Notes for the Lifebox, the Seashell, and the Soul - Rudy Rucker

Notes for The Lifebox, the Seashell, and the Soul, by Rudy Rucker
Fredkin’s Billiard Ball Computer
In the 1970s, the computer scientist Edward Fredkin gave a theoretical proof that, if
you idealize away all of the real-world crud, you can make a universal computer from billiard
balls bouncing around on a sufficiently large table. At the time this seemed like a surprising
result, but by now it’s beginning to seem likely that almost all of the physical systems we
encounter are universal, just as they are.
Now, in reality, you can’t actually build a Fredkin-style billiard ball computer, for the
balls’ large-scale motions will quickly display the slight inaccuracies in your starting
conditions, not to mention the influences of effects as small as gravitational forces from your
body as you walk around the table observing the experiment. The system won’t behave in
the repeatable digital fashion that you’d planned. It’s not realistic to expect the motions of
ordinary objects to behave like a digital computer. We live in an irredeemably analog world.
But Fredkin’s proof-in-principle encourages us to believe some form of Wolfram’s
Principle of Computational Equivalence, which claims that essentially all physical systems
embody universal computations fully rich enough to emulate anything that happens inside an
electronic machine.
Schrödinger’s Wave Equation
I’m not going to try and explain Schrödinger’s Wave Equation in detail, but I do feel
I owe you a few comments.
• Since a system’s wave function ψ(x) changes with time, we write it as ψ(x, t).
• We put an arrow over the x to indicate that this symbol stands in for an arbitrarily long
list of state variables
• Multiplying by i on the left corresponds to a ninety-degree rotation of phase.
• h is a the tiny number called Planck’s constant.
• The quotient with the two backwards sixes means “the change with respect to time”.
• The H on the right is the so-called Hamiltonian operator, which corresponds to the total
energy of the system.
In practice it’s difficult or even impossible to write out a precise formula for a
complicated system’s Hamiltonian operator ⎯ which fact tends to limit the real-world
usefulness of Schrödinger’s Wave Equation.
Randomness
When you measure a system, its wave function undergoes an abrupt change ⎯
sometimes called the collapse of the wave function. The resulting simplified wave function
is called an eigenstate of the measurement. The particular measurement eigenstate is picked
wholly at random, although in accordance with probabilities that can be calculated from the
pre-collapse wave function.
Regarding the randomness of quantum mechanics, where would the randomness
come in? One might say that the world contains two kinds of computation: the computation
of quantum mechanical physics and the computation of classical physics. Each of these is
deterministic, but there is an unavoidable element of randomness in switching from the
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