帕斯库尔·约尔丹Pascual Jordan

帕斯库尔·约尔丹的父母是Ernst Pasqual 约尔丹（1858-1924）和Eva Fischer。本传记传主的父亲Ernst Jordan是一位画家，以其肖像画和风景画闻名。他儿子出生时，他是汉诺威技术大学的美术副教授。这个家族姓氏原本是Jorda，源自西班牙。长子都取名为Pasqual或约尔丹这一变体。1815年拿破仑在滑铁卢战役中战败后，家族定居于汉诺威，并在某个时候将姓氏从Jorda改为约尔丹。Ernst Jordan于1892年与Eva Fischer结婚。约尔丹从父母那里学到了许多东西。在1963年的一次采访中，他谈到了从他们那里学到的东西：-

我父亲过去常读关于自然科学的通俗书籍，尤其是本世纪初流行的“Kosmos”丛书。这套丛书每年出版几本，作者包括像Eilhelm Bölsche这样知名的通俗作家。所有这些“Kosmos”书籍都属于我最早的自然科学读物。……我母亲对植物、花卉、鸟类和星辰了解很多。我从她那里得知光从太阳到地球需要八分钟。她还对数字和计算感兴趣；我从她那里学会了算术。

大约十一岁时，约尔丹被绘画和建筑吸引，这些是他父亲非常擅长的领域。他父亲大力鼓励他追随这些兴趣，但不久约尔丹就被生物学吸引了。他在1963年的采访中说：-

很早的时候，当我还是个小男孩时，我就非常喜欢动物。我常被带到动物园，并开始收集已灭绝动物的图片。

他参加了在汉诺威举行的改革文理中学，到十四岁时，他已经在阅读奥古斯特·保利1905年的著作Darwinismus und Lamarckismus. Entwurf einer psychophysischen TeleologieⓉ（达尔文主义与拉马克主义。心理物理目的论草稿）。在那里，他了解到查尔斯·高尔顿·达尔文和让·巴蒂斯特·拉马克的思想。他还阅读了Friedrich Albert朗格的Geschichte des Materialismus und Kritik seiner Bedeutung in der GegenwartⓉ（唯物主义史及其当前重要性的批判）（1866年），并开始深入思考一些基本问题，例如有机生命是否可以从纯粹机械和物理的角度来理解。他担心达尔文的理论与圣经之间是否存在矛盾，但他在文理中学的宗教老师使他相信基督教教义与科学之间没有矛盾。为了获得进一步的理解，他学习了经典物理学和力的理论。他认定，没有理由阻止人类建造一艘可以飞往月球的火箭。这些想法使他对学习科学充满热情。

十六岁时，他通过阅读 Walther Nernst 和 阿图尔·舍恩弗利斯 Einführung in die Mathematische Behandlung der Naturwissenschaften Ⓣ（《科学数学处理导论》）（1895）这本经典著作，以及同两位作者的著作 Kurzgefasstes Lehrbuch Der Differential Und Integralrechnung Ⓣ（《微积分简明教科书》）（1895），对高等数学产生了兴趣。关于后一本书，他（在1963年）评论道：-

它介绍了微分学以及积分学的开端，还有一些关于多重积分之类内容的思考，并具体应用于科学问题。……我无法一下子全部读完；起初我只读了前几章，然后停了下来，但后来我又继续读了下去。因此，我花了几年时间才完全读完，直到我大学学习的头一年。

在中学期间他读过的其他数学书籍中，我们提及 康拉德·克诺普 的 Funktionentheorie Ⓣ（《复分析》）（1913）。

违背父亲的意愿，约尔丹于1921年从文理中学毕业后，当年夏天进入汉诺威技术高等学校学习数学和物理。他的父亲曾极力敦促他学习建筑，但约尔丹抵制了，并追随自己的意愿。技术高等学校的物理教学令约尔丹失望，他主要学习数学，修读了微分方程、积分方法和代数课程。他还参加了一门电气工程课程、一门物理化学课程和动物学课程。由于物理教学未达到他所希望的水平，他离开了技术高等学校，于1922年夏天入读哥廷根大学。他的父亲仍然敦促约尔丹留在技术高等学校学习建筑，但约尔丹的母亲说服丈夫让步，接受他们的儿子将学习物理。然而，当约尔丹开始在哥廷根学习时，他发现物理讲座都在早上9点之前。这不合他的心意，因为他已养成晚睡晚起的习惯，所以约尔丹再次主要修读数学课程。他特别选修了理查·科朗特的微分方程课程，理查·科朗特很快意识到他有一个才华出众的学生。他帮助理查·科朗特撰写理查·科朗特和大卫·希尔伯特的Methoden der mathematischen Physik Ⓣ（数学物理方法）（1924年）。两人每天会面讨论问题，理查·科朗特在该书的前言中向约尔丹表达了感谢。

