华丽的科普~华丽的翻译~《古墓丽影瀚量子力学》！

看到一篇强文：Quantum mechanics and Tomb Raider
原文地址，看到wordpress.com这个域名，你可能就知道这个BLOG需要用代理访问了……

原文我也看不明白，然则NTRPG论坛上的Ross老爷就华丽的把它翻译出来了哑！赞！经译者授权，转帖于此：

我们要讨论的是，在这样一款游戏中，存在一个“里世界”，即Lara所存在的游戏世界；以及一个 “表世界”，即玩家通过电脑操纵Lara的现实世界。当游戏运行的时候，这两个世界或多或少是相互平行的。但是某种操作，特别是“保存游戏”和“读取游戏”的行为，会破坏表世界和里世界的关系。这种行为对处于表世界的我们来说是再正常不过的事情了。但是，如果我们从Lara的角度，即里世界的角度来看，就会发现事情会变得有趣起来。

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看到一篇强文：Quantum mechanics and Tomb Raider
原文地址：http://terrytao.wordpress.com/2007/02/26/quantum-mechanics-and-tomb-raider/
看到wordpress.com这个域名，你可能就知道这个BLOG需要用代理访问了……

原文我也看不明白，然则NTRPG论坛上的Ross老爷就华丽的把它翻译出来了哑！赞！经译者授权，转帖于此：

量子力学导致了很多匪夷所思的结论，这里我们只着重讨论其中三点：

一、物体可以同时表现粒子性和波动性；
二、那些描述量子力学的方程式是确定的,但关于这些方程的解的标准解释却是概然的；
三、如果用量子力学法则来解释宏观现象,那就得要求宇宙本身必须分割成为多个互不相同的“世界”的叠加体。


为了通过传统的概念模型来描绘量子力学的非常现象，我们决定使用电脑游戏来帮助我们进行分析。具体选择哪款游戏并不重要，就让我们用古墓丽影—一款十多年以前就开始流行的游戏来进行我们的讨论吧。在这款游戏中，Lara Croft是女主人公，她致力于在各种古墓和地城中探险，经受迷题和陷阱的考验，只为得到她所心仪的宝贝。要知道，在游戏中，Lara死掉是一件稀松平常的事情，很可能一个陷阱就要了她的性命。

我们要讨论的是，在这样一款游戏中，存在一个“里世界”，即Lara所存在的游戏世界；以及一个 “表世界”，即玩家通过电脑操纵Lara的现实世界。当游戏运行的时候，这两个世界或多或少是相互平行的。但是某种操作，特别是“保存游戏”和“读取游戏”的行为，会破坏表世界和里世界的关系。这种行为对处于表世界的我们来说是再正常不过的事情了。但是，如果我们从Lara的角度，即里世界的角度来看，就会发现事情会变得有趣起来。

首先想象一下，Lara碰到了游戏中的某个迷题，此时操作Lara的玩家选择在尝试之前先保存一下游戏。接下来会发生的事情可能是：Lara的第一次尝试失败，死掉了；玩家读取之前的存档，这次Lara成功了。

现在，从Lara的角度来看，情况会有什么不同呢？在游戏的记录点，Lara的世界变成了两种互不干扰的可能分支的重叠，在其中一个分支里，她尝试失败而死掉了；在另一个分支里，她成功地活下来。（就像薛定谔的猫）她的未来变得不确定起来。如果她在记录点以前向先知询问她是否能活下来，先知能给她的唯一的正确回答就是50%可能死掉，50%可能活下来。

这个简单的例子表明，表世界中的结果十分确定，但里世界的结果却可以是不确定的。然而，这个例子无法完全说明量子力学的匪夷所思之处。在这两个分支世界中的Lara都无法了解关于另一个分支世界中的情况，因此Lara只能得到她是活在一个“传统的、结果确定的世界”中这样一个结论。

那么，让我们把这个游戏变得更有趣一些。我们假设，Lara每次死后，都会在死亡地点留下尸体，而且未来的Lara可以碰到这些尸体。那么Lara就会注意到以下现象：无论她何时开始解决游戏中的迷题，她都会碰到一定数量的自己的尸体。我们无法用传统的确定的现实模型来向Lara解释这些奇怪的现象。我们能给她的最简单的解释就是现实由多元世界构成，而包含每各种可能性的世界之间部分关联。另一个可能的解释是：每当Lara通过一个记录点的时候，她的确定的粒子性发散成为各种可能性的波形叠加，然后在她通过迷题的瞬间，描述她的命运的各种可能性的波函数立刻坍缩成为一个确定的结果。

