This post is derived from an interesting conversation I had several years ago with my friend Jason Newquist on trying to find some intuitive analogies for the non-classical nature of quantum mechanics. It occurred to me that this type of informal, rambling discussion might actually be rather suited to the blog medium, so here goes nothing…

Quantum mechanics has a number of weird consequences, but here we are focusing on three (inter-related) ones:

1. Objects can behave both like particles (with definite position and a continuum of states) and waves (with indefinite position and (in confined situations) quantised states);
2. The equations that govern quantum mechanics are deterministic, but the standard interpretation of the solutions of these equations is probabilistic; and
3. If instead one applies the laws of quantum mechanics literally at the macroscopic scale, then the universe itself must split into the superposition of many distinct “worlds”.

In trying to come up with a classical conceptual model in which to capture these non-classical phenomena, we eventually hit upon using the idea of using computer games as an analogy. The exact choice of game is not terribly important, but let us pick Tomb Raider – a popular game from about ten years ago (back when I had the leisure to play these things), in which the heroine, Lara Croft, explores various tombs and dungeons, solving puzzles and dodging traps, in order to achieve some objective. It is quite common for Lara to die in the game, for instance by failing to evade one of the traps. (I should warn that this analogy will be rather violent on certain computer-generated characters.)

The thing about such games is that there is an “internal universe”, in which Lara interacts with other game elements, and occasionally is killed by them, and an “external universe”, where the computer or console running the game, together with the human who is playing the game, resides. While the game is running, these two universes run more or less in parallel; but there are certain operations, notably the “save game” and “restore game” features, which disrupt this relationship. These operations are utterly mundane to people like us who reside in the external universe, but it is an interesting thought experiment (which others have also proposed :-) ) to view them from the perspective of someone like Lara, in the internal universe. (I will eventually try to connect this with quantum mechanics, but please be patient for now.) Of course, for this we will need to presume that the Tomb Raider game is so advanced that Lara has levels of self-awareness and artificial intelligence which are comparable to our own.

Imagine first that Lara is about to navigate a tricky rolling boulder puzzle, when she hears a distant rumbling sound – the sound of her player saving her game to disk. Let us suppose that what happens next (from the perspective of the player) is the following: Lara navigates the boulder puzzle but fails, being killed in the process; then the player restores the game from the save point and then Lara successfully makes it through the boulder puzzle.

Now, how does the situation look from Lara’s point of view? At the save point, Lara’s reality diverges into a superposition of two non-interacting paths, one in which she dies in the boulder puzzle, and one in which she lives. (Yes, just like that cat.) Her future becomes indeterministic. If she had consulted with an infinitely prescient oracle before reaching the save point as to whether she would survive the boulder puzzle, the only truthful answer this oracle could give is “50% yes, and 50% no”.

This simple example shows that the internal game universe can become indeterministic, even though the external one might be utterly deterministic. However, this example does not fully capture the weirdness of quantum mechanics, because in each one of the two alternate states Lara could find herself in (surviving the puzzle or being killed by it), she does not experience any effects from the other state at all, and could reasonably assume that she lives in a classical, deterministic universe.

So, let’s make the game a bit more interesting. Let us assume that every time Lara dies, she leaves behind a corpse in that location for future incarnations of Lara to encounter. (This type of feature was actually present in another game I used to play, back in the day.) Then Lara will start noticing the following phenomenon (assuming she survives at all): whenever she navigates any particularly tricky puzzle, she usually encounters a number of corpses which look uncannily like herself. This disturbing phenomenon is difficult to explain to Lara using a purely classical deterministic model of reality; the simplest (and truest) explanation that one can give her is a “many-worlds” interpretation of reality, and that the various possible states of Lara’s existence have some partial interaction with each other. Another valid (and largely equivalent) explanation would be that every time Lara passes a save point to navigate some tricky puzzle, Lara’s “particle-like” existence splits into a “wave-like” superposition of Lara-states, which then evolves in a complicated way until the puzzle is resolved one way or the other, at which point Lara’s wave function “collapses” in a non-deterministic fashion back to a particle-like state (which is either entirely alive or entirely dead).

Now, in the real world, it is only microscopic objects such as electrons which seem to exhibit this quantum behaviour; macroscopic objects, such as you and I, do not directly experience the kind of phenomena that Lara does and we cannot interview individual electrons to find out their stories either. Nevertheless, by studying the statistical behaviour of large numbers of microscopic objects we can indirectly infer their quantum nature via experiment and theoretical reasoning. Let us again use the Tomb Raider analogy to illustrate this. Suppose now that Tomb Raider does not only have Lara as the main heroine, but in fact has a large number of playable characters, who explore a large number deadly tombs, often with fatal effect (and thus leading to multiple game restores). Let us suppose that inside this game universe there is also a scientist (let’s call her Jacqueline) who studies the behaviour of these adventurers going through the tombs, but does not experience the tombs directly, nor does she actually communicate with any of these adventurers. Each tomb is explored by only one adventurer; regardless of whether she lives or dies, the tomb is considered “used up”.

