约翰·斯图尔特·贝尔John Stewart Bell

约翰·斯图尔特·贝尔的伟大成就是，在20世纪60年代，他能够为量子理论的基础注入新的、令人兴奋的活力，这一主题似乎因三十年前尼尔斯·玻尔-阿尔伯特·爱因斯坦辩论的结果而耗尽，并且在此期间几乎被所有used量子理论的人所忽视。贝尔能够表明，对“实在论”、“决定论”和“定域性”等概念的讨论可以精炼为严格的数学陈述，“贝尔不等式”，这能够进行实验检验。在过去三十年中，此类检验一直在进行，其能力和精度稳步提高。

事实上，几乎完全由于贝尔的开创性努力，量子基础这一主题，无论是实验方面还是理论和概念方面，已成为许多国家科学家主要兴趣的焦点，并教给了我们许多具有根本重要性的东西，不仅关于量子理论，而且关于物理宇宙的本质。

此外，甚至在20世纪90年代中期——贝尔去世几年之后——也几乎无法预料的是，贝尔所研究的概念以及发展其工作的人们所研究的概念，许多已成为量子信息论这一新学科领域的基础，该领域包括量子计算和量子密码学等主题。过去几年里，对量子信息论的关注急剧增加，这一学科似乎必定会成为21世纪科学最重要的增长领域之一。

贝尔的父母双方都曾在爱尔兰北部生活了好几代。他的父亲也叫弗瑞兹·约翰，所以约翰 贝尔在家里一直被称为贝尔。他的母亲安妮鼓励孩子们专注于教育，她认为教育是充实而有尊严生活的关键。然而，在她的四个孩子中——约翰有一个姐姐鲁比和两个弟弟弗洛伦斯·南丁格尔·大卫和罗伯特——只有约翰能够在十四岁之后继续上学。他们家并不富裕，当时还没有普及中等教育，从贝尔家族这样的背景进入大学是极其不寻常的。

贝尔本人对书籍感兴趣，尤其从小就对科学感兴趣。他在最初就读的阿尔斯特维尔大道学校和费恩街学校极为成功，十一岁时轻松通过了升入中等教育的考试。不幸的是，贝尔法斯特一所著名文法学校的学费令人望而却步，但凑够了钱让贝尔进入贝尔法斯特技术高中，在那里，完整的学术课程使他具备大学入学资格，同时还有职业课程。

随后，贝尔在贝尔法斯特女王大学物理系做了一年技术员，该系的高级教职员卡尔·埃梅勒斯教授和罗伯特·斯隆博士给予了极大的帮助，借书给贝尔并允许他旁听一年级的课程。1945年，贝尔得以作为学生进入该系。他的进步极为成功，1948年以实验物理学一等荣誉学位毕业。他又多读了一年，在那一年获得了第二个学位，同样是一等荣誉学位，这次是数学物理学。在数学物理学方面，他的主要老师是彼得·保罗·埃瓦尔德教授，他以X射线晶体学创始人之一而闻名；埃瓦尔德是来自纳粹德国的难民。

贝尔已经在深入思考量子理论，不仅仅是如何使用它，还有它的概念意义。在接受杰里米·伯恩斯坦的采访中，这次采访是在他生命末期进行的，并被引用在伯恩斯坦的书[1]中，贝尔报告说，他对维尔纳·海森堡不确定性或测不准原理的通常表述感到困惑（$\Delta x \Delta p ≥ \hbar$，其中$\Delta x$和$\Delta p$分别是位置和动量中的不确定性或测不准量，取决于一个人的哲学立场，而ℏ是（约化）马克斯·普朗克常数）。

看起来你可以取这个大小，然后位置就确定了，或者取那个大小，然后动量就确定了。听起来好像你可以随意选择。只是慢慢地我才意识到，这不是你希望什么的问题。这实际上是关于什么仪器产生了这种情况的问题。但对我来说，要理解这一点有点费劲。在我可用的书籍和课程中，这并没有很清楚地阐述。我记得为此和我的一个教授，一个叫Sloane的博士争论过。我变得非常激动，或多或少地指责他不诚实。他也变得非常激动，说：‘你太过分了。’