约尔丹 参加了 埃德蒙·朗道 关于数论的讲座，并在数学学会与 埃米·诺特 进行了讨论。他还参加了由 马克斯·玻恩、大卫·希尔伯特 和 James Franck 主持的关于物质结构的讨论班。约尔丹 成为 马克斯·玻恩 的学生，并协助 马克斯·玻恩 撰写其关于晶体动力学的百科全书文章。百科全书文章完成后，马克斯·玻恩 对分子理论产生了浓厚兴趣，并试图让他的学生 约尔丹 将此主题作为其博士学位论文。然而，约尔丹 对这一领域并不特别感兴趣，觉得它相当专门化，而他想攻克更基本的问题。他读了 阿尔伯特·爱因斯坦 和 保罗·埃伦费斯特 1923年关于光量子的一篇论文，并在一个讨论班上就该论文作了报告。约尔丹（在1963年）说：-

听完我关于光量子的讨论班报告后，马克斯·玻恩 说，如果我更喜欢这个题目，我可以把它作为我的学位论文题目。当时对光量子普遍持怀疑态度，并且认为需要一种不同的方法来处理这个问题。马克斯·玻恩 对此的看法并不固定，他说既然我提出了这个问题，我就可以继续研究它。

约尔丹 的博士学位论文于1924年发表在《物理学杂志》上。阿尔伯特·爱因斯坦 认为 约尔丹 的学位论文是一项巧妙的工作，但不同意其基本假设。阿尔伯特·爱因斯坦 于1925年在《物理学杂志》上发表了一则短讯，陈述了他的反对意见。约尔丹 很高兴 阿尔伯特·爱因斯坦 对他的工作产生了兴趣，并于1925年遵循 马克斯·普朗克、阿尔伯特·爱因斯坦 和 保罗·埃伦费斯特 的方法发表了另一篇关于光量子的论文。在这篇论文中，约尔丹 感谢 阿尔伯特·爱因斯坦“对这项研究给予的善意关注”。然而，此时 约尔丹 正在帮助 James Franck 撰写 Anregung von Quantensprungen durch Stöβe Ⓣ（《通过碰撞激发量子跃迁》）一书。与他之前协助撰写书籍不同，这次当该书于1926年出版时，他作为合著者出现。该书研究了光谱学问题，但 约尔丹 没有继续这一领域，因为他已经对量子力学产生了兴趣。

1925年夏天，曾担任马克斯·玻恩助手的维尔纳·海森堡从哥本哈根访问归来，他在那里与尼尔斯·玻尔合作。回到哥廷根后，他向马克斯·玻恩展示了他的新矩阵力学，即量子力学的第一个版本。然而，他并非将这些概念作为矩阵代数来发明，而是将注意力集中在一组量子化概率幅上。这些概率幅构成了一个非交换代数。马克斯·玻恩阅读了维尔纳·海森堡的手稿，并试图理解它。他认为维尔纳·海森堡的量子对象必定与矩阵有关，但觉得需要一位具有深厚数学背景的年轻合作者来帮助他将自己的想法形式化。约尔丹提出帮助马克斯·玻恩攻克这个问题，他在几天内将马克斯·玻恩关于矩阵的模糊想法形式化了。马克斯·玻恩和约尔丹于1925年9月27日提交了他们的论文Zur Quantenmechanik Ⓣ（论量子力学II），关于矩阵力学。这篇论文几乎完全是约尔丹的工作，因为马克斯·玻恩在撰写期间大部分时间都在瑞士度假。这篇论文包含了著名的$pq - qp$方程，这是这个基本量子理论方程的首次出现。约尔丹的贡献迅速涌现，1925年11月，约尔丹、马克斯·玻恩和维尔纳·海森堡提交了他们的三作者论文Zur Quantenmechanik II Ⓣ（论量子力学II），该论文于1926年发表。年底之前，约尔丹提交了一篇单作者论文，但遭遇了不幸的命运。

