物理学家将虚数从量子力学中剔除
内容总结：
量子力学百年争议迎来突破：复数并非描述量子世界所必需
量子力学自一个世纪前诞生以来，其核心方程中一直包含虚数单位i（即-1的平方根）。这一数学工具虽被物理学家视为“虚构”，却在描述原子和粒子行为时展现出惊人准确性。薛定谔在1926年提出著名波动方程时就曾坦言其中存在“粗糙之处”，希望未来能发展出完全基于实数的理论版本。
2021年，学界曾通过精密实验宣称“量子理论必须使用复数”，但今年多项研究彻底颠覆了这一结论。德国理论物理学家团队于3月率先提出与标准量子理论完全等效的实数表述，法国两个研究组随后独立构建出另一套实数理论框架。9月，谷歌量子计算专家更从算法角度证明：通过消除T逻辑门，量子计算同样无需依赖复数。
尽管新理论在数学表达上避免了虚数，却仍保留着复数运算的典型特征。正如威廉姆斯学院量子信息专家比尔·伍特斯所言：“即便将量子理论转化为实数表述，复数算术的印记依然清晰可见。”中国科学技术大学参与相关实验的陆朝阳教授也指出，虽然理论上可行，但基于复数的量子理论“在简洁性与数学优雅性上仍具有无可替代的优势”。
这场持续百年的数学工具之争，不仅关乎理论表达的简繁，更触及对物理世界本质的理解。正如罗格斯大学科学哲学家吉尔·诺斯所说：“数学表述形式确实影响着我们对物理世界本质的推断。”目前学界普遍认为，虽然复数并非量子力学的数学必需品，但其在描述量子纠缠、粒子自旋等特性时展现的天然适配性，仍值得深入探究。
中文翻译：
物理学家从量子力学中剔除了虚数
引言
一个世纪前，原子与基本粒子的奇异行为促使物理学家构建了全新的自然理论。这套被称为量子力学的理论甫一问世便大获成功，通过精确计算氢原子的光辐射与吸收证明了自身价值。然而其中存在一个隐患——量子力学的核心方程中出现了虚数i（即-1的平方根）。
物理学家深知i是数学虚构的概念。质量、动量这类真实物理量经平方运算后永远不会出现负值。但这个满足i²=-1的虚幻数字，却仿佛扎根于量子世界的核心。
埃尔温·薛定谔在推导出这个充满虚数的方程（本质上即量子实体的运动定律）后，曾期望它能被完全基于实数的版本取代（他在1926年写道："当前方程形式无疑存在某种粗糙之处"）。尽管薛定谔心存芥蒂，虚数i依然留存下来，新一代物理学家沿用他的方程时也并未过多质疑。
直到2021年，虚数在量子理论中的角色引发了新一轮关注。某研究团队提出可通过实验验证i究竟是量子理论不可或缺的要素，抑或仅是数学工具。两支团队随即完成精密实验，声称获得了量子理论需要虚数的明确证据。
但今年的一系列论文推翻了这一结论。
三月，德国理论学家团队反驳了2021年的研究，提出与标准版本完全等效的实数化量子理论。法国两位理论学家随后也构建出各自的实数化量子理论体系。九月，另有研究者从量子计算视角切入，得出相同结论：描述量子现实根本无需虚数i。
尽管实数化理论避免直接使用i，却仍保留着其独特算法的印记。这令部分学者怀疑：量子力学（乃至现实本身）的虚数特性是否真正被消除？
"数学表述确实指引着我们对物理世界本质的推断。"罗格斯大学物理哲学家吉尔·诺斯指出。
虚幻的数值
1637年，勒内·笛卡尔居住于正值郁金香狂热巅峰的阿姆斯特丹（当时荷兰人对花朵的痴迷导致郁金香球茎出现荒谬估值），他正在钻研那些解值看似不可能的方程。以x³-6x²+13x-10=0为例，笛卡尔写道其解"并非总是实数；有时仅是虚数……有时不存在与设想对应的量"。该方程的三个解为2、2-i和2+i。后两者以a+ib形式同时包含实部与虚部，后来被称作复数。
笛卡尔对此嗤之以鼻，但复数后来因其在几何学、光学及信号分析等领域的实用性而被广泛采纳。
薛定谔在量子理论中不情愿地承认了复数的便捷性。他的方程主导着波函数的演化，这个数学实体描述物体的可能量子状态（这些状态会像波一样发生相消与相长干涉）。尽管对量子系统的实际测量总是返回实数值，薛定谔的波函数却采用复数值。"量子理论确实是首个让复数看似居于理论核心的物理理论。"威廉姆斯学院量子信息理论学家比尔·伍特斯表示。
将a+ib这类复数表示为平面上的点是一种常见方式：a对应x轴坐标（可视为实数轴），b对应虚数y轴坐标。每个复数都是始于原点指向复坐标(a,b)的矢量。这些复矢量遵循特殊的复数运算法则：例如乘以i会使矢量旋转90度。
这些特性使其天然契合波函数的量子态——同样遵循奇特组合规则的矢量。
物理学家曾多次尝试用实数定义等效矢量。1960年，瑞士物理学家恩斯特·施图克伯格发展出实数化量子力学，通过若干技巧将波函数从复值空间映射到实值空间，使实数能模拟绕虚数轴的旋转。但在复数值理论简洁之处，实数值理论却显得冗繁。描述两个粒子的波函数涉及四个复数；将施图克伯格方案推广到双粒子体系时，所需实数增至16个。