在本科学习结束时，贝尔本想攻读博士学位。他也本想更彻底地研究量子理论的概念基础。然而，经济考虑意味着他不得不暂时放弃量子理论，去找工作，于是在1949年，他加入了位于哈维尔的英国原子研究机构，尽管他很快转到了位于马尔文的加速器设计组。

正是在这里，他遇到了他未来的妻子玛丽·罗斯，她带着来自苏格兰的数学和物理学学位。他们于1954年结婚，婚姻长久而成功。玛丽将在她的职业生涯中一直从事加速器设计；在约翰生命的末期，他回到了加速器设计的问题上，他和玛丽共同写了一些论文。在他的职业生涯中，他从与玛丽的讨论中获益良多，当1987年他的量子理论论文被收集出版[21]时，他包括了以下的话：-

我在此特别再次向Mary Bell致以热烈的感谢。当我再次翻阅这些论文时，我到处都能看到她的身影。

加速器设计当然是一个相对较新的领域，贝尔在马尔文的工作包括追踪带电粒子通过加速器的路径。在计算机出现之前的那些日子里，这需要对电磁学有严格的理解，以及必要的洞察力和判断力，以进行必要的数学简化，使问题在机械计算器上可处理，同时保留物理的本质特征。贝尔的工作非常精湛。

1951年，贝尔获得一年休假，前往伯明翰大学与物理学教授鲁道夫·佩尔斯合作。在伯明翰期间，贝尔做出了极为重要的工作，提出了他对著名的量子场论CPT定理的版本。这一定理表明，在三个算符对物理事件的联合作用下：$P$，即执行反射的宇称算符；$C$，即用反粒子替换粒子的电荷共轭算符；以及$T$，即执行时间反演的算符，结果将是另一个可能的物理事件。
不幸的是，Gerhard Lüders和沃尔夫冈·泡利比贝尔稍早证明了同一定理，他们获得了全部荣誉。

贝尔又增加了一项工作，并于1956年获得博士学位。他还获得了佩尔斯极为宝贵的支持，从伯明翰返回后，他前往哈维尔，加入一个为研究理论基本粒子物理而新成立的小组。他在哈维尔一直待到1960年，但他和Mary逐渐担心哈维尔正从基础工作转向更应用性的物理学领域，于是他们双双搬到了CERN，即日内瓦的欧洲核子研究中心。他们在这里度过了余下的职业生涯。

贝尔在高能物理和量子场论领域发表了约80篇论文。其中一些与CERN的实验物理项目相当密切相关，但大多数属于一般理论领域。

最重要的工作是1969年的工作，导致了量子场论中的Adler-贝尔-Jackiw（ABJ）反常。这源于贝尔和Ronan Jackiw的合作工作，随后由Stephen Adler加以澄清。他们表明，标准的流代数模型包含一个歧义。量子化导致了模型的对称性破缺。这项工作解决了粒子物理学中的一个突出问题；理论似乎预测中性π介子不能衰变为两个光子，但实验上衰变确实发生，正如ABJ所解释的那样。在随后的三十年中，对此类反常的研究在粒子物理学的许多领域变得重要。埃拉斯穆斯·赖因霍尔德 Bertlmann本人曾与贝尔做过重要工作，他写了一本题为Anomalies in Quantum Field Theory [10]的书，而ABJ两位仍在世的成员Adler和Jackiw因他们的工作共同获得了1988年的里雅斯特国际理论物理中心保罗·狄拉克奖章。

虽然粒子物理学和量子场论是贝尔受薪所做的工作，并且他做出了卓越的贡献，但他最大的热爱是量子理论，正是他在这里的工作将使他被人们铭记。正如我们所见，他从本科时代起就关心该理论的基本意义，他的许多重要论证都以那时为基础。

这些概念问题可以用自旋-$\large\frac{1}{2}\normalsize$系统来概述。我们可以说，当态矢量分别为$\alpha_{+}$或$\alpha_{-}$时，$s_{z}$分别等于$\large\frac{1}{2}\normalsize \hbar$和$-\large\frac{1}{2}\normalsize \hbar$，但是，如果把自己限制在埃尔温·薛定谔方程内，$s_{x}$和$s_{y}$根本就没有值。人们所能说的只是，例如，如果对$s_{x}$进行测量，所得结果为$\large\frac{1}{2}\normalsize \hbar$或$-\large\frac{1}{2}\normalsize \hbar$的概率都是$\large\frac{1}{2}\normalsize$。

