Configurations and Amplitude

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So the universe isn’t made of little billiard balls, and it isn’t made of crests and troughs in a pool of aether… Then what is the stuff that stuff is made of?

In Figure 1, we see, at A, a half-silvered mirror, and two photon detectors, Detector 1 and Detector 2.

Early scientists, when they ran experiments like this, became confused about what the results meant. They would send a photon toward the half-silvered mirror, and half the time they would see Detector 1 click, and the other half of the time they would see Detector 2 click.

The early scientists—you’re going to laugh at this—thought that the silver mirror deflected the photon half the time, and let it through half the time.

Ha, ha! As if the half-silvered mirror did different things on different occasions! I want you to let go of this idea, because if you cling to what early scientists thought, you will become extremely confused. The half-silvered mirror obeys the same rule every time.

If you were going to write a computer program that was this experiment— not a computer program that predicted the result of the experiment, but a computer program that resembled the underlying reality—it might look sort of like this:

At the start of the program (the start of the experiment, the start of time) there’s a certain mathematical entity, called a configuration. You can think of this configuration as corresponding to “there is one photon heading from the photon source toward the half-silvered mirror,” or just “a photon heading toward A.”

A configuration can store a single complex value—“complex” as in the complex numbers , with i defined as . At the start of the program, there’s already a complex number stored in the configuration “a photon heading toward A.” The exact value doesn’t matter so long as it’s not zero. We’ll let the configuration “a photon heading toward A” have a value of .

All this is a fact within the territory, not a description of anyone’s knowledge. A configuration isn’t a proposition or a possible way the world could be. A configuration is a variable in the program—you can think of it as a kind of memory location whose index is “a photon heading toward A”—and it’s out there in the territory.

As the complex numbers that get assigned to configurations are not positive real numbers between 0 and 1, there is no danger of confusing them with probabilities. “A photon heading toward A” has complex value −1, which is hard to see as a degree of belief. The complex numbers are values within the program, again out there in the territory. We’ll call the complex numbers amplitudes.

There are two other configurations, which we’ll call “a photon going from A to Detector 1” and “a photon going from A to Detector 2.” These configurations don’t have a complex value yet; it gets assigned as the program runs.

We are going to calculate the amplitudes of “a photon going from A toward 1” and “a photon going from A toward 2” using the value of “a photon going toward A,” and the rule that describes the half-silvered mirror at A.

Roughly speaking, the half-silvered mirror rule is “multiply by 1 when the photon goes straight, and multiply by i when the photon turns at a right angle.” This is the universal rule that relates the amplitude of the configuration of “a photon going in,” to the amplitude that goes to the configurations of “a photon coming out straight” or “a photon being deflected.”[1]

So we pipe the amplitude of the configuration “a photon going toward A,” which is