尽管实数化量子理论如此笨重，2008年与2009年有两个研究组证明可用该理论重现贝尔测试的标准结果——这是检验量子理论特性的关键探针。"对于许多情况，实数值理论确实能解决问题。"伍特斯说。但实数值理论总能给出相同结果吗？
关键假设
2021年，日内瓦大学物理学家尼古拉·吉桑等人意识到，可通过增加贝尔测试复杂度来检验实数值理论的局限。
传统贝尔测试需要制备一对"纠缠"粒子：其可能状态相互关联，例如偏振相关的光子对。粒子分离后分别送往代号为爱丽丝与鲍勃的两位参与者，由他们测量偏振态并比对结果。
吉桑团队则设计了一种定制版贝尔测试，包含两个独立纠缠粒子源和三位参与者：爱丽丝、鲍勃与查理。经计算发现，实数量子理论中纠缠粒子偏振关联度存在上限，而复数值理论的上限更高。这不再关乎计算便捷性或哲学思辨：存在可证伪实数量子力学的实验方案。
不久后，合肥中国科学技术大学团队实施该方案，发现观测到的纠缠光子关联度远超实数值理论极限。复数对于描述这些量子态似乎不可或缺。
但统计学上的压倒性结果并未平息质疑。
"复数不过是两个实数加上特定运算规则。"德国航空航天中心物理学家、新近德国论文合著者迈克尔·埃平表示，"为何不能仅用实数描述量子力学？"
里昂高等师范学院米沙·伍兹与巴黎萨克雷大学蒂莫西·霍夫勒蒙（新近法国论文合著者）同样存疑。在2021年的论文中，吉桑团队对"张量积"（一种将描述爱丽丝与查理粒子的复矢量整合为纠缠态的数学运算）做了关键假设。他们假定实数值量子理论会采用相同形式的张量积来组合态。
但法德团队指出，那种张量积形式不适用于实数值理论。类比而言，在平直空间中直角三角形始终满足a²+b²=c²，但该法则对弯曲空间（如球面）上的三角形并不成立。两支团队采纳的新近观点是：标准张量积属于更广义矢量组合规则的特例。他们开发出不同的组合规则，构建出与复数量子理论预测完全一致的实数值量子理论。
量子计算领域的新进展也展示了避开复数的方法。量子计算机通过"逻辑门"操纵量子比特。常见的T门能使代表量子比特状态的矢量在复平面旋转。九月，谷歌量子人工智能团队的量子计算专家克雷格·吉德尼找到从任何量子算法中剔除T门的方法，数值验证了量子计算无需复数。
自然之选
实数值量子理论的可行性引发深刻追问。其中最核心的是：为何它如此复杂？自量子力学诞生之初该问题便如影随形；薛定谔曾尝试构建实数值波方程，但最终转向复数值版本，因为正如他在笔记中所言"计算目的而言极为简便"。
如今看来量子理论虽不显式需要i，但薛定谔所见的简洁性或许仍蕴含自然之理。"采用自然张量积的复数量子理论依然远更凝练、优雅且数学直观。"中国科学技术大学实验物理学家陆朝阳评价，他参与了执行吉桑团队定制贝尔测试的实验。
"即使将量子理论转化为实数形式，你依然能看见复数运算的印记。"伍特斯指出。
就连那些将理论从复数中解放出来的学者也承认，复数仍是更自然的选择。实数值理论虽不包含i，却复制了其旋转矢量的能力。"我们通过实数来模拟复数。"杜塞尔多夫海因里希·海涅大学物理学家、德国论文合著者安东·特鲁谢奇金表示。
物理哲学家诺斯赞同陆朝阳的观点："即便复数非必需，它们确实催生了特别契合量子力学的表述形式。"她的目标是"精准定位促成这种契合的特有量子机制"。粒子自旋这种经典物理中无对应物的量子特性或许是答案之一。
实数值理论中挥之不去的复数本质令研究者审慎；关于i被淘汰的论断可能言之过早。"你可以随意选择表述形式，但它们必须如同复数般进行乘法运算，这是无法回避的。"牛津大学物理学家弗拉特科·韦德拉尔表示。他更期待为量子力学找到更简洁的公理——能令理论家以全新形式彻底重构该理论的直观原理。
"我们至今没有真正替代百年前量子力学表述的方案。"他坦言，"问题在于：为何我们无法超越现有框架？"
英文来源：
Physicists Take the Imaginary Numbers Out of Quantum Mechanics
Introduction
A century ago, the strange behavior of atoms and elementary particles led physicists to formulate a new theory of nature. That theory, quantum mechanics, found immediate success, proving its worth with accurate calculations of hydrogen’s emission and absorption of light. There was, however, a snag. The central equation of quantum mechanics featured the imaginary number i, the square root of −1.
