It was an unassuming blue-grey volume tucked away in the popular science section of the Siskiyou County Library. “Spacetime Physics” it announced proudly in gold letters across the front of the book. Published in 1965, the book looked as if it hadn’t been touched in the decades since 1965. A quick opening of the book revealed diagrams of dogs floating beside rocket ships, infinite cubic lattices, and buses orbiting the Earth, all interspaced with a mathematical equations containing symbols the likes of which I’d never seen before. What was this strange book, and what, exactly, did those equations mean? How could there be equations and dogs and buses all in the same book? Answering these questions would be the beginning, for me, of a lifelong love of physics. It would also inspire in me a deep love of science books which make you smile, and, more importantly perhaps, led me to works of the physicist John Archibald Wheeler, who would serve as the model of the researcher I have always wanted to be.
Special relativity, the subject of the book “Spacetime Physics”, is very much about geometry. In fact, it is very much about a particularly odd kind of geometry. Now, as a child I had always been fascinated with geometry. As a young boy in a rural area with the infinite time of youth at my disposal, I recall spending hours drawing squares and measuring the sides of these squares and then measuring the length of the diagonal connecting the corner of the squares. Strangely, the ratio of the lengths of the sides of the squares to the diagonal was a very odd number: 1.4 something or other. What was this strange number? Sadly, I never thought to square the number and hence notice that the ratio was the square root of two. But I was fascinated by the fact that no matter what size square I drew, this strange number always appeared.
At about the same time as I discovered “Spacetime Physics” I was also entering a phase of my life where I was moving beyond typing programs from the backs of computer magazines into my computer, and beginning instead to write my own programs. One of the first programs I remember trying to write was a program to simulate the planetary dynamics of two objects interacting gravitationally. Of course, at this time, I didn’t know anything about trigonometry, so while I knew about velocities and accelerations and how you might simulate a differential equation (yeah, don’t ask me how I knew that and didn’t know trig!) I didn’t know how to break a vector down into components. By that time, I did know how to compute the length of the hypotenuse of a triangle, but this alone was only enough to allow me to calculate the force due to Newton’s law of gravitation. So my first computer programs for simulating two bodies interacting via Newton’s law of gravitation, because I did not know how to break down a vector into components, basically used a good guess for how this should work. My guess, however, was not good enough, and so I got these strange interacting planets where the curvature of the trajectories would sharpen along left-right and up-down directions on my computer screen.
So, when I discovered, “Spacetime Physics” I was in the middle of trying to figure out how to do trigonometry correctly. If you take a two dimensional surface and start doing geometry on figures and diagrams drawn on the surface, the rules you get involve measuring angles and taking these angles and using functions like sine and cosine and tangent to figure out relationship between lengths in the diagrams. A first way to begin talking about special relativity is to consider a two dimensional surface where one dimension represents a one dimensional space (flatter land, you might say), and the other dimension represents time. Such a “spacetime” picture then can have lines representing the history of a particle as it moves in space as a function of time.
Now what is interesting about special relativity, is that to measure “distances” in spacetime, you don’t use an ordinary ruler. What you do instead is you measure the change in the spatial dimension of an object, square it, and you measure the change in the temporal dimension of the object, square it, and then you subtract these two numbers. Finally, one takes this number, makes it positive, and computes the square root of this number. This, you may be thinking, is a lot like the Pythagorean theorem of ordinary geometry, and it is-except for the fact that you subtract the spatial and temporal distances squared. That subtraction makes a big difference, however!
So in a world where distances obey this strange subtraction instead of addition, you can imagine, you need to develop a different set of rules and regulations for doing things like trigonometry on these spacetime diagrams. And indeed, in “Spacetime Physics” this is exactly what I learned to do. In particular I learned to take the hyperbolic cosine and hyperbolic sine and use this number to do geometry on the diagrams I could draw in spacetime. That’s right, I actually learned to do hyperbolic trigonometry before I learned how to do ordinary trigonomety. That a young kid with a hunger for mathematics, but not even knowledge of basic trig, could pick up “Spacetime Physics” and learn special relativity from it, I think, is a testament to the quality of this first edition of the book.
After learning about hyperbolic sines and cosines in special relativity, a quick trip down the mathematics section of my local library eventually led me to books on trig and led me to be able to complete my program for simulating gravity on my TRS-80 color computer. Soon to follow I would simulate electric interactions between objects attempting, of course, to see if I could simulate chemistry and then artificial life. While my endeavors into doing this wouldn’t make much progress towards that goal, much of my early programming was spent trying to simulate physics or at least things that looked a bit like physics.