另一方面，如果初始态矢量具有$c_{+}\alpha_{+}+ c_{-}\alpha_{-}$的一般形式，那么我们只能说，在测量$s_{z}$时，得到$\large\frac{1}{2}\normalsize \hbar$值的概率是$|c_{+}^{2} |$，而得到$-\large\frac{1}{2}\normalsize \hbar$值的概率是$|c_{-}^{2} |$。在任何测量之前，$s_{z}$根本没有值。

这些陈述与我们两个基本观念相矛盾。我们拒绝realism，它告诉我们一个量有一个值，更宏大地说——物理世界具有独立于任何观察者行为的存在。阿尔伯特·爱因斯坦对这种实在论的抛弃特别不安——他坚持observer-free realm的存在。我们也拒绝determinism，即相信如果我们完全知道系统的状态，就能准确预测它将如何行为。在这种情况下，我们知道系统的态矢量，但无法预测测量$s_{z}$的结果。

显然，如果我们允许这样的观点，即埃尔温·薛定谔方程以及波函数或态矢量可能不包含关于系统可获得的全部信息，我们就可以试图恢复实在论和决定论。可能有其他量提供额外信息——hidden variables。作为一个简单的例子，上面的态矢量可能适用于许多系统的系综，但此外每个系统的一个隐变量可能说明$s_{z}$的实际值可能是什么。实在论和决定论都将被恢复；$s_{z}$在任何时候都有一个值，并且，在完全知道系统状态（包括隐变量的值）的情况下，我们可以预测测量$s_{z}$的结果。

一个完整的隐变量理论实际上必须比这更复杂——我们必须记住，我们希望预测的不仅是测量$s_{z}$的结果，还有$s_{x}$和$s_{y}$，以及$s$的任何其他分量。然而，用隐变量补充埃尔温·薛定谔方程的可能性本来似乎很自然会被认真对待。但实际上，尼尔斯·玻尔和维尔纳·海森堡确信人们不应追求实在论。因此，当冯·诺伊曼证明了一个定理，声称严格表明不可能将隐变量添加到量子理论的结构中时，他们很高兴。这在三十多年里被非常普遍地接受。

尼尔斯·玻尔提出了他（也许相当晦涩）的complementarity框架，试图解释为什么人们不应期望同时测量$s_{x}$和$s_{y}$（或$x$和$p$）。这就是他的量子理论Copenhagen interpretation。然而阿尔伯特·爱因斯坦拒绝了这一点，并旨在恢复实在论。物理学家几乎一致赞成尼尔斯·玻尔。

阿尔伯特·爱因斯坦最有力的论据，尽管这在几十年里没有变得非常普遍明显，在于1935年著名的阿尔伯特·爱因斯坦-Podolsky-Rosen（EPR）论证，该论证由阿尔伯特·爱因斯坦在其两位年轻合作者Boris Podolsky和Nathan Rosen的协助下构建。在这里，如通常所做的那样，我们讨论该论证的一个更简单的版本，这是后来由大卫 Bohm想出的。

考虑两个自旋为$\large\frac{1}{2}\normalsize$的粒子；它们由一个自旋为$\large\frac{1}{2}\normalsize$的粒子衰变产生，并从这个衰变点沿相反方向向外运动。组合态矢量可以写成$(\large\frac{1}{√2}\normalsize )(\alpha_{1-}\alpha_{2+} - \alpha_{1-} \alpha_{2+})$，其中粒子1和粒子2的$\alpha_{1}$和$\alpha_{2}$与上面的$\alpha$有关。这个态矢量有一种奇怪的形式。这两个粒子并不是独立地出现在其中；相反，粒子1的任一状态都与粒子2的某一特定状态相关联。这个态矢量被称为entangled。

现在想象测量$s_{1}z$。如果我们得到$+\large\frac{1}{2}\normalsize v$，我们就知道立即测量$s_{2}z$必定得到$-\large\frac{1}{2}\normalsize \hbar$，反之亦然，尽管至少按照哥本哈根解释，在任何测量之前，任一自旋都没有一个分量具有特定值。

