Saturday, October 25, 2008

Martin Gardner Mathematical Library

I have just received a flyer from the Cambridge University Press advertising The New Martin Gardner Mathematical Library. Of course, the name Martin Gardner immediately attracted my attention (isn't it so useful to have a widely recognised name?), because I immediately thought of his excellent Mathematical Games column that used to appear in Scientific American. It says here that his column stopped being published in 1981 - was it that long ago?

Anyway I clicked through to The New Martin Gardner Mathematical Library to discover that it is exactly what I thought it might be, i.e. an updated version of his Mathematical Games column. The library is described thus:

The books based on Martin Gardner's enormously popular Scientific American columns and puzzles continue to challenge and fascinate readers. In these new editions, the author, in consultation with experts, has written updates to all the chapters, including new game variations, new mathematical proofs, and connections to recent developments and discoveries. New diagrams and illustrations have been added and old ones improved, and the bibliographies have been greatly expanded throughout.
The web page looks unfinished, but it gives at least some of the titles that will be in the library, which I list below with links that I have added for convenience:

1. Hexaflexagons, Probability Paradoxes [I have linked to the Monty Hall problem as an example of this genre], and the Tower of Hanoi
2. Origami, Eleusis, and the Soma Cube
3. Sphere Packing, Lewis Carroll, and Reversi

It looks like the sort of good stuff that will provoke those familiar mental gymnastics of yore.

Thursday, October 23, 2008

Many Worlds Theory

Nova has a nice collection of information here about Hugh Everett's so-called Many Worlds Theory of quantum mechanics. The package includes a letter from Everett to Bryce DeWitt explaining the basic concepts underlying his theory, and the published version of Everett's PhD dissertation. It's fascinating stuff that I highly recommend.

For a long time I have had an affinity for Everett's theory, but I didn't find out about Everett's work until long after I had discovered "Many Worlds Theory" for myself whilst doing my PhD work (circa 1980) in high energy particle physics. The reasoning that led me to this theory was to try to see the world from the "point of view" of a simple QM system (e.g. a fundamental particle), and to then work upwards in complexity towards ever larger QM systems.

The only way a fundamental particle can "see" the world is to exchange particles with it, and QM does this by progressively applying (the infinitesimal version of) the evolution operator exp(i H t), which is a unitary operator that rotates the system state (e.g. scattering/creating/annihilating particles) in a norm-preserving way (i.e. probability conserving). This leads to a QM description of the world in which there are physical processes going on "in parallel", where all the alternative processes that can be generated by exp(i H t) actually do occur simultaneously. QM (unlike classical physics) automatically does parallel processing at each and every point of space-time, which is where the processing power of a quantum computer comes from.

Working upwards towards larger QM systems involves no change in the theory (that we know of, that is) because the evolution operator exp(i H t) can be applied to any state no matter how complicated it is. There is no system "size" above which the physics is fundamentally different from what is already known to be correct at the level of elementary particles. This includes the use of effective degrees of freedom, because these are still governed by the underlying exp(i H t) although many of the details are usually hidden from view; I reserve the right to revise my opinion here having now seen the paper More Really is Different.

Carried on to physically large system sizes (e.g. human brains), this line of reasoning inevitably leads to a "Many Worlds Theory" point of view, where it is QM all the way up from the bottom to the top. We are inside a QM universe, not outside it looking in.

I need direct experimental evidence for "non-QM physics" (i.e. evidence that exp(i H t) is not the whole story) in order to discard my assumption that it is QM all the way from bottom to top. Isn't that the way science should normally be done (I innocently ask), where you preserve the status quo until experimental evidence contradicts it?

Circa 1980 I was on the receiving end of a lot of criticism from physicists around me, but in the interests of self-preservation I then decided to keep quiet about my contrarian thoughts on QM. It was only many years after completing my PhD (and moving to another research field outside QM, but continuing to think about QM) that I finally realised that I had not been the first person to think of these ideas. Duh!

Monday, October 13, 2008

Virtual Forbidden City

IBM and Palace Museum announce the opening of the Forbidden City Virtual World celebrating 600 years of Chinese culture (see here). The Virtual Forbidden City website says:

The Virtual Forbidden City is a 3-dimensional virtual world where visitors from around the world can experience the Forbidden City in Beijing. You can explore the magnificient palace as it was during the Qing dynasty, which ruled from 1644 until 1912, the end of the Imperial period in China.

The image above shows a location that I "photographed" on my first visit to the Virtual Forbidden City. There is much more than can be seen in this single photograph.

This is not a fully featured virtual world (e.g. Second Life), but it is good enough for visiting and familiarising yourself with the Forbidden City. This virtual reconstruction of the Forbidden City has been done quite carefully. The in-world objects have been "painted" with textures that appear to have been derived from photographs of their real-world counterparts, which adds to the realism. This is quite hard work to do properly, especially for irregularly shaped objects, as I have found when creating virtual copies of real-world objects in Second Life.