Physicists knew i was a mathematical fiction. Real physical quantities like mass and momentum never yield a negative amount when squared. Yet this unreal number that behaves as i2 = −1 seemed to sit at the heart of the quantum world.
After deriving the i-riddled equation — essentially the law of motion for quantum entities — Erwin Schrödinger expressed the hope that it would be replaced by an entirely real version. (“There is undoubtedly a certain crudeness at the moment” in the equation’s form, he wrote in 1926.) Schrödinger’s distaste notwithstanding, i stuck around, and new generations of physicists took up his equation without much concern.
Then, in 2021, the role of imaginary numbers in quantum theory attracted newfound interest. A team of researchers proposed a way to empirically determine whether i is essential to quantum theory or a mere mathematical convenience. Two teams quickly followed up to perform the intricate experiments and found supposedly unequivocal evidence that quantum theory needs i.
This year, however, a series of papers has overturned that conclusion.
In March, a group of theorists based in Germany rebutted the 2021 studies, putting forward a real-valued version of quantum theory that’s exactly equivalent to the standard version. Two theorists in France followed up with their own formulation of a real-valued quantum theory. And in September, another researcher approached the question from the perspective of quantum computing and arrived at the same answer: i isn’t necessary for describing quantum reality after all.
Although the real-valued theories avoid explicit use of i, they do retain hallmarks of its distinct arithmetic. This leads some to wonder whether the imaginary aspect of quantum mechanics — or even reality itself — is truly vanquished.
“The mathematical formulation does guide what we infer about the nature of the physical world,” said Jill North, a philosopher of physics at Rutgers University.
Impossible Values
Living in Amsterdam in 1637 at the peak of tulip mania (the Dutch frenzy for flowers which led to impossibly valued tulip bulbs), René Descartes grappled with equations whose solutions also seemed to have impossible values. Using x3 − 6x2 + 13x − 10 = 0 as an example, Descartes wrote that its solutions “are not always real; but sometimes only imaginary. … There is sometimes no quantity that corresponds to what one imagines.” The three numbers you can plug in for x are 2, 2 − i and 2 + i. The latter two numbers, each of which has both a real part and an imaginary part in the form a + ib, came to be called complex numbers.
Descartes viewed them with derision, but complex numbers were later adopted for their utility in fields as diverse as geometry, optics and signal analysis.
Schrödinger grudgingly acknowledged their ease of use in quantum theory. His equation governs the evolution of the wave function, an entity representing the possible quantum states of an object. (These states can interfere destructively and constructively like waves.) Schrödinger’s wave function was complex-valued, even though actual measurements of quantum systems always return real values. “Quantum theory really is the first physical theory where the complex numbers seem to be right smack in the middle of the theory,” said Bill Wootters, a quantum information theorist at Williams College.
One way to represent a complex number like a + ib is as a point on a plane, where a is the position on the x-axis (which can be thought of as the real number line) and b is the position on an imaginary y-axis. Each complex number is an arrow, called a vector, pointing from the origin to the complex coordinate (a, b). These complex vectors obey the unusual math of complex numbers: Multiplying by i, for example, rotates the vector 90 degrees.
These properties made them a natural fit for the quantum states of the wave function — also vectors obeying odd combination rules.
Physicists tried now and again to define equivalent vectors with real numbers. In 1960 the Swiss physicist Ernst Stueckelberg developed a real-valued quantum mechanics that mapped the wave function from a complex-valued space to a real one, using a few tricks to get real numbers to mimic the rotations around an imaginary axis. But where complex-valued theory was compact, the real-valued theory was cumbersome. The wave function for two particles involves four complex numbers; extending Stueckelberg’s formulation to two particles increases the description to 16 real numbers.
The clunkiness of real-valued quantum theories notwithstanding, in 2008 and 2009, two groups showed it was possible to use these theories to re-create the standard results of the Bell test — a crucial probe of quantum theory’s properties. “For a lot of things, you actually can get away with the real theory,” Wootters said. But would the real-valued theory always produce the same results?