这个论证的结果是，三个陈述中至少有一个必须为真：

(1) 粒子必须瞬时交换信息，即超光速；
(2) 存在隐变量，因此实验结果是预先注定的；或
(3) 量子理论在这些相当特殊的实验中并非完全正确。

第一种可能性可被描述为对locality原理的放弃，即信号不能以超光速从一个粒子传递到另一个粒子。这一建议对阿尔伯特·爱因斯坦来说是令人厌恶的。因此，他得出结论，如果量子理论是正确的，那么排除了可能性(3)，则(2)必须为真。用阿尔伯特·爱因斯坦的话说，量子理论不是complete，而是需要用隐变量来补充。

贝尔将自己视为阿尔伯特·爱因斯坦的追随者。他告诉Bernstein [1]：-

我觉得，在这个例子中，阿尔伯特·爱因斯坦在智力上对尼尔斯·玻尔的优越性是巨大的；一个清楚看到需要什么的人与一个蒙昧主义者之间存在着巨大的鸿沟。

贝尔因此支持以隐变量形式的实在论。他对1952年大卫 Bohm创建的一种包含隐变量的量子理论版本感到高兴，这似乎无视了冯·诺伊曼的结果。贝尔写道[21]：-

1952年，我看到了不可能之事被完成。

1964年，贝尔对量子理论做出了自己的重大贡献。首先，他构建了自己关于任意自旋分量测量的隐变量解释。这比Bohm的工作简单得多，因而更难以被忽视。随后，他比Bohm走得更远，非常清晰地证明了冯·诺伊曼论证究竟错在哪里。

冯·诺伊曼不合法地把量子理论变量中的一个结果推广到了他假定的隐变量上，即$A + B$的期望值等于$A$的期望值与$B$的期望值之和。（一个变量的期望值是其可能的实验结果按其发生概率加权的平均值。）一旦认识到这个错误，就很清楚量子理论的隐变量理论是可能的。

贝尔随后证明了隐变量理论必定具有的某些不受欢迎的性质。最重要的是它们必须是非局域的。他通过扩展EPR论证证明了这一点，允许在装置的每一翼测量任意自旋分量，而不仅仅是$s_{z}$。他发现，即使允许隐变量，在某些情况下，一翼得到的结果也必须取决于另一翼测量的是哪个自旋分量；这违反了局域性。阿尔伯特·爱因斯坦会喜欢的那种解决EPR问题的方法，即拒绝(1)但保留(2)，是不合法的。即使保留(2)，只要坚持(3)，也就必须保留(1)。

贝尔严格地表明，人们不可能拥有量子理论的局域实在论。Henry Stapp称这一结果为[18]：-

科学史上最深刻的发现。

贝尔所证明的隐变量的另一个性质是，它们必须是contextual。除最简单的情形外，测量一个变量所得到的结果必定取决于同时测量了哪些其他量。因此，不能把隐变量理解为说明一个量‘具有’什么值，而只能说明如果我们测量它将会得到什么值。

让我们回到定域性问题。于是人们一直假设量子理论是严格正确的，但当然这一点永远无法知道。约翰 Clauser、Richard Holt、Michael Horne和Abner Shimony改造了贝尔的工作，以给出对局域实在论的直接实验检验。于是就有了著名的CHHS-贝尔不等式[19]，通常只称为贝尔不等式。在EPR型实验中，局域隐变量满足这个不等式，但其他理论，包括量子理论，可能违反它。

贝尔已经达到了所谓的experimental philosophy；从实验中可以获得具有相当哲学重要性的结果。近三十年来，贝尔不等式以日益精密的方式受到检验，实验检验实际上使用具有纠缠偏振的光子，这在数学上等价于上面讨论的纠缠自旋。虽然许多科学家参与其中，但最重要者中的一部分会包括Clauser、Alain Aspect和Anton Zeilinger。

