In Hilbert Space, All Things Are Quantumly Possible
Quantum objects have multiple possible futures before they are observed.
量子物体在被观察之前有多种可能的未来。
Hilbert space allows physicists to explore those possibilities.
希尔伯特空间允许物理学家探索这些可能性。
A t the heart of quantum mechanics lie a few sacred rules for how to use the theory.
量子力学的核心在于如何使用该理论的一些神圣规则。
First and foremost is, roughly, that thou shalt not think about ordinary objects presently whizzing through ordinary space.
首先也是最重要的是，粗略地说，你不应该想到目前在普通空间中呼啸而过的普通物体。
Rather, quantum mechanics predicts — in exquisite detail — all the possible ways that an object might turn out to be in the future.
相反，量子力学以极其详细的方式预测了一个物体在未来可能呈现的所有可能形态。
Exploring those possible futures requires tracking an entirely different mathematical object — an arrow known as a vector, one oriented in an expansive, alien domain.
探索这些可能的未来需要追踪一个完全不同的数学对象——一个被称为向量的箭头，它指向一个广阔的、陌生的领域。
These arrows aren’t pointing at locations.
这些箭头没有指向位置。
“It’s a much more abstract space than that,” said Lucien Hardy , a physicist at the Perimeter Institute for Theoretical Physics in Waterloo, Canada.
“这是一个比这更抽象的空间，”加拿大滑铁卢周边理论物理研究所的物理学家卢西安·哈迪 (Lucien Hardy) 说。
They’re “really pointing in a direction in a possibility space.”
它们“确实在可能性空间中指明了方向”。
This possibility space is called Hilbert space, and it acts as the primary arena for quantum physics.
这个可能性空间称为希尔伯特空间，它是量子物理学的主要舞台。
The early quantum pioneers didn’t realize — at first — that the arcane math that strikingly captured the conduct of atoms had left the real world behind.
早期的量子先驱们一开始并没有意识到，惊人地捕捉原子行为的神秘数学已经把现实世界抛在了后面。
It took a visionary mathematical physicist, John von Neumann, to recognize and define the quantum world as a Hilbert space.
一位富有远见的数学物理学家约翰·冯·诺依曼将量子世界认识并定义为希尔伯特空间。
Once he did, exploring the ins and outs of Hilbert space would lead physicists to a deeper, more unified understanding of quantum physics.
一旦他这样做了，探索希尔伯特空间的来龙去脉将引导物理学家对量子物理学有更深入、更统一的理解。
Here’s how von Neumann’s first commandment of quantum physics came to be, and how to understand it.
以下是冯·诺依曼量子物理学的第一条戒律是如何产生的，以及如何理解它。
Von Neumann’s commandments, or axioms, were his way of making sense of the two distinct forms of quantum mechanics developed back-to-back in the 1920s.
冯·诺依曼的戒律或公理是他理解 20 世纪 20 年代连续发展的两种不同形式的量子力学的方式。
First came Werner Heisenberg’s “matrix mechanics” in 1925.
1925 年，维尔纳·海森堡 (Werner Heisenberg) 首次提出了“矩阵力学”。
It used inscrutable tables and, in later formulations, interminable towers of numbers to calculate the odds that an electron circling an atom would jump to a higher or lower orbit.
它使用难以理解的表格，以及后来的公式中无休止的数字塔来计算绕原子运行的电子跳跃到更高或更低轨道的几率。
The next year, Erwin Schrödinger introduced his “wave mechanics.” It used waves to track, for instance, the probability of a particle being found at a certain location in space.
第二年，埃尔文·薛定谔介绍了他的“波动力学”。例如，它使用波来跟踪在空间中某个位置发现粒子的概率。
While the pictures evoked by these two physicists looked completely distinct, they yielded identical predictions.
虽然这两位物理学家所描绘的图像看起来完全不同，但他们得出了相同的预测。
Heisenberg and Schrödinger had come up with two radically different incarnations of one theory.
海森堡和薛定谔提出了一种理论的两种截然不同的体现。
But what was that theory?
但那个理论是什么？
The question fascinated David Hilbert, a renowned mathematician who had devoted much of his life to rebuilding physics on a sturdy foundation of crisp axioms.