这篇论文[29]：-

……包含了如今被称为恩里科·费米-保罗·狄拉克统计的东西；然而，它在提交后遭遇了极其不幸的命运，因为在一次前往美国的长期讲学之旅前夕，它落到了马克斯·玻恩（当时他担任《物理学杂志》编辑）的一个手提箱底部，并在那里待了大约半年。当马克斯·玻恩发现这一失误时，保罗·狄拉克和恩里科·费米的论文已经在出版过程中了。

约尔丹的这篇论文从未发表，但他进一步发展了其内容，这项扩展的工作，Zur Quantenmechanik der Gasentartung Ⓣ（论气体简并的量子力学），由约尔丹于1927年发表。他在1927年还发表了其他极其重要的论文，如Über eine neue Begründung der Quantenmechamik Ⓣ（论量子力学的新基础）（分两部分），涉及今天所谓的“变换理论”。他在这里的想法接近保罗·狄拉克的想法。1926年12月24日，当时正在哥本哈根拜访尼尔斯·玻尔的保罗·狄拉克写信给约尔丹（例如见[16]）：-

维尔纳·海森堡博士给我看了你寄给他的工作，据我所见，它在所有要点上都与我自己的工作等价。获得结果的方式可能相当不同……我希望你不介意我获得了与你相同的结果，而且（我相信）是在与你相同的时间。此外，Royal Society发表论文比《物理学杂志》更快，我想我的论文会出现在他们的一月号上。我预计二月初去哥廷根，非常期待在那里见到你和马克斯·玻恩教授。

确实，两篇论文都在1927年1月发表，即提交后一个月，保罗·狄拉克的论文如他所想的那样先出现。约尔丹的论文是我们上面给出标题的两部分论文的第一部分。约尔丹在这些年对量子理论还有许多其他贡献，但我们必须看看他做出的另一项对数学发展产生重大影响的杰出贡献。1929年，约尔丹被任命为罗斯托克大学物理学编外教授。尼尔斯·玻尔很高兴，写信给约尔丹说，由于罗斯托克离哥本哈根相当近，他希望他们能合作。此时，约尔丹开始对将量子理论的思想应用于生物学问题感兴趣。例如，约尔丹发表了Quantenmechanik und die Grundprobleme der Biologie und Psychologie Ⓣ（量子力学与生物学和心理学的基本问题）（1932年）和Quantenphysikalische Bemerkungen zu Biologie und Psychologie Ⓣ（关于生物学和心理学的量子物理观察）（1934年）。尼尔斯·玻尔也对这些想法感兴趣。

1932年，为了将量子理论置于一个新的代数框架中，约尔丹试图建立量子理论矩阵解释中所满足的基本规则。他在Über die Multiplikation quantenmechanischer Gröβen Ⓣ（《论量子量的乘法》）（1934）中发表了这些想法，并使冯·诺伊曼和尤金·维格纳相信这些是重要的想法。约尔丹、冯·诺伊曼和尤金·维格纳在三作者论文On the algebraic generalization of the quantum mechanical formalism（1934）中发表了他们关于此的联合想法。通过他的方法，约尔丹引入了今天所谓的约尔丹代数。更多细节见THIS LINK。

正如我们所见，约尔丹是量子理论发展的主要贡献者之一。然而，他从未获得其他人（其贡献似乎较少）所获得的认可。现在大多数人认为这是由于约尔丹的政治观点及其对纳粹党的支持。因此，我们必须看看约尔丹生活的这一方面，部分出于兴趣，但也部分因为它与他的科学和数学观点的联系。约尔丹的本科岁月是在哥廷根度过的，当时他的大多数同学都因为第一次世界大战后盟国强加的严重财政负担对德国的影响而感到非常委屈。然而，I. Jordan的观点远远超出了他的大多数同学。从1920年代末起，他以笔名Ernst Domeier发表带有极端极右政治观点的文章。他匿名发表这些文章的事实确实表明，在那个阶段，他想将自己的政治观点与科学工作分开。我们引用约尔丹在1930年以Ernst Domeier之名写的一篇文章中的一小段话：-