虽然[2002年8月]至少还有一个漏洞有待堵住，但几乎可以肯定局域实在论被违反，并且量子理论能够预言所有实验的结果。

在他的余生中，贝尔继续批评量子理论中通常的测量理论。逐渐地，质疑尼尔斯·玻尔和冯·诺伊曼至少变得稍微更可接受，而对量子理论意义的研究已经成为一项体面的活动。

贝尔本人早在1972年就成为皇家学会的会士，但很久以后他才获得他应得的奖项。在他生命的最后几年中，他被授予皇家学会的休斯奖章、物理学会的保罗·狄拉克奖章以及美国物理学会的海涅曼奖。在1988年7月的两周内，他获得了都柏林女王学院和三一学院两所学院的荣誉学位。他曾被提名诺贝尔奖；如果他多活十年，他肯定会获得该奖。

但事与愿违。贝尔于1990年10月1日因中风突然去世。自那以后，人们对他工作及其在量子信息论中应用的兴趣一直在稳步增长。

John Bell's great achievement was that during the 1960s he was able to breathe new and exciting life into the foundations of quantum theory, a topic seemingly exhausted by the outcome of the Bohr-Einstein debate thirty years earlier, and ignored by virtually all those who used quantum theory in the intervening period. Bell was able to show that discussion of such concepts as 'realism', 'determinism' and 'locality' could be sharpened into a rigorous mathematical statement, 'Bell's inequality', which is capable of experimental test. Such tests, steadily increasing in power and precision, have been carried out over the last thirty years.

Indeed, almost wholly due to Bell's pioneering efforts, the subject of quantum foundations, experimental as well as theoretical and conceptual, has became a focus of major interest for scientists from many countries, and has taught us much of fundamental importance, not just about quantum theory, but about the nature of the physical universe.

In addition, and this could scarcely have been predicted even as recently as the mid-1990s, several years after Bell's death, many of the concepts studied by Bell and those who developed his work have formed the basis of the new subject area of quantum information theory, which includes such topics as quantum computing and quantum cryptography. Attention to quantum information theory has increased enormously over the last few years, and the subject seems certain to be one of the most important growth areas of science in the twenty-first century.

John Stewart Bell's parents had both lived in the north of Ireland for several generations. His father was also named John, so John Stewart has always been called Stewart within the family. His mother, Annie, encouraged the children to concentrate on their education, which, she felt, was the key to a fulfilling and dignified life. However, of her four children - John had an elder sister, Ruby, and two younger brothers, David and Robert - only John was able to stay on at school much over fourteen. Their family was not well-off, and at this time there was no universal secondary education, and to move from a background such as that of the Bells to university was exceptionally unusual.

Bell himself was interested in books, and particularly interested in science from an early age. He was extremely successful in his first schools, Ulsterville Avenue and Fane Street, and, at the age of eleven, passed with ease his examination to move to secondary education. Unfortunately the cost of attending one of Belfast's prestigious grammar schools was prohibitive, but enough money was found for Bell to move to the Belfast Technical High School, where a full academic curriculum which qualified him for University entrance was coupled with vocational studies.

Bell then spent a year as a technician in the Physics Department at Queen's University Belfast, where the senior members of staff in the Department, Professor Karl Emeleus and Dr Robert Sloane, were exceptionally helpful, lending Bell books and allowing him to attend the first year lectures. Bell was able to enter the Department as a student in 1945. His progress was extremely successful, and he graduated with First-Class Honours in Experimental Physics in 1948. He was able to spend one more year as a student, in that year achieving a second degree, again with First-Class Honours, this time in Mathematical Physics. In Mathematical Physics, his main teacher was Professor Peter Paul Ewald, famous as one of the founders of X-ray crystallography; Ewald was a refugee from Nazi Germany.

Bell was already thinking deeply about quantum theory, not just how to use it, but its conceptual meaning. In an interview with Jeremy Bernstein, given towards the end of his life and quoted in Bernstein's book [1], Bell reported being perplexed by the usual statement of the Heisenberg uncertainty or indeterminacy principle ($\Delta x \Delta p ≥ \hbar$, where $\Delta x$ and $\Delta p$ are the uncertainties or indeterminacies, depending on one's philosophical position, in position and momentum respectively, and ℏ is the (reduced) Planck's constant).

It looked as if you could take this size and then the position is well defined, or that size and then the momentum is well defined. It sounded as if you were just free to make it what you wished. It was only slowly that I realized that it's not a question of what you wish. It's really a question of what apparatus has produced this situation. But for me it was a bit of a fight to get through to that. It was not very clearly set out in the books and courses that were available to me. I remember arguing with one of my professors, a Doctor Sloane, about that. I was getting very heated and accusing him, more or less, of dishonesty. He was getting very heated too and said, 'You're going too far'.