Jacqueline observes several types of trapped tombs in her world, and gathers data as to how likely an adventurer is to survive any given type of tomb. She learns that each type of tomb has a fixed survival rate – e.g. a tomb of type A has a 20% survival rate, while a tomb of type B has a 50% survival rate – but that it seems impossible to predict with any certainty whether any given adventurer will survive any given type of tomb. So far, this is something which could be explained classically; each tomb may have a certain number of lethal traps in them, and whether an adventurer survives these traps or not may entirely be due to random chance.

But then Jacqueline encounters a mysterious “quantisation” phenomenon: the survival rate for various tombs are always one of the following numbers:

$100\%, 50\%, 33.3\ldots\%, 25\%, 20\%, \ldots$;

in other words, the “frequency” of success for a tomb is always of the form 1/n for some integer n. This phenomenon would be difficult to explain in a classical universe, since the effects of random chance should be able to produce a continuum of survival probabilities.

Here’s what is going on. In order for Lara (say) to survive a tomb of a given type, she needs to stack together a certain number of corpses together to reach a certain switch; if she cannot attain that level of “constructive interference” to reach that switch, she dies. The type of tomb determines exactly how many corpses are needed – suppose for instance that a tomb of type A requires four corpses to be stacked together. Then the player who is playing Lara will have to let her die four times before she can successfully get through the tomb; and so from her perspective, Lara’s chances of survival are only 20%. In each possible state of the game universe, there is only one Lara which goes into the tomb, who either lives or dies; but her survival rate here is what it is because of her interaction with other states of Lara (which Jacqueline cannot see directly, as she does not actually enter the tomb).

A familiar example of this type of quantum effect is the fact that each atom (e.g. sodium or neon) can only emit certain wavelengths of light (which end up being quantised somewhat analogously to the survival probabilities above); for instance, sodium only emits yellow light, neon emits blue, and so forth. The electrons in such atoms, in order to emit such light, are in some sense clambering over skeletons of themselves to do so; the more commonly given explanation is that the electron is behaving like a wave within the confines of an atom, and thus can only oscillate at certain frequencies (similarly to how a plucked string of a musical instrument can only exhibit a certain set of wavelengths, which incidentally are also proportional to 1/n for integer n). Mathematically, this “quantisation” of frequency can be computed using the bound states of a Schrödinger operator with potential. (Now, I am not going to try to stretch the Tomb Raider analogy so far as to try to model the Schrödinger equation! In particular, the complex phase of the wave function – which is a fundamental feature of quantum mechanics – is not easy at all to motivate in a classical setting, despite some brave attempts.)

The last thing we’ll try to get the Tomb Raider analogy to explain is why microscopic objects (such as electrons) experience quantum effects, but macroscopic ones (or even mesoscopic ones, such as large molecues) seemingly do not. Let’s assume that Tomb Raider is now a two-player co-operative game, with two players playing two characters (let’s call them Lara and Indiana) as they simultaneously explore different parts of their world (e.g. via a split-screen display). The players can choose to save the entire game, and then restore back to that point; this resets both Lara and Indiana back to the state they were in at that save point.

Now, this game still has the strange feature of corpses of Lara and Indiana from previous games appearing in later ones. However, we assume that Lara and Indiana are entangled in the following way: if Lara is in tomb A and Indiana is in tomb B, then Lara and Indiana can each encounter corpses of their respective former selves, but only if both Lara and Indiana died in tombs A and B respectively in a single previous game. If in a previous game, Lara died in tomb A and Indiana died in tomb C, then this time round, Lara will not see any corpse (and of course, neither will Indiana). (This entanglement can be described a bit better by using tensor products: rather than saying that Lara died in A and Indiana died in B, one should instead think of $\hbox{Lara } \otimes \hbox{ Indiana}$ dying in $\left|A\right> \otimes \left|B\right>$, which is a state which is orthogonal to $\left|A\right> \otimes \left|C\right>$.) With this type of entanglement, one can see that there is going to be significantly less “quantum weirdness” going on; Lara and Indiana, adventuring separately but simultaneously, are going to encounter far fewer corpses of themselves than Lara adventuring alone would. And if there were many many adventurers entangled together exploring simultaneously, the quantum effects drop to virtually nothing, and things now look classical unless the adventurers are somehow organised to “resonate” in a special way.

One might be able to use Tomb Raider to try to understand other unintuitive aspects of quantum mechanics, but I think I’ve already pushed the analogy far beyond the realm of reasonableness, and so I’ll stop here. :-)