I just learned the very sad news that John A. Wheeler has passed away. Wheeler was one of my heroes and inspired me in many ways to be where I am today. I’m buried under a heap of work today, but will write more when I can come up for air. Below I’ve pasted a post from my old blog describing a result I first learned about by reading a Wheeler paper.
Note: This post originally appeared at my old blog site.
When I was an undergraduate at Caltech visiting Harvard for the summer I stumbled upon Volume 21 of the International Journal of Theoretical Physics (1982). What was special about this volume of this journal was that it was dedicated to papers on the subject of physics and computation (I believe it was associated with the PhysComp conference?) Now for as long as I can remember I have been interested in physics and computers. Indeed one of the first programs I ever wrote was a gravitational simulator on my TRS-80 Color Computer (my first attempt failed because I didn’t know trig and ended up doing a small angle approximation for resolving vectors…strange orbits those.) Anyway, back to Volume 21. It contained a huge number of papers that I found totally and amazingly interesting. Among my favorites was the plenary talk by Feynman in which he discusses “Simulating Physics with Computers.” This paper is a classic where Feynman discusses the question of whether quantum systems can be probabilistically simulated by a classcal computer. The talk includes a discussion of Bell’s theorem without a single reference to John Bell, Feynman chastizing a questioner for misusing the word “quantizing”, and finally Feynman stating one of my favorite Feynman quotes
The program that Fredkin is always pushing, about trying to find a computer simulation of physics, seem to me an excellent program to follow out. He and I have had wonderful, intense, and interminable arguments, and my argument is always that the real use of it would be with quantum mechanics, and therefore full attention and acceptance of the quantum mechanical phenomena-the challenge of explaining quantum mechanical phenomena-has to be put into the argument, and therefore these phenomena have to be understood ver well in analyzing the situation. And I’m not happy with all the analyses that go with just the classical theory, because nature isn’t classical dammit, and if you want to make a simulation of nature, you’d better make it quantum mechanical, and by golly it’s a wonderful problem, becuase it doesn’t look so easy.
(That, by the way, is how he ends the paper. Talk about a way to finish!)
Another paper I found fascinating in the volume was a paper by Marvin Minsky in which he points out how cellular automata can give rise to relativistic and quantum like effects. In retrospect I dont see as much amazing about this paper, but it was refreshing to see things we regard as purely physics emerging from simple computational models.
But the final paper, and which to this day I will go back and read, was “The Computer and the Universe” by John Wheeler. Of course this being a Wheeler paper, the paper was something of a poetic romp…but remember I was a literature major so I just ate that style up! But the most important thing I found in that paper was a description by Wheeler of the doctoral thesis of Wootters. Wootters result, is I think, one of the most interesting result in the foundations of quantum theory that you’ve never heard of (unless you’ve read one of the versions of the computation and physics treatise that Wheeler has published.) Further it is one of those results which is hard to find in the literature.
So what is this result that I speak of? What Wootters considers is the following setup. Suppose a transmitter has a machine with a dial which can point in any direction in a plane. I.e. the transmitter has a dial which is an angle between zero and three hundred and sixty degrees. Now this transmitter flips a switch and off goes something…we don’t know what…but at the other end of the line, a receiver sits with another device. This device does one simple thing: it receives that something from the transmitter and then either does or does not turn on a red light. In other words this other device is a measurment aparatus which has two measurment outcomes. Now of course those of you who know quantum theory will recognize the experiment I just described, but you be quiet, I don’t want to hear from you…I want to think, more generally, about this experimental setup.
So we have transmitter with an angle and a reciever with a yes/no measurement. Now yes/no measurements are interesting. Suppose that you do one hundred yes/no measurements and find that yes occurs thirty times. You will conclude that the probability of the yes outcome is then roughly thirty percent. But probabilities are finicky and with one hundred yes/no measurements you can’t be certain that the probability is thirty percent. It could very well be twenty five percent or thirty two percent. Now take this observation and apply it to the setup we have above. Suppose that the transmitter really really wants to tell you the angle he has his device set up at. But the receiver is only getting yes/no measurements. What probability of yes/no measurements should this setup have, such that the the receiver gains the most information about the angle being sent? Or expressed another way, suppose that a large, but finite, number of different angles are being set on the transmitter. If for each of these angles we get to choose a probability distribution, then this probability distribution will have some ability to distinguish from other probability distributions. Suppose that we want to maximizes the number of distinguishable settings for the transmitter. What probability distribution should occur (i.e. what probability of yes should there be as a function of the angle)?
And the answer? The answer is p(t)=cos^2(nx/2) where n is an integer and x is the angle. Look familiar? Yep thats the quantum mechanical expression for a setup where you send a spin n/2 particle with its amplitude in a plane, and then you measure along one of the directions in that plane. In other words quantum theory, in this formulation, is set up so that the yes/no distribution maximizes the amount of information we learn about the angle x! Amazing!
You can find all of this in Wootters’s 1980 thesis. A copy of which I first laid my hands on because Patrick Hayden had a copy, and which I subsequently lost, but now have on loan for the next two weeks! Now, of course, there are caviots about all of this and you should read Wootters’s thesis, which I do highly recommend. But what an interesting result. Why haven’t we all heard of it?