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:

- 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);
- The equations that govern quantum mechanics are deterministic, but the standard interpretation of the solutions of these equations is probabilistic; and
- 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:

;

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 dying in , which is a state which is orthogonal to .) 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. :-)

## 74 comments

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10 February, 2008 at 5:32 pm

Tiempo finito y logarítmico » Quantum Mario[…] Quantum mechanics and Tomb Raider propuso originalmente un concepto interesante. Al estar en el save point de un videojuego, desde el punto de vista del personaje éste se encuentra en una superposición de estados; a partir de ahí puede ser que continúe hacia caer a un pozo o hacia terminar el juego; justo como “ese gato”. […]

7 August, 2008 at 1:32 am

AzzanNice example Quantum mechanics I love Tomb Raider its my best game

11 December, 2008 at 5:03 am

ugechi peace N.What do u mean pls help & explain more

14 April, 2009 at 7:13 am

Johan FalkReally cool post. Thanks a lot!

(It reminds me of a previous discussion about quantum suicide, http://en.wikipedia.org/wiki/Quantum_suicide. Don’t try that!)

//Johan Falk, Sweden

20 May, 2009 at 11:53 pm

Daniel FordThere’s a great flash game called Chronotron which relies on interacting with past incarnations of yourself to solve puzzles. I highly recommend it.

http://www.kongregate.com/games/Scarybug/chronotron

22 August, 2009 at 6:45 am

Why is many-worlds winning the foundations debate? | Matt Leifer[…] every time the foundations of quantum theory are mentioned in another science blog, the comments contain […]

22 August, 2009 at 6:46 am

Tao on Many-Worlds and Tomb Raider | Matt Leifer[…] Tao has an interesting post on why many-worlds quantum theory is like Tomb Raider. I think it’s de Broglie-Bohm theory that is more like Tomb Raider though, as you can see […]

5 December, 2009 at 7:45 pm

古墓丽影瀚量子力学[…] 原文：Quantum mechanics and Tomb Raider 作者：Terence Tao（UCLA数学系的华人教授陶哲轩） 翻译：Ross […]

23 January, 2010 at 6:07 pm

Jeff BurdgesIt’ll be interesting if someone devises a game to let players “experience” some of the unintuitive effects of quantum mechanics. A turn based board game would help keep the superposition small, but you’d surely want a computer tracking the superposition.

p.s. It appears a real-time strategy game Achron incorporates time travel by using the fact that such real-time strategy games are mostly just simulations using preprogramed unit behaviors. So players may edit the orders they gave at an earlier point in time, which prompts the game to rerun the simulation from that point on. Of course, the opposing player may also adjust this past commands too.

24 January, 2010 at 2:46 am

discrete transformTheir Resequence Engine allowing for minmax strategies must be from the future too.

25 March, 2010 at 10:39 pm

AnonymousDear Dr.Tao..In an entirely different context can I ask you a question?

General Relativity says that mass deforms the shape of space-time. There are different shapes for the same mass that can produce this deformation.

Is there a way to construct a schrodinger type of equation whose solutions are the possible shapes for a given mass that can give rise to the same curvature in space-time?

Thanks,

Ganesh Raghavan

2 May, 2010 at 3:02 pm

Ron MaimonThe analogy between a classical duplication event for a conscious observer and the quantum mechanical many-worlds interpretation is a good one, but it has philosophical fine points regarding the probability measure, which you glossed over.

It is not at all clear that you are free to conclude that the probability of survival is 1/N for N duplications. If you restore and play again, once a day, you are constantly increasing N, and so if you look at it from Lara’s point of view, should your probability of survival depend on the N on tuesday or on wednesday? What if you run on a processor that sometimes stores a backup copy in a cache, and so internally duplicates some data? Should you count that double? What if you run on 2-petahertz processor vs a 1-petahertz, should the internal probability measure include the duration of existence of the Lara copies?

The only way I know to stop being endlessly confused on this point is to formulate this positivistic way— to ask “what answers to questions that I ask will the Lara’s give” (just as Everett did for quantum mechanics). This is after all the only way to acquire data on the internal experience of the Lara’s. Then the probability measure that you observe when asking questions of Laras depends on the details of which copies of Lara you choose to ask question of, and with what probability.

For example, you could just erase all the Lara events where Lara does not survive, and only talk to survivors. In this coupling between the external and internal world, the chance of survival is certainty, since if you ask the Lara’s whether they always survive, they will answer “yes”. You could talk to all the injured Lara’s, and they would have a different opinion on the danger of the puzzle then if you talk to the uninjured ones. If you keep all the Lara’s, then the great majority would associate some risk to each puzzle, but this again depends on the selection measure for who you choose to end up communicate with. So it is not at all clear that the question of internal experience in this model world has a unique right answer.

But in quantum mechanics the probability measure does have a unique right answer: it’s psi-squared. It’s not determined by an outside agent looking in, or at least, it’s the same for all the outside agents looking in, if you take the Copenhagen interpretation and consider us to be the outside agents. The probabilities in QM are nothing like the reciprocal of an integer, and that effect, which I don’t think should be thought of as necessarily true in the Tomb Raider world, has nothing to do with what physicists call “quantization”.