这个问题引起了著名数学家戴维·希尔伯特的兴趣，他一生的大部分时间都致力于在清晰的公理的坚实基础上重建物理学。
He got von Neumann thinking about the problem in the mid-1920s.
他让冯·诺依曼在 20 世纪 20 年代中期开始思考这个问题。
In 1927 the 23-year-old prodigy — building on insights from Paul Dirac — solved it in a single-author trilogy of papers .
1927 年，这位 23 岁的神童在保罗·狄拉克 (Paul Dirac) 的见解的基础上，通过单作者论文三部曲解决了这个问题。
The mathematician David Hilbert sought a mathematical structure that would unify the different forms of quantum mechanics.
数学家大卫·希尔伯特寻求一种能够统一不同形式的量子力学的数学结构。
“The ideas specifically were von Neumann’s, but the inspiration — why do you axiomatize and what for — this is something that he took from Hilbert,” said Leo Corry , a historian of mathematics.
“这些想法具体来说是冯·诺依曼的，但灵感——为什么公理化以及为什么要公理化——这是他从希尔伯特那里得到的，”数学历史学家利奥·科里说。
These papers laid out the rules for quantum mechanics, carefully defining the theory’s central objects and how they behaved.
这些论文奠定了量子力学的规则，仔细定义了该理论的中心对象及其行为方式。
Von Neumann showed that Heisenberg’s towers and Schrödinger’s waves were reflections of the same entity, just as 0.5 and ½ indicate the same point on the number line.
冯·诺依曼表明，海森堡的塔和薛定谔的波是同一实体的反射，就像 0.5 和 1/2 表示数轴上的同一点一样。
They both represented the main character in quantum mechanics: the quantum state.
它们都代表了量子力学的主要特征：量子态。
Everything has a state.
一切都有一个状态。
A coin can read heads or tails.
硬币可以读出正面或反面。
A grandfather clock’s bob can take on any number of positions as it swings.
老爷钟的摆锤在摆动时可以采取任意多个位置。
Most of physics amounts to capturing an object’s state and predicting how it will change.
大多数物理学都是捕捉物体的状态并预测它会如何变化。
Von Neumann’s quantum rules complicate the notion of a state.
冯·诺依曼的量子规则使状态的概念变得复杂。
Before you observe a quantum object, it does not have a fixed set of properties, such as a specific position.
在观察量子物体之前，它没有一组固定的属性，例如特定位置。
Instead it has a combination of possible properties unique to quantum mechanics — a “quantum superposition.” A superposition combines, for instance, all possible places the particle might end up being.
相反，它具有量子力学特有的可能属性的组合——“量子叠加”。例如，叠加结合了粒子最终可能出现的所有可能位置。
Those possibilities can be precise and informative; perhaps there is a 99% chance you’ll find your particle to your left and a 1% chance you’ll find it to your right.
这些可能性可以是精确且信息丰富的；也许有 99% 的机会你会在你的左边找到你的粒子，而有 1% 的机会你会在你的右边找到它。
You know how to place your bets, but you can’t know for sure if you’ve won until you check.
您知道如何下注，但在检查之前您无法确定是否赢了。
Von Neumann rendered the quantum state as a mathematical arrow called a vector.
冯·诺依曼将量子态呈现为称为矢量的数学箭头。
This arrow points in some direction through a space capturing all the possible futures of any quantum object — a Hilbert space.
这个箭头指向某个方向，穿过一个捕捉任何量子物体所有可能的未来的空间——希尔伯特空间。
Imagine a quantum traffic light with three possible states — red, yellow, or green.
想象一个具有三种可能状态的量子交通灯——红色、黄色或绿色。
Its arrow exists in a three-dimensional Hilbert space, where the three axes represent the three possible future colors.
它的箭头存在于三维希尔伯特空间中，其中三个轴代表三种可能的未来颜色。
Until the moment the light is observed, it doesn’t have a color, but rather a mixture of possible colors.
在观察光的那一刻之前，它并没有颜色，而是可能颜色的混合。
So its arrow points into the space’s central region.