自1918年以来……我们不是世界政治的主体，而是世界政治的客体……所有与我们的帝国政府有任何关系的人……内心都必须充满一种感情，这种感情由那种紧迫的无能为力的依赖感以类似规律般的必然性产生出来。……自卑情结制造出不惜一切代价进行补偿的必要性。……官僚式的浮夸和官僚对其本国民众的恐怖统治产生了所需的补偿。

此外，作为一个德国民族主义者，他非常反共，并将苏联的崛起视为对世界的重大危险。这些观点当然使他成为纳粹宣传的容易目标，事实上他在1933年，即希特勒上台的那一年，加入了纳粹党。然而，事情并不像表面看起来那么简单，人们不能因为约尔丹是一个坚定而热情的纳粹支持者，就认为他相信所有纳粹政策。例如，约尔丹写了Physikalisches Denken in der neuen ZeitⓉ（《现代物理思维》）一书，该书于1935年出版。这本书包含了对路德维希·比贝尔巴赫的“德意志数学”观念的抨击，尽管约尔丹没有点名提到路德维希·比贝尔巴赫。

你可以在THIS LINK阅读约尔丹攻击“德意志数学”思想的书中摘录。

有趣的是，他不仅不赞同所有纳粹政策，还积极试图从纳粹党内部改变它们。

纳粹的另一项“大”政策是他们的反犹太主义。没有证据表明约尔丹有任何反犹太观点，他拥有的大量犹太朋友似乎证实了这一点。纳粹要求所有德国科学家在其学科发展中省略提及犹太科学家，但约尔丹强烈主张政治和科学应保持分离。他写了关于20世纪物理学史的书籍，如Die Physik des 20sten Jahrhunderts Ⓣ（20世纪物理学）（1936年），该书充分赞扬了所有犹太科学家。当然，约尔丹的观点给纳粹领导人带来了一些困难。一方面，他发表了他们喜欢的强烈亲纳粹宣传，但他与犹太科学家的持续联系意味着约尔丹不被当局信任。当然，具有一定讽刺意味的是，约尔丹几乎肯定因为他的纳粹观点而没有获得诺贝尔奖，而他自己曾与纳粹争论政治不应影响对科学价值的看法。

1939年第二次世界大战爆发时，约尔丹加入了德国空军。他没有参加实战，而是在波罗的海的佩讷明德担任天气分析师，那里是V-2火箭研发和生产的地方。然而，他仍然对自己的想法更感兴趣，而不是分配给他的工作。Peter P Wegener写道[9]：-

约尔丹整天坐在打字机前撰写一本代数教科书，从不查阅笔记。通常到了傍晚，他会突然想起他在佩讷明德任务的原本目的。他从未研究过流体动力学，但他不像我那样阅读书籍，而是着手推导超音速流动的运动方程。他乐于发现诸如激波之类的现象。

1944年，战争结束前，约尔丹被任命为柏林大学理论物理学普通教授，填补了Max von Laue先前担任的讲席，后者于1943年被宣布为荣誉退休，比他预定退休早一年。当然，德国战败后，作为坚定的纳粹支持者，约尔丹陷入了一些困境。他的几位同事，如维尔纳·海森堡和沃尔夫冈·泡利，为他辩护。有一个有趣的小插曲，战后沃尔夫冈·泡利和约尔丹相遇。沃尔夫冈·泡利引用了约尔丹的一些“纳粹宣传”并问道：“约尔丹先生，你怎么能写出这样的东西？”约尔丹回答说：“沃尔夫冈·泡利先生，你怎么能读这样的东西？”在1945-47这两年，约尔丹被解除了大学职位，但在1947年，在诸如维尔纳·海森堡这样的声明之后，他被恢复为访问教授，维尔纳·海森堡说：-

我从未考虑过约尔丹可能是一个真正的国家社会主义者的可能性……

约尔丹曾以最喧嚣的方式宣扬自己是纳粹，维尔纳·海森堡的这番话只能意味着他希望所有纳粹分子都复职。约尔丹也请求尼尔斯·玻尔支持他复职，但尼尔斯·玻尔不那么客气，他只是回复约尔丹一份被纳粹杀害的亲友名单。1947年，约尔丹获得了弗赖堡和汉堡的临时教授职位。他选择了后者，1953年该职位转为永久职位，约尔丹获准指导博士生。Wolfgang Kundt写道[24]：-