在现实世界中，只有微观物体，例如电子，才会体现出量子性。而宏观物体，比如你和我，则无法体会 Lara在游戏世界中所遭遇的那种现象，我们也无法去采访单个电子，询问它的切身体会。不过，通过对微观物体行为的统计，我们可以经由实验和理论推测间接得出它们的量子性质。让我们再一次用古墓丽影来分析一下这个问题。设想在古墓丽影中并非只有Lara一个主人公，而有很多角色可以使用。他们都像Lara 一样喜欢冒险。我假设，在游戏世界中也有一个科学家——就让我们叫她Jacqueline——专门研究这些冒险者的行为。但是她不会亲自去体验在古墓中探险活动，她甚至也不跟这些冒险者有任何接触。每一座古墓只有一个冒险者进入，无论这个冒险者死掉或是活着完成探索，都不会有其他冒险者再进入这座古墓。

Jacqueline 尝试考察她世界中的一些不同类型的古墓，并且收集了冒险者门在不同类型的古墓中幸存下来的数据。她发现，对于一个给定类型的古墓，都有一个确定的幸存概率 ——比如类型A的古墓所对应的幸存率固定为20%，而类型B则固定为50%——但是却无法确定，一个给定的冒险者在一个给定的古墓中是否能幸存下来。到目前为止，这些现象都可以通过传统理论来解释：每一座墓地中都有若干致命的陷阱，冒险者能否成功通过这些陷阱都由一个随机几率决定。

但是，Jacqueline此刻会遇到神秘的量子现象：不同类型古墓的幸存几率不外乎以下一些数字：

100%、50%、33.3%、25%、20%……

换句话说就是，每一座古墓的幸存几率固定为某个整数的倒数1/n。这种现象在传统理论中难以解释，因为就随机效应来说，幸存率的分布应该是一条连续的曲线。

这就是说，Lara为了能在某种类型的古墓中幸存下来，她需要叠加一定数量的自己的尸体来到达某个临界点。如果她无法跃迁到一定的干涉能级来抵达那个临界点，她就会在冒险中死掉。古墓的类型决定了具体需要多少尸体才能抵达临界点——比如A类型的墓地需要四具尸体。这样控制Lara的玩家就必须让Lara失败四次才能成功通过这个古墓。以Jacqueline的角度来看，Lara的幸存纪律就是20%。在承载着每一个可能性的游戏世界中，只有一个Lara进入了古墓中。但是Lara的幸存几率却是因为多个重叠世界相互干涉而决定的。

这种量子效应的一个类似的例子是，每一种原子只能发射出特定波长的可见光波，比如钠原子的光谱线就是黄色的，而氖原子的光谱线是蓝色的。在这些原子中的电子为了能发射出这样的光波，从某种意义上来说就是爬着它们自己的尸体抵达被激发的临界点。通常的解释是，在原子内部的电子以波的形式存在，并以某个固定的频率震荡。

最后我们再次利用古墓丽影来解释一下为什么微观物体能体现出量子效应而宏观物体却没有。我们假设古墓丽影是一个双人合作游戏，两个玩家分别控制两个人物（这两个人物就叫Lara和 Indiana），她们可以同时探索她们世界的不同部分。玩家可以保存和读取游戏。读取游戏的时候Lara和Indiana都会回到记录点的状态。

此时，游戏依然会有Lara和Indiana遭遇到她们自己尸体的奇怪现象。然而，我们假设，Lara和Indiana通过以下方式纠缠：如果Lara探索墓地A而Indiana探索墓地B，只有Lara和Indiana两人都在上一次记录游戏中分别死在墓地A和墓地B，那么Lara和Indiana才会各自碰到自己在上一次记录游戏中的尸体。如果在上一次记录游戏中，Lara死在墓地A而Indiana死在墓地C，那么她们这次就不会看到任何尸体。因为这种纠缠，量子效应减少了。Lara和Indiana如果同时在这个世界中进行不同的探险活动，将比Lara单独进行探险活动要遇到更少的尸体。如果同时有大量的冒险者分别进行探索，那么量子效应基本上为0。

原文：

Quantum mechanics and Tomb Raider


Monday, February 26th, 2007 in non-technical




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. From the perspective of the player, what happens next 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. :-)