In the book Good and Real, author Gary Drescher, who received his PhD from MIT’s AI lab, defends the view that determinism is a consistent and coherent view of the world. In doing so, he enters many different arenas: ethics, decision theory, and physics.

In his chapter on quantum mechanics, he defends the “many-worlds” interpretation (although he doesn’t think the term accurately describes the concept) versus the Copenhagen interpretation. In the process of doing so, he does something I thought was extraordinary: he comes up with a simple model of quantum mechanics in which all of the standard concepts you read about: the two-slit experiment, the Heisenberg uncertainty principle, etc., are represented. This model requires no prerequisites from physics and actually uses almost totally discrete mathematics!

(Edit: I somehow missed this when originally writing this post, but Drescher also outlines quantish physics in an online paper.)

I’ll sketch it below.

The first step is to define the “classical” version of our physics, which we will then tweak to get the quantum version. The “topology” of our universe will be given by a finite directed graph where each vertex has three edges coming in, and three edges going out. There is given a bijection such that if edge is directed in to vertex , edge is directed out of vertex . Given this bijection, you can think of each edge as actually a piece of a wire: a directed loop in the graph. Finally, we require that each edge is labeled either , or so that each triple of in-edges to a given vertex gets a distinct label.

We can picture vertices like this:

The vertex is represented by the big box in the center. We will always put the control edge at the top, and the two switch edges at the bottom. Note that the edge need not have the same label as .

Particles inhabit edges. If is the set of particles and is the set of edges, then a point in determines the position of each particle. The set is thus called (classical) configuration space. Time in this universe is discrete; to describe how the system evolves, we just have to define the successor function which tells how the system progresses one time step (the superscript stands for “classical”).

For any edge and label , we let be the edge with same destination vertex as and with label . For a configuration and edge , we let if edge does not have a particle in configuration , and we let it be if it does. Now we define as follows: For every edge labeled , . For every edge labeled ,

In other words, particles on a control edge always go straight along whatever loop they are on. However, particles on a switch edge may or may not cross over to the loop of the other switch edge, depending on whether or not there is a particle on the control edge (hence the names).

For example, this configuration:

turns into this configuration:

and this configuration:

turns into this configuration:

Now for the “quantum” variation, which Drescher calls quantish physics. In this case, each particle now has a sign ( / ) attached to it. Furthermore, each vertex has an angle associated with it, called ‘s measurement angle. The classical configuration space was ; the quantum configuration space will be the set of all formal linear combinations of states . Given a state , the number can reasonably be interpreted as the probability of being in state (see Drescher for more comment on this).

The task now is to describe the successor function describing how the universe evolves through time. First some preliminary definitions:

Given a nonzero complex number , an angle , and , let be the component of which is parallel to if and the component of which is perpendicular to if . Note that and, due to the Pythagorean theorem, . If, furthermore, , let be the component of which is parallel to (note: , not as above) if and perpendicular to if . As above , and . Similarly, and . Finally, note that for a given , and , the split function is simply multiplication by a complex number independent of (and similarly for the two-argument split function).

First off, is -linear; it therefore suffices to define for classical configurations .

If a particle is on edge labeled , it always passes straight through to .

If a particle is on a switch edge, the successor state will be the sum of (up to) 4 non-zero classical configurations corresponding to whether or not it stays on its loop or crosses over to the loop of the other switch edge, and whether or not it changes sign. Let for denote the classical state where the particle stays on its own loop iff and the particle keeps the same sign iff . Then the weight given to is (the is not a typo). The use of is just because we are assuming that we are starting from a pure classical configuration; if it’s a classical configuration time a weight , the would be replaced with .

When there are particles in the classical configuration , each is split separately; may be the sum of up to classical configurations. Since the splitting is simply multiplication by a complex number, it doesn’t matter in what order the splittings are performed.

I’ll now briefly describe some quantum phenomena which can be interpreted in the quantish world. For much more insight, meaning, and many more examples, please see Drescher’s book!

The Two-Slit Experiment

Suppose we have the following configuration:

where the measurement angles of and are equal and oblique to the measurement angle of , and the sign of the particle is positive. Then, it is always the case that after three timesteps, the particle is in the middle rightmost edge (i.e., applied three times to the initial state yields the linear combination consisting of the sum of the single classical state where the particle is in the middle rightmost edge). It is never in the bottom rightmost edge.

However, if we remove one of the ways the particle can get to the bottom edge:

now the particle arrives at the bottom rightmost edge with positive probability (i.e., nonzero weight). (The ellipsis simply means that this edge goes somewhere else; we don’t care what happens there.) This is because of destructive interference in the first case, which was removed in the second.

Suppose we try to investigate what’s going on, and observe if the particle is on one particular edge:

What’s going on here? Well first of all, the way we observe things is by, e.g., having the particles we want to observe interact with particles in our eye. In this example, we’ll take the red particle at the bottom to be a particle “in our eye” and observe the blue particle by having it interact with the red. Second of all, there are various “delay” gates throughout (at the bottom left, the delay gate is explicit, at the top, two wires are labeled “delay”, which means that they pass through a delay gate, although I haven’t drawn it). These aren’t really significant; they’re just to synchronize things.

Note that this gate has exactly the same behavior as our first setup, except that we are observing when the blue particle is on the bottom edge by having it interact with the red particle (and syncing things up). However, the results are as in the second case: with nonzero probability, the red particle appears on the bottom rightmost edge! The observation blocks the destructive interference of the first case.

Heisenberg Uncertainty

Say that a particle in a quantish state is definite with respect to a measurement angle if passing it through a switch wire of a vertex with measurement angle and no other particles entering the vertex will yield the particle always emerging on one specific edge. Heisenberg uncertainty is represented in quantish physics by the fact that whenever a particle is definite with respect to some measurement angle , it is always indefinite with respect to measurement angles oblique to .

The Einstein-Podolsky-Rosen Experiment

Unfortunately, the diagrams for this setup are beyond my poor figure-making abilities. However, I can substitute a poor description for poor figure-making. It is possible to entangle two quantish particles by sending both of them through gates which have related measurement angles, then setting up two further gates through which a third particle is sent. In the first of these gates, one of the switch wires from the first particle’s measurement is the control wire, and in the second of these gates, one of the switch wires from the second particle’s measurement is the control wire. You then observe if the third particle emerges from the second of the two gates at the same place where it entered.

In that case, you can then measure the two particles both with respect to any fixed angle, and you will get the same results for both. (Let me reiterate that this was a terrible description; see Drescher’s book for more).