Key Assumptions
In 2021, a group of researchers including Nicolas Gisin, a physicist at the University of Geneva, realized that they could test the limits of real-valued theories by making a Bell test more complicated.
Canonically, Bell tests involve the creation of a pair of “entangled” particles: particles whose possible states are linked, such as photons with correlated polarizations. The particles are separated and sent to two participants nicknamed Alice and Bob, who measure their polarizations and compare notes.
Gisin’s team instead considered a bespoke Bell test with two separate sources of entangled particles and three participants: Alice, Bob and Charlie. Running the numbers, they found that there was a ceiling on how correlated the polarizations of the entangled particles could be for a real-valued quantum theory, and a different, higher ceiling for a complex-valued quantum theory. This was no longer a matter of calculational ease or philosophy: An empirical test existed that could rule out real-valued quantum mechanics.
Soon after, a group at the University of Science and Technology of China (USTC) in Hefei ran the protocol and found that the observed correlations between entangled photons far exceeded the limit for the real-valued theory. Complex numbers seemed essential for describing these quantum states.
But the statistically overwhelming result didn’t quell the questions.
“Complex numbers are just two real numbers with some calculation rules,” said Michael Epping, a physicist at the German Aerospace Center and a co-author on the new German paper. “Why shouldn’t you be able to describe quantum mechanics just using real numbers?”
Mischa Woods of the École Normale Supérieure in Lyon and Timothée Hoffreumon of Paris-Saclay University, co-authors on the new French paper, were also dubious. In the 2021 paper, Gisin and his colleagues made a critical assumption about the “tensor product,” a mathematical operation that wrangles the complex vectors describing Alice’s particle and Charlie’s particle into one entangled state. Gisin and his co-authors assumed that a real-valued version of quantum theory would use the same mathematical formulation of the tensor product to combine states.
But the French and German teams argue that that form of tensor product is the wrong rule for a real-valued theory. By way of analogy, in flat space, the hypotenuse of a right triangle is always a2 + b2 = c2. But that rule doesn’t hold for a triangle in curved space, like one that’s stamped on the surface of a sphere. A recent argument, adopted by the two teams, is that that the standard tensor product is a specific case of a more general class of vector-combination rules. They developed different combination rules to create real-valued quantum theories that give exactly the same predictions as a complex quantum theory.
A new development in quantum computing also shows how to avoid complex numbers. Quantum computers use “logic gates” to manipulate quantum bits. One common logic gate, called a T gate, rotates the vector representing the quantum bit’s state around the complex plane. In September, Craig Gidney, a quantum computing expert at Google Quantum AI, found a way to eliminate T gates from any quantum algorithm — numerically proving that quantum computing doesn’t require complex numbers.
What Comes Naturally
The feasibility of real-valued quantum theory raises provocative questions. Foremost among them: Why is it so much more complicated? The question has been with us since the birth of quantum mechanics; Schrödinger attempted to work with a real-valued wave equation but turned to a complex one because it was “extraordinarily much simpler for computational purposes,” as he described it in his notes.
Today it seems that quantum theory does not explicitly need i, but there may still be something natural about the simplicity Schrödinger found. “Complex quantum theory, with its natural tensor product, remains far more concise, elegant and mathematically straightforward,” said Chao-Yang Lu, an experimental physicist at USTC who was part of the team that carried out the bespoke Bell test proposed by Gisin’s team.
“Even when you translate quantum theory into real numbers, you still see the hallmark of complex-number arithmetic,” Wootters said.
Even those who emancipated the theory from complex numbers admit that the latter are a natural fit. The real-valued theories do not contain i, but they copy its ability to rotate vectors. We “simulate complex numbers by means of real numbers,” Anton Trushechkin, a physicist at Heinrich Heine University Düsseldorf and co-author on the German paper.
North, the philosopher of physics, agrees with Lu. “Even if complex numbers aren’t truly necessary, they do give rise to a formulation that seems particularly well suited to quantum mechanics,” she said. Her goal is to “pinpoint something peculiarly quantum mechanical” that contributes to that good fit. One possibility might be spin, a property of quantum particles that has no classical counterpart.
The lingering essence of complex numbers in the real-valued theories gives some researchers pause; reports of i’s demise may be somewhat exaggerated. “You can write them down whichever way you like, but it’s unavoidable that they have to multiply exactly as though they were complex numbers,” said Vlatko Vedral, a physicist at the University of Oxford. His preference would be to find simpler axioms for quantum mechanics — intuitive principles that would let theorists re-derive the theory in a new form altogether.
“We really don’t have a single alternative to how quantum mechanics was already done 100 years ago,” he said. “And the question is, why? Why can’t we go beyond this?”