所以它的箭头指向空间的中心区域。
The more closely the arrow aligns with, say, the red axis, the more likely the light is to shine red.
箭头与红轴对齐得越紧密，光线就越有可能发出红色光。
The state of any object, from an electron to a galaxy, can be captured by such a vector, pointing in some direction through such a Hilbert space.
任何物体的状态，从电子到星系，都可以通过这样的矢量来捕获，通过这样的希尔伯特空间指向某个方向。
This is von Neumann’s first rule of quantum mechanics.
这是冯·诺依曼的量子力学第一条规则。
An arrow moves through Hilbert space in one of two ways.
箭头以两种方式之一穿过希尔伯特空间。
Von Neumann’s other commandments specify how.
冯·诺依曼的其他戒律具体说明了如何进行。
The first possibility corresponds to what happens before an observation.
第一种可能性对应于观察之前发生的情况。
As the world influences the object, changing its state, the arrow turns smoothly through Hilbert space.
当世界影响物体并改变其状态时，箭头在希尔伯特空间中平稳转动。
It might get closer to the green axis, which would make our traffic light more likely to be measured as green, or to red or yellow.
它可能会更接近绿轴，这将使我们的交通信号灯更有可能被测量为绿色、红色或黄色。
The point is that all this happens smoothly and predictably.
关键是这一切都顺利且可预测地发生。
Then, if you actually observe the system, the vector will instantly and randomly snap onto either the red, yellow, or green axis.
然后，如果您实际观察该系统，矢量将立即随机地捕捉到红色、黄色或绿色轴上。
The more aligned it is with one axis, the more likely it is to snap to that axis instead of the others, but its fate is ultimately unpredictable.
它与一个轴越对齐，它就越有可能捕捉到该轴而不是其他轴，但它的命运最终是不可预测的。
Let’s say it goes green.
假设它变绿了。
You’ll observe a green light, and there is now a 100% chance that it will still be green in subsequent measurements, because the arrow is fully aligned with the green axis.
您将观察到绿光，并且现在有 100% 的机会在后续测量中它仍然是绿色的，因为箭头与绿轴完全对齐。
The quantum superposition is no more.
量子叠加不再存在。
The more possible futures an object has, the bigger its Hilbert space.
一个物体的未来可能越多，它的希尔伯特空间就越大。
A coinlike particle with two possible futures is a “qubit,” the computational building block of quantum computers.
具有两种可能未来的硬币状粒子是“量子位”，是量子计算机的计算构建块。
It has a two-dimensional Hilbert space.
它有一个二维希尔伯特空间。
Our three-color traffic light has a three-dimensional Hilbert space.
我们的三色交通灯具有三维希尔伯特空间。
But that’s just the beginning.
但这只是开始。
A freely floating particle could be found in any location in the universe, so its Hilbert space must span an infinite number of dimensions.
自由漂浮的粒子可以在宇宙的任何位置找到，因此它的希尔伯特空间必须跨越无限多个维度。
This size — whether it’s two dimensions or an infinite number — is the only fundamental feature of a Hilbert space, according to von Neumann’s rules.
根据冯·诺依曼的规则，这个大小——无论是二维还是无限数——是希尔伯特空间的唯一基本特征。
The axes are arbitrary and imagined by us; they aren’t intrinsic to the space.
轴是任意的，是我们想象的；它们不是空间固有的。
Consider an electron.
考虑一个电子。
It has one state, one arrow, pointing in a vast Hilbert space.
它有一种状态，一根箭头，指向广阔的希尔伯特空间。
Its Hilbert space spans all possible measurements — energy, position, momentum, etc. If you are curious where the electron might be, you can mark the space with the axes that represent possible positions.
它的希尔伯特空间跨越了所有可能的测量——能量、位置、动量等。如果您好奇电子可能在哪里，您可以用代表可能位置的轴来标记该空间。
If you are wondering where the particle might be going, you apply a different set of axes, those representing possible momenta.
如果您想知道粒子可能会去哪里，您可以应用一组不同的轴，这些轴代表可能的动量。
No matter which measurement you intend to make, the underlying Hilbert space remains the same.