我科学生涯的最初二十年，即1951—71年，深受约尔丹的塑造。他是我在汉堡大学理论物理课程中占主导地位的讲师，是我文凭、博士阶段的导师，也是教授资格论文（Habilitation），后来担任每周一次的“相对论讨论班”的主持人，还是一位同样关心其学者精神财富与物质财富的科学之父。

约尔丹于1971年从汉堡退休，但在此之前他曾有过一段政治生涯，1957年至1961年间担任基督教民主联盟的联邦议院议员。这样做时，他无视了沃尔夫冈·泡利的忠告，因为当他为约尔丹复职发声时，沃尔夫冈·泡利曾建议他不要涉足政治。在联邦议院期间，他频繁呼吁回归‘真正的前线精神’。他继续信奉军事化，主张允许德国发展核弹，始终是一名坚定的反共分子，并于1965年主张德国不应承认奥得-尼斯河线为德国与波兰之间的边界。他继续大力主张德国应更加相信自己，1973年写道：-

……德国人民今天最迫切需要的是克服其民族自卑情结。

他反对这样的事实：当他发声支持一个强大的德国时，政治左派人士便称他为法西斯分子。

让我们引用[21]中的话来结束我们的传记：-

约尔丹从未获得诺贝尔物理学奖，这令人费解。有人归咎于他因口吃而无法发表优雅的演讲；有人归咎于他的亲纳粹政治立场，或他在二战后对德国核武器计划的支持；有人归咎于马克斯·玻恩弄丢了约尔丹1925年的手稿，其中首次提出了恩里科·费米-保罗·狄拉克统计，从而使谦逊的约尔丹失去了对沃尔夫冈·泡利的优先权。但事实仍然是，他对现代量子理论发展的贡献与许多因成就而获得诺贝尔奖的人一样根本和深远。正是约尔丹，比任何人都更多地发展了矩阵力学的数学上优雅的表述。正是约尔丹，继续将矩阵力学与保罗·狄拉克的替代算子演算和埃尔温·薛定谔的波动力学表述整合为被称为统计变换理论的综合形式体系。正是约尔丹，除保罗·狄拉克之外比任何人都更多地开创了量子场论纲领，例如发展了二次量子化方法，并第一个发现了量子场论中的发散问题。正是约尔丹，与冯·诺伊曼和尤金·维格纳一起，为量子力学发展了更抽象的代数框架。约尔丹被描述为“量子力学创造者中未被歌颂的英雄”并非没有道理。

Pascual Jordan's parents were Ernst Pasqual Jordan (1858-1924) and Eva Fischer. Ernst Jordan, the father of the subject of this biography, was a painter who was renowned for his portraits and landscapes. He was an associate professor of art at Hanover Technical University when his son was born. The family name was originally Jorda and it was of Spanish origin. The first born sons were all given the name Pasqual or the version Pascual. The family settled in Hannover after Napoleon's defeat at the Battle of Waterloo in 1815 and at some stage changed their name from Jorda to Jordan. Ernst Jordan married Eva Fischer in 1892. Pascual leant much from his parents. In an interview in 1963 he spoke about things he learnt from them:-

My father used to read popular books about natural science especially the series of 'Kosmos' books which was popular at the beginning of this century. Each year several books came out in this series, written by people such as Eilhelm Bölsche who was well-known as a popular author. All these 'Kosmos' books belonged to my first reading in natural science. ... My mother knew a good deal about plants, flowers, birds and stars. I learnt from her that light takes eight minutes in coming from the sun to the earth. She was also interested in numbers and calculation; I learnt arithmetic from her.

When he was around eleven years old Jordan was attracted to painting and architecture, topics in which his father was highly skilled. His father strongly encouraged him to follow these interests but soon Jordan became attracted to biology. He said in the 1963 interview:-

Very early when I was a little boy, I was very fond of animals. I was often taken to the zoological gardens and I began collecting pictures of extinct animals.