At the conclusion of his undergraduate studies Bell would have liked to work for a PhD. He would also have liked to study the conceptual basis of quantum theory more thoroughly. Economic considerations, though, meant that he had to forget about quantum theory, at least for the moment, and get a job, and in 1949 he joined the UK Atomic Research Establishment at Harwell, though he soon moved to the accelerator design group at Malvern.

It was here that he met his future wife, Mary Ross, who came with degrees in mathematics and physics from Scotland. They married in 1954 and had a long and successful marriage. Mary was to stay in accelerator design through her career; towards the end of John's life he returned to problems in accelerator design and he and Mary wrote some papers jointly. Through his career he gained much from discussions with Mary, and when, in 1987, his papers on quantum theory were collected [21], he included the following words:-

I here renew very especially my warm thanks to Mary Bell. When I look through these papers again I see her everywhere.

Accelerator design was, of course, a relatively new field, and Bell's work at Malvern consisted of tracing the paths of charged particles through accelerators. In these days before computers, this required a rigorous understanding of electromagnetism, and the insight and judgment to make the necessary mathematical simplifications required to make the problem tractable on a mechanical calculator, while retaining the essential features of the physics. Bell's work was masterly.

In 1951 Bell was offered a year's leave of absence to work with Rudolf Peierls, Professor of Physics at Birmingham University. During his time in Birmingham, Bell did work of great importance, producing his version of the celebrated CPT theorem of quantum field theory. This theorem showed that under the combined action of three operators on a physical event: $P$, the parity operator, which performed a reflection; $C$, the charge conjugation operator, which replaced particles by anti-particles; and $T$, which performed a time reversal, the result would be another possible physical event.
Unfortunately Gerhard Lüders and Wolfgang Pauli proved the same theorem a little ahead of Bell, and they received all the credit.

However, Bell added another piece of work and gained a PhD in 1956. He also gained the highly valuable support of Peierls, and when he returned from Birmingham he went to Harwell to join a new group set up to work on theoretical elementary particle physics. He remained at Harwell till 1960, but he and Mary gradually became concerned that Harwell was moving away from fundamental work to more applied areas of physics, and they both moved to CERN, the Centre for European Nuclear Research in Geneva. Here they spent the remainder of their careers.

Bell published around 80 papers in the area of high-energy physics and quantum field theory. Some were fairly closely related to experimental physics programmes at CERN, but most were in general theoretical areas.

The most important work was that of 1969 leading to the Adler-Bell-Jackiw (ABJ) anomaly in quantum field theory. This resulted from joint work of Bell and Ronan Jackiw, which was then clarified by Stephen Adler. They showed that the standard current algebra model contained an ambiguity. Quantisation led to a symmetry breaking of the model. This work solved an outstanding problem in particle physics; theory appeared to predict that the neutral pion could not decay into two photons, but experimentally the decay took place, as explained by ABJ. Over the subsequent thirty years, the study of such anomalies became important in many areas of particle physics. Reinhold Bertlmann, who himself did important work with Bell, has written a book titled Anomalies in Quantum Field Theory [10], and the two surviving members of ABJ, Adler and Jackiw shared the 1988 Dirac Medal of the International Centre for Theoretical Physics in Trieste for their work.

While particle physics and quantum field theory was the work Bell was paid to do, and he made excellent contributions, his great love was for quantum theory, and it is for his work here that he will be remembered. As we have seen, he was concerned about the fundamental meaning of the theory from the time he as an undergraduate, and many of his important arguments had their basis at that time.

The conceptual problems may be outlined using the spin-$\large\frac{1}{2}\normalsize$ system. We may say that when the state-vector is $\alpha_{+}$ or $\alpha_{-}$ respectively, $s_{z}$ is equal to $\large\frac{1}{2}\normalsize \hbar$ and $-\large\frac{1}{2}\normalsize \hbar$ respectively, but, if one restricts oneself to the Schrödinger equation, $s_{x}$ and $s_{y}$ just do not have values. All one can say is that if a measurement of $s_{x}$, for example, is performed, the probabilities of the result obtained being either $\large\frac{1}{2}\normalsize \hbar$ or $-\large\frac{1}{2}\normalsize \hbar$ are both $\large\frac{1}{2}\normalsize$.