As I grew older and began to think about what I wanted to do with my life, I was fascinated by by computers and physics. On the physics side, I early learned about this very mysterious subject called quantum theory, which had a huge representation in the popular science section of my local library. Motivated by the fact that quantum physics seemed to be connect with questions of predestination, fate, and causality, I taught myself calculus so that I could read the quantum theory book I had purchased. As an undergraduate at Caltech, I got my first real taste of quantum theory via an introductory quarter of quantum mechanics. So excited was I about the stuff I was learning, that I applied to the Santa Fe Insitute to study quantum computing (this would have been in early 1994) and also an REU at the Harvard-Smithsonian Center for Astrophysics. The later is where I ended up for my summer, doing a project involving calculating two photon absorption cross sections in molecular hydrogen. Yeah: throw the kid directly into the think of things with only a quarters worth of quantum theory. Needless to say, thanks to the kindness and patience of my advisers, I spent much of that summer just learning how quantum theory works (there were so many different types of angular momentum and spin!) It was an awesome experience, doing physics calculations relating to a crazy theory of how some diffuse interstellar absorption bands might come about. But the most amazing aspect, for me, was when I discovered in the library of the center, volume 21 of the International Journal of Theoretical Physics (1982). In this volume there is an article which, low and behold, was written by one of the guys whose wrote the book I’d learned special relativity from: John Archibald Wheeler.
As I write this blog entry, today, many many years after my summer at Harvard, the photocopy of the article by Wheeler is, quite literally within arms grasp, sitting in a folder of my own personal photocopy of volume 21 of the IJTP. “The reasons are briefly recalled why (1) time cannot be a primordial category int he description of nature, but secondary, approximate and derived, and (2) the laws of physics could not have been engraved for all time upon a tablet of granite, but had to come into being by a higgedly-piggedly mechanism” reads the first line of the abstract for Wheeler’s entry “The Computer and the Universe.” The abstract continues: “It is difficult to defend the view that existence is built at bottom upon particles, fields of force or space and time. Attention is called to the `elementary quantum phenomenon’ as potential building element for all that is. The task of construction of physics from such elements is compared and contrasted with the problems of constructing a computer out of `yes, no’ devices.”
What madness was this paper! Here was a master of physics, whose offspring included Richard Feynman, Kip Thorne, and others, the man who coined the word “black hole,” the man who sat at the feet of Bohr, and the man who coauthored the perfectly named behemoth textbook “Gravitation,” writing
An unfamiliar computer from far away stands at the center of the exhibition hall. Some of the onlookers marvel at its unprecedented power; others gather in animated knots trying, but so far in vain, to make out its philosophy, its logic, and its architecture. The central idea of the new device escapes them. The central idea of the universe escapes us.
No real computer, of course, ever springs full blown from the brow of Minerva. We start wit the elements and analyze how to achieve structure. For the universe we start with the structure and try to analyze it into elements. Computer science and basic physics mark two of the frontiers of the civilization of this age. One seeks to build complexity out of simplicity. The other tries to unravel complexity into simplicity. No one, it has been said, is better at taking a puzzle apart than the person who put it together and no one is better at putting a puzzle together than the one who took it apart. Can it be that there is a little of the flavor of the physics enterprise of interest for computer science? And something of use for the unraveling of the universe to be learned from the philosophy of computer design?
To this days these words still resonate around in my small cranium. Here were the two subjects I most dearly loved, computers and physics, put together, not as computers simulating physics, or the physics of semiconductors, but instead as two separate intellectual fields who intimate entwining could, in Wheeler’s opinion, lead to the holy grails of understanding physics and understanding computer science. What an idea!
Needless to say the idea that physics and computer science might have something to say to each other beyond “more transistors please” and “can you fix my computer?” is a point of view I hold near and dear. Quantum computing is often exciting simply for this reason: who knew that postselected quantum computation is equal to the complexity class PP? Who knew that studying entanglement could lead you to new ideas in density matrix renormalization group methods? Who knew that studying quantum information theory could lead you to a deeper understanding of the black hole information paradox? I’m not sure how much Wheeler knew of quantum computing, but I’m certain that there are many quantum computing researchers who were inspired by his intuition, that physics and computer science are not as far apart as one would at first imagine.
Feynman said of Wheeler: “Some people think Wheeler’s gotten crazy in his later years, but he’s always been crazy.” In my research, and in my life, I think I would like to be known in just the same way as Wheeler. Because he had a fascinating kind of craziness for physics. A kind which tried to come to grips with how the world works and wasn’t afraid to branch and learn and connect.
In Wheeler’s “The Computer and the Universe” he discusses a method for deriving certain rules of quantum mechanics due to Bill Wootters. This model, is, for me, one of the most fascinating results I’ve ever encountered. As a postdoc at Caltech, one of the subjects I was fortunate enough to work on had a similar flavor to this result. With Ben Toner, we were investigating how much communication is needed to augment a local realistic theory in order to simulate quantum theory. Surprisingly, at least to me, we were able to show that only a single bit of communication was needed in addition to shared randomness to simulate the most basic experiment on distant entangled quantum states. This result, I’ve often thought, was for me exactly the kind of result that Wheeler would have enjoyed, and indeed I, at least, was inspired to work on this problem by Wootters results and Wheelers writings. We sent Wheeler a copy of the paper.
I guess I’ll never know if John Wheeler ever read my crazy paper with Ben Toner, as sadly, Wheeler pasted away a few days ago, on April 12, 2008, at his home in New Jersey. But I’m certain, from reading his writings, thinking about his ideas, and admiring his achievements, that it would have been exactly the kind of crazy that he would have thoroughly enjoyed. For making me a little more crazy, and more, I thank you John Wheeler, and may the “it” and the “bit” forever dance in our minds when we reread your greatest works.