He attended the Reform Gymnasium in Hannover and, by the age of fourteen, he was reading books such as August Pauly's 1905 work Darwinismus und Lamarckismus. Entwurf einer psychophysischen Teleologie Ⓣ. There he learnt of the ideas of Charles Darwin and Jean Baptiste Lemarck. He also read Friedrich Albert Lange's Geschichte des Materialismus und Kritik seiner Bedeutung in der Gegenwart Ⓣ (1866) and began to think deeply about fundamental questions such as whether organic life was understandable from a purely mechanical and physical perspective. He worried whether there was a contradiction between Darwin's theories and the Bible but his teacher of religion at the Gymnasium convinced him that there was no contradiction between Christian teaching and the sciences. In order to gain further understanding he studied classical physics and the theory of forces. He decided that there was no reason to prevent man building a rocket which could travel to the moon. These thoughts made him enthusiastic about studying science.

At the age of sixteen, he became interested in advanced mathematics through reading the classic book by Walther Nernst and Arthur Schönflies Einführung in die Mathematische Behandlung der Naturwissenschaften Ⓣ (1895) and the same two authors' book Kurzgefasstes Lehrbuch Der Differential Und Integralrechnung Ⓣ (1895). Concerning this last book he remarked (in 1963):-

It presents differential calculus and the beginnings of integral calculus, also some considerations about multiple integrals and such things, with specific application to scientific problems. ... I could not read it all at once; at first I read only the first few chapters, then stopped, but later I read on further. It thus took me several years to read completely, up to the first year of my university studies.

Among other mathematics books he read while at the Gymnasium, we mention Konrad Knopp's Funktionentheorie Ⓣ (1913).

Against his father's wishes, after Jordan graduated from the Gymnasium in 1921, he entered the Hanover Technische Hochschule in the summer of that year to study mathematics and physics. His father had pressed him hard to study architecture but Jordan resisted and followed his own wishes. The teaching of physics at the Technische Hochschule disappointed Jordan and he studied mostly mathematics taking courses on differential equations, methods of integration, and algebra. He also attended an electrical engineering course, a physical chemistry course and zoology courses. Since the physics teaching was not up to the standard he hoped for, he left the Technische Hochschule and enrolled at the University of Göttingen in the summer of 1922. His father still pressed for Jordan to remain at the Technische Hochschule and study architecture there but Jordan's mother persuaded her husband to relent and accept that their son would study physics. However, when Jordan began studying at Göttingen he discovered that the physics lectures were all before 9 o'clock in the morning. This was not to his liking since he had developed the habit of going to bed late and rising late in the mornings, so again Jordan took mainly mathematics courses. In particular he enrolled in Richard Courant's differential equations course and Courant quickly realised that he had an exceptionally talented student. He helped Courant with the writing of Courant and Hilbert's Methoden der mathematischen Physik Ⓣ (1924). The two would meet daily to discuss problems and Courant expressed his thanks to Jordan in the Foreword of the book.

At Göttingen, Jordan attended lectures by Edmund Landau on number theory and had discussions with Emmy Noether at the Mathematische Gesellschaft. He also attended the seminar on the structure of matter run by Max Born, David Hilbert and James Franck. Jordan became Max Born's student and assisted Born with his encyclopaedia article on crystal dynamics. After the encyclopaedia article was completed, Born became deeply interested in molecular theory and tried to interest his student Jordan in this topic for his doctoral dissertation. However, Jordan was not particularly interested in this area finding it rather specialised while he wanted to attack more fundamental questions. He read a 1923 paper by Einstein and Ehrenfest on light quanta and gave a talk on the paper in a seminar. Jordan said (in 1963):-

After listening to my seminar on light-quanta Born said that I could make that the topic of my dissertation if I liked it better. There was general scepticism about light-quanta and a belief that a different approach to the problem was needed. Born's opinion on this was not fixed and he said that since I had raised the question I could pursue it.

Jordan's doctoral dissertation was published in 'Zeitschrift für Physik' in 1924. Einstein thought that Jordan's dissertation was an ingenious piece of work but disagreed with his basic hypotheses. Einstein published a note in 'Zeitschrift für Physik' in 1925 stating his objections. Jordan was pleased that Einstein had taken an interest in his work and published a further paper on light quanta in 1925 following the approach by Planck, Einstein and Ehrenfest. In this paper Jordan thanked Einstein "for the kind interest he has taken in this investigation." However, by this time Jordan was helping James Franck in writing the book Anregung von Quantensprungen durch Stöβe Ⓣ. Unlike the assistance he had given previously in writing books, this time he appeared as a joint author when the book was published in 1926. The book studied spectroscopic problems but Jordan did not continue with this area for he had already become interested in quantum mechanics.

