pennies are which.  Another shake, and it might be 74/26, a configuration that there are even more possibilities for.  And so on.  Since at each turn there are a huge number of possible disordered states and a smaller number of ordered ones, each time you perturb the system, you are much more likely to decrease orderliness than to increase it.  You might shake a 50/50 distribution of pennies and end up with all heads -- but it's so fantastically unlikely that the probability might as well be zero.  This push toward disorder gives an arrow to the direction of time.

Well, that's all well and good, but there's also the problem I wrote about last week, about physical processes being symmetrical -- there are a great many of them that are completely time-reversible.  Consider, for example, watching a ten-second clip of a single billiard ball bouncing off the side of a pool table.  Could you tell if you were watching the clip backward or forwards?  It's unlikely.  Such interactions look as sensible physically in real time or time-reversed.

So what time actually is, and why there's an arrow of time, is still a mystery.  Because we certainly feel the passage of time, don't we?  And not from any probabilistic perception of "well, I guess it's more likely time's flowing this way today because things have gotten more disorderly."  It feels completely real -- and completely fixed and invariable.

As Einstein put it, "The distinction between past, present, and future is an illusion, but it is a stubbornly persistent one."

Anyhow, that's our bizarre scientific discovery of the day.  But I better get this post finished up.  Time's a wasting.

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This week's Skeptophilia book recommendation of the week is a classic -- Martin Gardner's wonderful Did Adam and Eve Have Navels?

Gardner was a polymath of stupendous proportions, a mathematician, skeptic, and long-time writer of Scientific American's monthly feature "Mathematical Games."  He gained a wonderful reputation not only as a puzzle-maker but as a debunker of pseudoscience, and in this week's book he takes on some deserving targets -- numerology, UFOs, "alternative medicine," reflexology, and a host of others.

Gardner's prose is light, lucid, and often funny, but he skewers charlatans with the sharpness of a rapier.  His book is a must-read for anyone who wants to work toward a cure for gullibility -- a cure that is desperately needed these days.

[Note: if you purchase this book using the image/link below, part of the proceeds goes to support Skeptophilia!]