Posts tagged with quantum mechanics

Mirror symmetry is an example of a duality, which occurs when two seemingly different systems are isomorphic in a non-trivial way. The non-triviality of mirror symmetry involves quantum corrections. It’s like the Fourier transform, where “local” in one domain translates to “global”—something requiring information from over the whole space—in the other domain.

Under a local/global isomorphism, complicated quantities get mapped to simple ones in the dual domain. For this reason the discovery of duality symmetries has revolutionized our understanding of quantum theories and string theory.

summer school on mirror symmetry (liberally edited)

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Thinking about local-global dualities gave me another idea about my model-sketch of knowledge, ignorance & expectation.

• Under physical limitations, at a fixed energy level, Fourier duality causes a complementary tradeoff between frequency and time domains—not both can be specific. Same with position & momentum, again at a fixed energy level.
• Under human limitations, at a fixed commitment of effort|time|concentration, you can either dive deep into a few areas of knowledge|skill, or swim broadly over many areas of knowledge|skill.

If I could come up with a specific implementation of that duality it would impose a boundary constraint on that model-sketch. Which would be great as optimal time|effort|concentration|energy could be computed from other parts of decision theory.

Playing around with polaroid screens at Google.

If you shine light through

• a light polariser `A`
• another light polariser `B` that’s perpendicular to `A` (i.e., `A ⊥B` or `A×B=0`)
• i.e., `AB` represents “shine the light through `A` then through `B` which` ⊥A`
• then no light comes through (it’s black.)

If you shine light through

• a polaroid `A`
• another polaroid `B ⊥ A`
• a third polaroid `C` that’s halfway between `A` and `B` (either halfway)

then no light comes through (it’s black.)

So far, formulaically, we have:

• `AB=0`
• `BA=0`
• whence it follows
• `ABC=(AB)C=(0)C=0`
• `CBA=C(0)=0`
• `BAC=0`
• `CBA=0`

But! This is surprising to watch and surprising to see the formula.

• If you shine through `A` then `C` then `B`, it’s kind of light!
• `ACB≠0`
• furthermore
• `BCA≠0`

Woo-hoo, Noncommutativity!

Earlier in the talk Ron Garret does a two-slit experient with two mechanical-pencil leads and a laser pointer. Wave-particle duality with an at-home science kit.

## philosophy & quantum mechanics

The isomorphismes links (last three tweets) might be worth sharing with everyone. (I’ve been accused that this site is hard to browse—sorry!)

@isomorphisms haha, yep.

— Clark (@jtc_19) April 15, 2013

By the way! OCW has course notes on the standard weird phenomena and also the standard interpretation of the physics. http://ocw.mit.edu/courses/linguistics-and-philosophy/24-111-philosophy-of-quantum-mechanics-spring-2005/lecture-notes/

hi-res

## Nice Pictures of Hamiltonian Dynamics

I had judged The Emperor’s New Mind by the negative reviews but never actually picked it up. It has a lot of great stuff, almost like an “early draft” of The Road to Reality.

All I knew about Emperor’s New Mind before was that it invokes quantum mechanics to explain free will, which was perceived as “icky” by people who study the brain. (Despite that, like quantum nonsense, the “greats” of QM—Bohr, Schrödinger—also weighed in with QM/free-will speculations (do you hear me, Conrad&Kochen? Quantum communication folks?) — because, let’s be real here, free will is a millennia-old conundrum and I think we’d all appreciate it if the people who understand compositions of Hilbert spaces weighed in on whether and what the latest “master theory” (bringer of semiconductors = transistors, LCD’s, lasers, MRI/PET and certain polymers/piezoelectrics/other materials) would say about the age-old question)

I got a bit more of the debate whilst reading about pi-1 sentences, which is a computability/knowability/logic dealio. But again, this was the level of “What’s RP’s argument in a nutshell?” rather than “Is here anything worth reading in the 400 pages?”. It’s a lot of good.

For most people, this:

is indistinguishable from this:

In both cases, the layperson needs an interpreter.

…[I]t takes years of dedicated study before scientific truth in its truest, mathematical and symbolic forms can be understood. The rest of us rely on experts to explain it, someone who has seen and understood the truth and can dumb it down for us….

The Last Psychiatrist

Black Hole Fragmentation

Image Credit: Burkhard Zink, Nikolaos Stergioulas, Ian Hawke, Christian D. Ott, Erik Schnetter, and Ewald Muller

hi-res

John Baez and James Dolan, From Finite Sets to Feynman Diagrams

• an explosion of ideas
• equality `x=x` is boring
• why is `6÷2=3` ?

(Source: arxiv.org)

What is the best interpretive program for making sense of quantum mechanics? Here is the way I would put it now. The question is completely backward. It acts as if there is this thing called quantum mechanics, displayed and available for everyone to see as they walk by it—kind of like a lump of something on a sidewalk. The job of interpretation is to find the right spray to cover up any offending smells. The usual game of interpretation is that an interpretation is always something you add to the pre-existing, universally recognized quantum theory.

What has been lost sight of is that physics [theory] is a dynamic interplay between storytelling and equation writing. Neither one stands alone, not even at the end of the day. But which has the more fatherly role? If you ask me, it’s the storytelling…. An interpretation is powerful if it gives guidance, and I would say the very best interpretation is the one whose story is so powerful it gives rise to the mathematical formalism itself (the part where nonthinking can take over)….

Take the nearly empty imagery of the many-worlds interpretation(s). Who could derive the specific structure of complex Hilbert space out of it if one didn’t already know the formalism? Most present-day philosophers of science just don’t seem to get this: If an interpretation is going to be part of physics, instead of a self-indulgent ritual to the local god, it had better have some cash value for physical practice itself.

One way to think about quantum operators is as Questions that are asked of a quantum system.

• Identity operator = "Who are you?"
• Energy operator = "How much do you weigh?"
• "What is your spin along the `z` axis?”
• and so on.

Statistical moments, letter values, and other verbs that are often just called “statistics” can be thought of the same way: asking questions of a data set.

For example, after you run the `∑/n` operation to get the mean happiness in Europe (2.0 / 3.0) versus the mean happiness in the US (1.2 / 2.0), you naturally would want to ask things like:

• What about the least happy people? Are there more people answering near 0.0 in the US or Europe?
• What’s the variance `√∑²/n`?
• What’s the skewness? (Blanchflower & Oswald’s data survey 45,000 Americans and 400,000 Europeans — enough degrees of freedom to meaningfully measure skew.)
• What’s the conditional value-at-risk at the 10% level? (average of the bottom 10% unhappiness.)
• Apply a smoothing kernel to pick up which country has the more least-happy people without choosing a particular cutoff. (And maybe a second kernel to deal with the different scales: should we assume `US1.0 = EUR1.5`? Or maybe count from the top, to `US1.8 = EUR2.8`?)

Running these operators on the dataset will tell you an answer to one question, just like in English.

One difference is that classical statistical operators typically spit out two numbers in reply to your question: an answer, and a confidence level in that answer. The confidence in the answer is computed based on experimental assumptions by people with names like Pearson, Fisher, and Chisquare.