In the summer of 1925 Werner Heisenberg, who was been Born's assistant, returned to Göttingen from a visit to Copenhagen where he had worked with Niels Bohr. Back in Göttingen he showed Born his new matrix mechanics, the first version of quantum mechanics. He did not invent these concepts as a matrix algebra, however, rather he focused attention on a set of quantised probability amplitudes. These amplitudes formed a non-commutative algebra. Born read Heisenberg's manuscript and tried to make sense of it. He thought that Heisenberg's quantum objects must be related to matrices but felt he needed a young collaborator with a strong mathematical background to help him formalise his ideas. Jordan offered to help Born attack the problem and he had formalised Born's vague ideas about matrices in a few days. Born and Jordan submitted their paper Zur Quantenmechanik Ⓣ on matrix mechanics on 27 September 1925. It was almost entirely the work of Jordan since Born was on holiday in Switzerland for most of the time that it was being written. This paper contains the famous $pq - qp$ equation, the first appearance of this fundamental quantum theory equation. Jordan's contributions were coming rapidly and in November 1925 Jordan, Born and Heisenberg submitted their three-author paper Zur Quantenmechanik II Ⓣ which was published in 1926. Before the end of the year Jordan had submitted a single author paper which suffered an unfortunate fate.

This paper [29]:-

... contained what is nowadays known as the Fermi-Dirac statistics; however it encountered an extremely unfortunate fate after its submission because it landed on the bottom of one of Max Born's (in his role as the editor of the 'Zeitschrift für Physik') suitcases on the eve of an extended lecture tour to the US, where it remained for about half a year. When Born discovered this mishap, the papers of Dirac and Fermi were already in the process of being published.

This paper by Jordan was never published, but he further developed its contents and this extended piece of work, Zur Quantenmechanik der Gasentartung Ⓣ, was published by Jordan in 1927. He published other, extremely significant, papers in 1927 such as Über eine neue Begründung der Quantenmechamik Ⓣ (in two parts) which deals with what today in called "transformation theory". His ideas here were close to those of Paul Dirac. On 24 December 1926 Dirac, who was at that time in Copenhagen visiting Niels Bohr, wrote to Jordan (see for example [16]):-

Dr Heisenberg has shown me the work you sent him, and as far as I can see it is equivalent to my own work in all essential points. The way of obtaining the results may be rather different though ... I hope you do not mind the fact that I have obtained the same results as you, at (I believe) the same time as you. Also, the Royal Society publishes papers more quickly than the Zeits. f. Phys., and I think my paper will appear in their January issue. I am expecting to go to Göttingen at the beginning of February, and I am looking forward very much to meeting you and Prof Born there.

Indeed both papers were published in January 1927, a month after being submitted, with Dirac's paper, as he had thought, appearing first. Jordan's paper was the first of the two part paper whose title we gave above. There are many other contributions that Jordan made to quantum theory in these years but we must look at another remarkable contribution he made which has had a major impact on the development of mathematics. In 1929 Jordan was appointed as extraordinary professor of physics at the University of Rostock. Bohr was delighted and wrote to Jordan saying that, since Rostock was quite close to Copenhagen, he hoped they could collaborate. Jordan was, by this time, becoming interested in applying ideas from quantum theory to biological matters. For example, Jordan published Quantenmechanik und die Grundprobleme der Biologie und Psychologie Ⓣ (1932) and Quantenphysikalische Bemerkungen zu Biologie und Psychologie Ⓣ (1934). Bohr was also interested in these ideas.

In 1932, in an attempt to put quantum theory into a new algebraic setting, Jordan tried to establish the basic rules satisfied in the matrix interpretation of quantum theory. He published these ideas in Über die Multiplikation quantenmechanischer Gröβen Ⓣ (1934) and convinced John von Neumann and Eugene Wigner that these were significant ideas. Jordan, von Neumann and Wigner published their joint ideas on this in the three-author paper On the algebraic generalization of the quantum mechanical formalism (1934). With his approach, Jordan had introduced what today are called Jordan algebras. For more details see THIS LINK.