无论您打算进行哪种测量，底层的希尔伯特空间都保持不变。
This freedom to carve up Hilbert space as we see fit is what allowed Heisenberg and Schrödinger to come up with two distinct versions of the same theory.
这种按照我们认为合适的方式划分希尔伯特空间的自由使得海森堡和薛定谔能够提出同一理论的两个不同版本。
Heisenberg’s picture essentially put in axes and let them rotate around the vector, while Schrödinger’s picture did the opposite: It put in a fixed set of axes and let the vector rotate relative to them.
海森堡的图本质上放置了轴，让它们绕矢量旋转，而薛定谔的图则相反：它放置了一组固定的轴，让矢量相对于它们旋转。
They were two completely different mathematical perspectives on the same arrows, in the same Hilbert spaces.
它们是在相同的希尔伯特空间中对相同箭头的两种完全不同的数学视角。
John von Neumann developed an underlying structure for quantum mechanics that involves arrows moving inside an abstract Hilbert space.
约翰·冯·诺依曼开发了量子力学的基础结构，其中涉及在抽象希尔伯特空间内移动的箭头。
In the first of his 1927 papers, von Neumann laid out two mathematical criteria that defined such a space.
在 1927 年的第一篇论文中，冯·诺依曼提出了定义这样一个空间的两个数学标准。
First, it had to be “complete.” It couldn’t be missing any regions or points.
首先，它必须是“完整的”。它不能缺少任何区域或点。
And second, you had to be able to calculate the alignment between a state and an axis, which you can visualize by imagining a light shining straight down onto an arrow so it casts a shadow on an axis.
其次，您必须能够计算状态和轴之间的对齐方式，您可以通过想象一束光直接照射到箭头上，使其在轴上投射阴影来可视化。
(The longer the shadow, the more aligned with that axis the arrow is.) The space had to allow for this operation, known as an inner product.
（阴影越长，箭头与该轴越对齐。）空间必须允许这种操作，称为内积。
Any space with these two features, no matter its size or origin, was a Hilbert space.
任何具有这两个特征的空间，无论其大小或起源，都是希尔伯特空间。
“This was a major step in creating what we call Hilbert space quantum mechanics,” said Miklós Rédei , a philosopher of physics at the London School of Economics.
伦敦经济学院物理学哲学家米克洛什·雷代 (Miklós Rédei) 表示：“这是创建我们所说的希尔伯特空间量子力学的重要一步。”
“It’s a beautiful example of how mathematical generalization or abstraction takes place.”
“这是数学概括或抽象如何发生的一个很好的例子。”
Von Neumann referred to these abstract spaces as Hilbert spaces because his mentor Hilbert had been the first mathematician to explore specific spaces with infinite dimensions in the early 1900s.
冯·诺依曼将这些抽象空间称为希尔伯特空间，因为他的导师希尔伯特是 1900 年代初期第一位探索无限维度特定空间的数学家。
Hilbert’s work had relied on those spaces being complete and having an inner product, but he didn’t think of them as examples of a more general class of spaces until his protégé grouped them together.
希尔伯特的工作依赖于这些空间的完整性和内积，但他并不认为它们是更一般的空间类别的例子，直到他的门徒将它们组合在一起。
The older mathematician may have been surprised to find his name attached to this new mathematical structure.
这位年长的数学家可能会惊讶地发现他的名字与这个新的数学结构联系在一起。
“Dr. von Neumann, I am really curious to know what these Hilbert spaces are, after all,” Hilbert reportedly asked during a 1929 lecture.
据报道，希尔伯特在 1929 年的一次演讲中问道：“冯·诺依曼博士，我真的很想知道这些希尔伯特空间到底是什么。”
A century after the birth of quantum mechanics, the theory has left physicists in an awkward position.
量子力学诞生一个世纪后，该理论让物理学家陷入了尴尬的境地。
We live in a world where objects change position as they move through three dimensions of physical space.
我们生活在一个物体在物理空间的三个维度中移动时改变位置的世界。
But our most fundamental theory takes place somewhere else, in von Neumann’s vast realm of possibilities.
但我们最基本的理论发生在其他地方，即冯·诺依曼广阔的可能性领域。
What does that imply about the reality of our world, or that of Hilbert space?