If, on the other hand, the initial state-vector has the general form of $c_{+}\alpha_{+}+ c_{-}\alpha_{-}$, then all we can say is that in a measurement of $s_{z}$, the probability of obtaining the value of $\large\frac{1}{2}\normalsize \hbar$ is $|c_{+}^{2} |$, and that of obtaining the value of $-\large\frac{1}{2}\normalsize \hbar$ is $|c_{-}^{2} |$. Before any measurement, $s_{z}$ just does not have a value.

These statements contradict two of our basic notions. We are rejecting realism, which tells us that a quantity has a value, to put things more grandly -- the physical world has an existence, independent of the actions of any observer. Einstein was particularly disturbed by this abandonment of realism -- he insisted in the existence of an observer-free realm. We are also rejecting determinism, the belief that, if we have a complete knowledge of the state of the system, we can predict exactly how it will behave. In this case, we know the state-vector of the system, but cannot predict the result of measuring $s_{z}$.

It is clear that we could try to recover realism and determinism if we allowed the view that the Schrödinger equation, and the wave-function or state-vector, might not contain all the information that is available about the system. There might be other quantities giving extra information -- hidden variables. As a simple example, the state-vector above might apply to an ensemble of many systems, but in addition a hidden variable for each system might say what the actual value of $s_{z}$ might be. Realism and determinism would both be restored; $s_{z}$ would have a value at all times, and, with full knowledge of the state of the system, including the value of the hidden variable, we can predict the result of the measurement of $s_{z}$ .

A complete theory of hidden variables must actually be more complicated than this -- we must remember that we wish to predict the results of measuring not just $s_{z}$, but also $s_{x}$ and $s_{y}$, and any other component of $s$. Nevertheless it would appear natural that the possibility of supplementing the Schrödinger equation with hidden variables would have been taken seriously. In fact, though, Niels Bohr and Werner Heisenberg were convinced that one should not aim at realism. They were therefore pleased when John von Neumann proved a theorem claiming to show rigorously that it is impossible to add hidden variables to the structure of quantum theory. This was to be very generally accepted for over thirty years.

Bohr put forward his (perhaps rather obscure) framework of complementarity, which attempted to explain why one should not expect to measure $s_{x}$ and $s_{y}$ (or $x$ and $p$) simultaneously. This was his Copenhagen interpretation of quantum theory. Einstein however rejected this, and aimed to restore realism. Physicists almost unanimously favoured Bohr.

Einstein's strongest argument, though this did not become very generally apparent for several decades lay in the famous Einstein-Podolsky-Rosen (EPR) argument of 1935, constructed by Einstein with the assistance of his two younger co-workers, Boris Podolsky and Nathan Rosen. Here, as is usually done, we discuss a simpler version of the argument, thought up somewhat later by David Bohm.

Two spin-$\large\frac{1}{2}\normalsize$ particles are considered; they are formed from the decay of a spin-$\large\frac{1}{2}\normalsize$ particle, and they move outwards from this decay in opposite directions. The combined state-vector may be written as $(\large\frac{1}{√2}\normalsize )(\alpha_{1-}\alpha_{2+} - \alpha_{1-} \alpha_{2+})$, where the $\alpha_{1}$s and $\alpha_{2}$s for particles 1 and 2 are related to the $\alpha$s above. This state-vector has a strange form. The two particles do not appear in it independently; rather either state of particle 1 is correlated with a particular state of particle 2. The state-vector is said to be entangled.

Now imagine measuring $s_{1}z$. If we get $+\large\frac{1}{2}\normalsize v$, we know that an immediate measurement of $s_{2}z$ is bound to yield $-\large\frac{1}{2}\normalsize \hbar$, and vice-versa, although, at least according to Copenhagen, before any measurement, no component of either spin has a particular value.

The result of this argument is that at least one of three statements must be true:

(1) The particles must be exchanging information instantaneously i.e. faster than light;
(2) There are hidden variables, so the results of the experiments are pre-ordained; or
(3) Quantum theory is not exactly true in these rather special experiments.

