物理与天文2021nature.com编辑精选
基于实数的量子理论可被实验证伪
Quantum theory based on real numbers can be experimentally falsified
专业 · 约 90 分钟读完前置知识：量子力学基础、贝尔不等式、希尔伯特空间
导读
这篇论文通过实验检验“量子力学是否必须使用复数”这一基础问题：作者构造了类似贝尔不等式的判据，证明基于实数的量子理论无法复现某些纠缠实验的预测，而标准复数量子理论可以。实验结果支持复数表述，从而在实验上排除了实数量子理论。
HN 关注点：量子力学基础问题——复数究竟是数学工具还是物理必需，讨论焦点集中在实数化改写的可能性、相位与射影希尔伯特空间等技术细节。
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这几条评论里：质疑 2 补充 2
补充复数只是表示方式，真正不可回避的是其算术结构，实数化改写无法消除它
注意，把实部和虚部拆成两个实数、用 90 度旋转矩阵代替 i，就能只用实数写出量子力学。量子力学里我们本来就到处用矩阵和向量——Dirac 当年为了对某个算符开平方引入 4×4 矩阵，结果预言了反物质的存在。复数的特殊之处在于，对最底层的量（态振幅、无自旋非相对论粒子的波函数）我们可以不用矩阵和向量。这篇文章说的是：无论你用什么表示方式，在最底层你都无法去掉这些性质。
doubleunplussed查看英文原文
Note that it is trivial to split the real and imaginary parts into two separate real-numbers and write quantum mechanics that way with only real numbers. Instead of i you get a 90 degree rotation matrix, instead of individual numbers you get a 2-element vector, etc. Lacking "numbers" with the right arithmetic properties for other things in quantum mechanics, we indeed use matrices and vectors for other stuff all the time. Dirac figured he needed some 4×4 matrices in order to be able to take the square root of some operator at some point, which is how his equation predicted the existence of antimatter (because it implied the wavefunction had to be a vector with more components - some of the other components turned out to be antimatter). But complex numbers happen to have the right properties that at least we can do without matrices and vectors for the lowest-level quantities in quantum mechanics: the state amplitudes or the values of wavefunctions of spinless, non-relativistic particles. This article is saying that you can't do away with these properties at the lowest level of quantum mechanics, whether or not you actually use complex numbers to represent them.
补充补充资料：从公理化角度可自然导出复数量子力学，且与纠缠密切相关
Lucien Hardy 有一篇论文，从几条合理的公理出发“自然地”导出量子理论，并说明这如何蕴含复数（同时排除四元数）。Scott Aaronson 有一篇更通俗幽默的讲解。纠缠已被证明与这一结果密切相关，考虑到纠缠已有大量实验证据支持，这很有意思。附相关论文链接。
magicalhippo查看英文原文
Lucien Hardy wrote a paper[1] showing how one "naturally" ends up with a quantum theory by demanding a few reasonable axioms. The paper also goes into how this implies complex numbers (and rules out quaternions). Scott Aaronson has a more accessible (and humorous) article on it here[2]. Entanglement has been shown to be intimately linked to this[3] result, which is interesting given the experimental evidence[4] for entanglement. Not my field, but I found this interesting at least. [1]: https://arxiv.org/abs/quant-ph/0101012 [2]: https://www.scottaaronson.com/democritus/lec9.html [3]: https://arxiv.org/abs/0911.0695 [4]: https://en.wikipedia.org/wiki/Quantum_entanglement#Notable_experimental_results_proving_quantum_entanglement
质疑质疑：论文忽略相位简并/射影希尔伯特空间，可能影响其论证
我惊讶文章没提相位和射影希尔伯特空间的概念。它给出的量子力学表述是众所周知的含糊的：<φ|φ>=1 并不能唯一确定 φ，因为 ψ=exp(iλ)φ 也满足该条件。所以物理系统的状态其实是由相差相位的所有态矢量的等价类（射线）描述的。不知道这是否会影响他们的推理和结论。
rnhmjoj查看英文原文
I'm surprised the article doesn't mention phases or the idea of projective Hilbert spaces. The QM formulation it gives is well known ambiguous: the condition <φ|φ>=1 does not determine φ, because it's also satisfied by any other state ψ=exp(iλ)φ with λ real. So, the state of a physical system is really described by the ray (equivalent class) of all state vectors differing by a phase. I wonder if this has any consequence on their reasoning and conclusions.
质疑指出摘要措辞不当：爱因斯坦并非量子力学奠基人，而是其怀疑者
摘要暗示爱因斯坦是量子力学奠基人，这很奇怪——事实更相反。爱因斯坦对量子力学基础始终深度怀疑，坚信隐变量理论（“上帝不掷骰子”），这一观点直到他去世后才被贝尔不等式的实验证伪。这篇论文描述的是另一组类似贝尔不等式的不等式，但检验的是量子力学是否需要复数，而非是否存在隐变量。也许我吹毛求疵，但科学家通常很斟酌摘要措辞，不知道他们为何这样写。
oofbey查看英文原文
It’s quite strange that the abstract implies Einstein is a founder of QM. More the opposite I think. Einstein remained deeply skeptical of many fundamental aspects of QM - believing in hidden variable theory through his famous statement “God does not play dice with the universe” which was only proven false after his death through experimental measurements of the Bell inequalities. This paper describes another set of inequalities similar to the Bell inequalities, but testing whether QM requires complex numbers or not, instead of whether there are hidden variables that can explain things like superposition. The statement in the abstract I’m referring to is this: > This has puzzled countless physicists, including the fathers of the theory, for whom a real version of quantum theory, in terms of real operators, seemed much more natural.[3] The reference [3] is to a letter by Einstein. Maybe I’m nitpicking words, but scientists tend to spend a lot of time getting the abstracts to their papers right. Any ideas why they would write it this way?
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