Bell’s Theorem explained

In 2009, I wrote an explanation of Bell’s Theorem that could be understood by popular audiences. I wanted to repost it, but ended up rewriting it completely.

Although the predictions of quantum theory are well-understood, its interpretation is famously difficult. Because quantum mechanics only makes probabilistic predictions, many people have desired a “hidden variable” interpretation, where quantum objects have definite states, despite appearances to the contrary. But hidden variable interpretations are generally not accepted.

It is certainly possible to create a hidden variable interpretation that agrees with the all the predictions of quantum theory, and de Broglie-Bohm theory is an example of such a interpretation. However, de Broglie-Bohm has a number of unsatisfying properties. Indeed there are a few theorems that prove that any hidden variable interpretation must have unsatisfying properties.

The most important of these is Bell’s Theorem, formulated in 1964. What follows is an explanation of the thought experiment, the mathematical proof, and its implications.

The setup

Bell’s Theorem considers a particular thought experiment, in which two electrons are emitted simultaneously from a single source in opposite directions. This source emits electrons that are “entangled”, meaning that their quantum states are correlated with one another. If you perform the same measurement on both electrons, both measurements will always produce the same result.1
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Expert caution, amateur engagement

In an earlier post, I talked about whether it is appropriate to judge historical people using today’s standards. I was surprised that commenters were so opinionated on an issue of arguably little importance.

On the “anti” side, multiple people argued that to truly understand history, you shouldn’t be so judgmental about it. But the thing is, I am not a historian, so why should I act like one? I do not perform any original historical research. The only way I might ever teach history is by sharing stuff I learned from Wikipedia or news articles. If I were to withhold judgment on historical people, I would not learn more about history, I would learn less because I would be less engaged.

Let’s switch to talking about my area of expertise, physics. I am a professional physicist, and most of my readers merely have an amateur interest in it. We have different attitudes towards physics, as well we should.
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The Physics Nobel this year

A few days ago, the Nobel Prize in Physics was awarded to Haldane, Kosterlitz, and Thouless, ”for theoretical discoveries of topological phase transitions and topological phases of matter”.  Hey, I recognize two of those names!  In my field of research, superconductivity, there’s something known as the Berezinskiĭ-Kosterlitz-Thouless transition.  (Berezinskiĭ didn’t win the Nobel because he died in 1980.)

As always, the Nobel provides an article explaining the background for popular audiences.  I’m just going to talk about it briefly to give you the gist.

Berezinskiĭ, Kosterlitz, and Thouless (BKT) explained the superconducting transition in flat, 2-dimensional materials.  Below a certain temperature, the material becomes a superconductor, conducting electricity with precisely zero resistance.  Above that temperature, it behaves as a normal material.  What changes?

Superconductors have topological defects called vortices (singular form is “vortex”), which I depict below:

On the left, a grid of atoms where all arrows point outwards from a topological defect. On the right, the arrows point horizontally outwards and vertically inwards.

Here, each blue circle is an atomic site, and the arrow represents the “phase” of the superconductor.  The two squares above show the two kinds of vortices, the “+” vortex and “-” vortex.  These vortices behave as electrical charges, with like repelling like, and opposite charges attracting. [Read more…]

Paper: The Sokal Hoax

In 1996, physicist Alan Sokal was unhappy with the tendency in academic postmodernism to dismiss scientific work. So he submitted a bogus paper called “Transgressing the Boundaries: Towards a Transformative Hermeneutics of Quantum Gravity” to Social Text, a journal in cultural studies. After it was published, he revealed it as a hoax. And now, it is one of the best known shots fired at academic anti-science.

While the hoax is a good conversation-starter, I would caution against thinking it’s a total slam-dunk. Journals are there to filter out shoddy work, rather than bad faith actors. Secondly, AFAICT Social Text is a journal of mediocre impact. Finally, Sokal himself said that academic postmodernism has now backed off from many of its previous excesses. (Sokal credits the Bush administration, which was more effective at satirizing academic postmodernism than he ever was.)

In any case, this is a paper report. To humorous ends, I will review Sokal’s paper as if it were a serious work.
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Paper: The Chaos within Sudoku

“The Chaos within Sudoku” is a paper about solving Sudoku puzzles with physics. They simulate an imaginary physical system, let it run, and when it stops the puzzle is solved. See the video below:

The thing about Sudoku is that Sudoku is hard. More specifically, when Sudoku is generalized to grids of arbitrary size, it’s NP-complete. What happens when you translate an NP-complete problem to a physical simulation? The authors find chaotic dynamics.  And in the process, they identify the hardest Sudoku puzzle…
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Paper: First observation of gravitational waves

This is a repost from February, when LIGO reported its first observation of gravitational waves.  This is relevant because last month LIGO reported its second observation, also resulting from inspiraling black holes.

Today, the Laser Interferometry Gravitational-Wave Observatory (LIGO) reported the first observation of gravitational waves. You can read about it in The New York Times (warning: autoplay), on Sean Carroll’s blog, or in comic form. I went straight to Physical Review Letters.

As an undergrad, I did some work on LIGO. Specifically, I was a data analyst looking for exactly the kinds of gravitational waves here observed. Anyway, I’m happy to play the role of your local expert, providing some context and answering any questions.

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Illustrating the Higgs mechanism, Part 2

Part 1 is a prerequisite to this post.

Global gauge symmetry

Previously, I talked about the “direction” of the Higgs field, but what does that even mean? It doesn’t refer to a direction in space. There’s hardly any meaning to the direction at all. In fact, I could go ahead and change all the directions and it wouldn’t matter at all.

Two rows of circles. In the top row, labeled A, the Higgs field is pointing to the upper right. In the bottom row, labeled B, the Higgs field is pointing down.Figure 6.
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