There is no naive way to go from copy-counting to the probability measure of quantum mechanics, but there is a naive way of identifying classical probability with copy-counts (which is the ensemble interpretation of probabilities). The measure in quantum mechanics is determined by Hilbert space massiveness of states, by a psi-squared measure on the states, not by any naive copy-counting. It is an interesting exercise to construct a copy-counting measure which reproduces quantum mechanical psi-squared probability (that’s DeBroglie Bohm theory, since a classical probabilistic ensemble which can be given a copy-counting frequentist interpretation, and DeBroglie Bohm gives a classical probabilistic ensemble which matches quantum mechanics)

You always run into the same philosophical question when you treat a physical system as self-contained correct model of reality. you have to somehow identify the experience of observers with the mathematical objects inside the theory. When the theory is either probabilistic, quantum mechanical, or duplicates observers, the identification of the correct probability measure from copy-counting is impossible a-priori, you need additional (mild) assumptions. The assumptions for a classical probabilistic theory is that there is an ensemble “underneath it all” and the number of worlds is proportional to the classical probabilities. For QM it’s that worlds with small hilbert space norm are unlikely. For duplicative theories, it requires knowing the way in which duplicates are most likely to talk to an outside observer.

This is the sticky point for many-worlds type interpretations, and it is present in duplicating classical theories in the exact same way, or an even worse way, depending on your point of view. It’s a philosophical problem, not a physical or mathematical one, and the resolution I think works is to adopt a more Platonic philsophy regarding the relation of the computational structures in the mind to the physical objects described by wavefunctions in the world.

Unfortunately, the philosophers who discuss this field don’t often take a functionalist philosophy of mind, so they don’t really get the interesting confusions. Pauli and Einstein discussed similar things first, although Everett brought in the duplication of course.

One thing though: I don’t understand Jaques’ Distler’s comments. The noncommutativity of observables is a non-sequitor for this philosophical issue. Non-commuting observables are just what happens when you describe an orthogonal collection of states and observable-values by a matrix. I don’t see why this algebraic property should be singled out— it’s an opaque algebraic way to restate the principle of superposition and the notion of orthogonality, which are all you need for a philosophical discussion.

14 March, 2011 at 11:51 pm

Weird Consequences of Quantum Mechanics |[…] From: https://terrytao.wordpress.com/2007/02/26/quantum-mechanics-and-tomb-raider/ […]

1 May, 2011 at 8:49 pm

.http://www.goorden.be/2010/12/why-your-understanding-of-quantum-mechanics-is-almost-certainly-wrong/

After reading the above blog, I feel how much I don’t understand what I understand. Do you have any opinion professor?

30 May, 2011 at 11:05 pm

Ultra Weekendok, I came up with a simple concept. Lets put a QRBG into a video game and see what happens. http://ultraweekend.blogspot.com/2011/05/many-worlds-hypothesis-video-game.html

3 June, 2012 at 2:39 am

1-1′

10 May, 2013 at 10:05 pm

Bird’s-eye views of Structure and Randomness (Series) | Abstract Art[…] Tomb raider: an analogy to quantum weirdness (adapted from “Quantum mechanics and Tomb Raider“) […]

11 May, 2013 at 7:24 am

Tomb raider: an analogy for quantum weirdness | Abstract Art[…] [1.1] Quantum mechanics and Tomb Raider […]

23 February, 2014 at 10:19 am

2PGHey Terry,

We are currently working on a cooperative game concept on quantum mechanics in a way that will assist us to build something that not only the external and internal universe has to cooperate but the 2 players that will co-ordinate and collaborate to perform various actions in to the internal environment will feel 100% brain stimulated. Not only they will need to behave as two players in 1 role but they will also have to perform hard advance estimations on possible scenarios the game will evolve.

One Example of what i mean is the game blocked out on our site ( 2pg.com ) . The two players are essentially working one against each other to block the other player “out”.

Given that they predict how the internal environment will behave as well as the 2nd players move’s. They manipulate inner game levers,buttons,traps to block the other guy out.

Thats a crude model and we are currently trying to realize this in a more complex scale.

Wish us luck!

27 March, 2014 at 9:27 am

Paul J. WerbosIt turns out that a lot of the paradoxes in quantum mechanics are a consequence of implicit classical assumptions that all noise is a function of initial conditions only (which is the effective meaning of “causality” as assumed in the classic “CHSH” theorem). At arxiv.org, I show how any of three Markov Random Field models across space-time can reproduce the observed results, even though they are local and realistic. To take this further, aside from cleanup, the REAL challenge is to actually prove stability for three-dimensional topological solitons with nonzero Higgs terms (which is claimed in a classic paper by Erich Weinberg, but not really proven), and then move on to some more realistic Lagrangians.

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4 November, 2015 at 1:18 pm

Tyrone Vasquez10/10 will play Tomb Raider to learn quantum mechanics