这对我们的世界或希尔伯特空间的现实意味着什么？
In quantum physics, Hilbert spaces use complex numbers, which involve the imaginary number i , and negative regions.
在量子物理学中，希尔伯特空间使用复数，其中涉及虚数 i 和负区域。
But von Neumann’s fourth commandment guarantees that the odds of any possibility will always come out as a real, positive number.
但冯·诺依曼的第四条戒律保证了任何可能性的可能性总是以真实的正数出现。
To Sean Carroll , a philosopher and physicist at Johns Hopkins University, the message is clear.
对于约翰·霍普金斯大学的哲学家和物理学家肖恩·卡罗尔来说，这个信息很明确。
If quantum mechanics is the fundamental theory of nature, then Hilbert space should be considered the fundamental theater of reality, he argued in a 2022 paper .
他在 2022 年的一篇论文中指出，如果量子力学是自然的基本理论，那么希尔伯特空间就应该被视为现实的基本剧场。
One of his lines of research seeks to distill our familiar world from the disorienting Hilbert space that encapsulates all the ways the universe could possibly be.
他的研究方向之一是从令人迷惑的希尔伯特空间中提炼出我们熟悉的世界，希尔伯特空间概括了宇宙可能存在的所有方式。
Other physicists take a more pragmatic stance.
其他物理学家则采取更为务实的立场。
Jonathan Sorce , a physicist at Princeton University, says that Hilbert space is a handy mathematical construction that is remarkably useful for describing many quantum systems — but not all of them.
普林斯顿大学物理学家乔纳森·索尔斯表示，希尔伯特空间是一种方便的数学结构，对于描述许多量子系统非常有用，但不是所有量子系统。
He belongs to a community of researchers searching for a mathematical construction that can describe the fabric of space and time as a quantum object.
他属于一个研究人员群体，正在寻找一种数学结构，可以将空间和时间的结构描述为量子对象。
Such a theory is a prerequisite for answering big questions such as what goes on at the heart of a black hole.
这样的理论是回答诸如黑洞中心发生了什么等重大问题的先决条件。
Physicists asking these questions have recently focused on an even more abstract space that seems especially well suited for their purposes.
提出这些问题的物理学家最近把注意力集中在一个更抽象的空间上，这个空间似乎特别适合他们的目的。
This kind of space is made up of the things you could do to a Hilbert space, such as slicing it up in different ways or rotating one slice into another.
这种空间由您可以对希尔伯特空间执行的操作组成，例如以不同的方式将其分割或将一个切片旋转到另一个切片中。
In this arena, they have found that black holes seem a bit less mysterious .
在这个舞台上，他们发现黑洞似乎不那么神秘了。
This sort of über-space is known today as a von Neumann algebra.
这种超级空间今天被称为冯·诺依曼代数。
Von Neumann himself helped develop it as a potential remedy for some logical inconsistencies with Hilbert space that troubled him.
冯·诺依曼本人帮助开发了它，作为解决困扰他的希尔伯特空间的一些逻辑不一致的潜在补救措施。
“I would like to make a confession which may seem immoral: I do not believe in Hilbert space anymore,” he wrote in a 1935 letter while exploring the virtues of algebras.
“我想做一个看似不道德的坦白：我不再相信希尔伯特空间，”他在 1935 年探索代数优点时的一封信中写道。
Sorce, for his part, doesn’t share von Neumann’s desire for one space to rule them all.
就索斯而言，他并不同意冯·诺依曼希望用一个空间来统治所有人的愿望。
He’s content to use whichever mathematical construction best suits the quantum object he’s studying.
他乐意使用最适合他正在研究的量子物体的数学构造。
Often it’s a Hilbert space.
通常它是希尔伯特空间。
Sometimes it’s a von Neumann algebra.
有时它是冯诺依曼代数。
And occasionally it might even be one of the many other spaces mathematicians have cooked up over the last century.
有时它甚至可能是数学家在上个世纪创造的许多其他空间之一。
“There’s a whole zoo of these things,” he said.
“这些东西就像一个动物园，”他说。
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辱骂、亵渎、自我推销、误导、语无伦次或偏离主题的评论将被拒绝。
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