The first possibility may be described as the renunciation of the principle of locality, whereby signals cannot be passed from one particle to another faster than the speed of light. This suggestion was anathema to Einstein. He therefore concluded that if quantum theory was correct, so one ruled out possibility (3), then (2) must be true. In Einstein's terms, quantum theory was not complete but needed to be supplemented by hidden variables.

Bell regarded himself as a follower of Einstein. He told Bernstein [1]:-

I felt that Einstein's intellectual superiority over Bohr, in this instance, was enormous; a vast gulf between the man who saw clearly what was needed, and the obscurantist.

Bell thus supported realism in the form of hidden variables. He was delighted by the creation in 1952 by David Bohm of a version of quantum theory which included hidden variables, seemingly in defiance of von Neumann's result. Bell wrote [21]:-

In 1952 I saw the impossible done.

In 1964, Bell made his own great contributions to quantum theory. First he constructed his own hidden variable account of a measurement of any component of spin. This had the advantage of being much simpler that Bohm's work, and thus much more difficult just to ignore. He then went much further than Bohm by demonstrating quite clearly exactly what was wrong with von Neumann's argument.

Von Neumann had illegitimately extended to his putative hidden variables a result from the variables of quantum theory that the expectation value of $A + B$ is equal to the sum of the expectation values of $A$ and of $B$. (The expectation value of a variable is the mean of the possible experimental results weighted by their probability of occurrence.) Once this mistake was realised, it was clear that hidden variables theories of quantum theory were possible.

However Bell then demonstrated certain unwelcome properties that hidden variable theories must have. Most importantly they must be non-local. He demonstrated this by extending the EPR argument, allowing measurements in each wing of the apparatus of any component of spin, not just $s_{z}$. He found that, even when hidden variables are allowed, in some cases the result obtained in one wing must depend on which component of spin is measured in the other; this violates locality. The solution to the EPR problem that Einstein would have liked, rejecting (1) but retaining (2) was illegitimate. Even if one retained (2), as long as one maintained (3) one had also to retain (1).

Bell had showed rigorously that one could not have local realistic theories of quantum theory. Henry Stapp called this result [18]:-

the most profound discovery of science.

The other property of hidden variables that Bell demonstrated was that they must be contextual. Except in the simplest cases, the result you obtained when measuring a variable must depend on which other quantities are measured simultaneously. Thus hidden variables cannot be thought of as saying what value a quantity 'has', only what value we will get if we measure it.

Let us return to the locality issue. So it has been assumed that quantum theory is exactly true, but of course this can never be known. John Clauser, Richard Holt, Michael Horne and Abner Shimony adapted Bell's work to give a direct experimental test of local realism. Thus was the famous CHHS-Bell inequality [19], often just called the Bell inequality. In EPR-type experiments, this inequality is obeyed by local hidden variables, but may be violated by other theories, including quantum theory.

Bell has reached what has been called experimental philosophy; results of considerable philosophical importance may be obtained from experiment. The Bell inequalities have been tested over nearly thirty years with increasing sophistication, the experimental tests actually using photons with entangled polarisations, which are mathematically equivalent to the entangled spins discussed above. While many scientists have been involved, a selection of the most important would include Clauser, Alain Aspect and Anton Zeilinger.

While at least one loophole still remains to be closed [in August 2002], it seems virtually certain that local realism is violated, and that quantum theory can predict the results of all the experiments.

For the rest of his life, Bell continued to criticise the usual theories of measurement in quantum theory. Gradually it became at least a little more acceptable to question Bohr and von Neumann, and study of the meaning of quantum theory has become a respectable activity.

Bell himself became a Fellow of the Royal Society as early as 1972, but it was much later before he obtained the awards he deserved. In the last few years of his life he was awarded the Hughes Medal of the Royal Society, the Dirac Medal of the Institute of Physics, and the Heineman Prize of the American Physical Society. Within a fortnight in July 1988 he received honorary degrees from both Queen's and Trinity College Dublin. He was nominated for a Nobel Prize; if he had lived ten years longer he would certainly have received it.

This was not to be. John Bell died suddenly from a stroke on 1st October 1990. Since that date, the amount of interest in his work, and in its application to quantum information theory has been steadily increasing.