Jordan was, as we have seen, one of the main contributors to the development of quantum theory. However, he never achieved the recognition for this that others, whose contributions appeared to be less, received. Most now believe that this was due to Jordan's political views and his support for the Nazi party. We must therefore look at this aspect of Jordan's life, partly out of interest, but also partly because of its connection with his scientific and mathematical views. Jordan's undergraduate years saw him in Göttingen at a time when most of his fellow students were feeling very hard done by because of the impact on Germany of the severe financial burdens imposed by the Allies following World War I. Jordan's views, however, went far beyond those of most of his fellow students. From the late 1920s he was publishing articles with extreme far-right political views under the pseudonym Ernst Domeier. The fact that he published these anonymously certainly shows that, at that stage, he wanted to keep his political views separate from his scientific work. We present a short quote from an article that Jordan wrote in 1930 under the name Ernst Domeier:-

Since 1918 ... instead of a subject we are an object of world politics ... All those who have anything to do with the government of our Reich ... must be inwardly filled with feelings which result with law-like necessity from the pressing sensation of powerless dependence .... Inferiority complexes create the necessity of a compensation at any price. ... Bureaucratic pomposity and bureaucrats' terrorizing of their own peoples produce the needed compensation.

In addition he being a German nationalist, he was very anti-Communist and viewed the rise of the Soviet Union as a major danger to the world. These views, of course, made him an easy target for the Nazi propaganda and indeed he joined the Nazi party in 1933, the year Hitler came to power. Things, however, are not quite as simple as they might appear and one must not think that because Jordan was a staunch and enthusiastic Nazi supporter, he believed in all the Nazi policies. For example, Jordan wrote the book Physikalisches Denken in der neuen Zeit Ⓣ which was published in 1935. This book contains an attack on Ludwig Bieberbach's idea of 'Deutsche Mathematik' although Jordan does not mention Bieberbach by name.

You can read an extract from the book in which Jordan attacks the idea of 'Deutsche Mathematik' at THIS LINK.

It is interesting that, as well as not subscribing to all the Nazi policies, he was actively trying to change them from inside the Nazi Party.

Another of the 'big' Nazi policies was their anti-Semitism. There is no evidence that Jordan had any anti-Semitic views and the large number of Jewish friends that he had seems to confirm this. The Nazis demanded that all German scientists omit mention of Jewish scientists in the development of their discipline but Jordan argued strongly that politics and science should be kept separate. He wrote books on the history of physics in the 20th century such as Die Physik des 20sten Jahrhunderts Ⓣ (1936) which gave full credit to all Jewish scientists. Of course, Jordan's views presented the leading Nazis with some difficulty. On the one hand he was putting out strong pro-Nazi propaganda which they loved, but his continued association with Jewish scientists meant that Jordan was not trusted by the authorities. There is, of course, a certain irony in the fact that Jordan was almost certainly not awarded a Nobel prize because of his Nazi views when he himself had argued with the Nazis that politics should not influence views of scientific merit.

When World War II began in 1939, Jordan enlisted in the Luftwaffe. He did not see active service but worked as a weather analyst at Peenemünde on the Baltic Sea where V-2 rockets were developed and produced. However, he continued to be more interested in his own ideas rather than in the work that was assigned to him. Peter P Wegener writes [9]:-

Jordan sat in front of his typewriter all day composing a textbook on algebra without ever consulting notes. Usually toward the end of the day he would suddenly recall the original purpose of his assignment at Peenemünde. He had never worked in fluid dynamics, but rather than read books as I did, he proceeded to derive the equations of motion of supersonic flows. He delighted in the discovery of such phenomena as shock waves.

In 1944, before the war ended, Jordan was appointed as an ordinary professor of theoretical physics at the University of Berlin to fill the chair previously held by Max von Laue who had been declared emeritus in 1943, one year before he was due to retire. Of course after Germany's defeat in the war, Jordan, as a strong Nazi supporter, was in some difficulty. Several of his colleagues, such as Werner Heisenberg and Wolfgang Pauli, spoke up in his favour. There is a nice episode when, after the war, Pauli and Jordan met. Pauli quoted some of Jordan's "Nazi propaganda" and asked, "Herr Jordan, how could you write such a thing?" Jordan replied, "Herr Pauli, how could you read such a thing?" For the two years 1945-47, Jordan was dismissed from his position at the university but in 1947 he was reinstated as a visiting professor after statements such as that by Heisenberg who said:-

I never reckoned with the possibility that Jordan could be a true National Socialist ...

Since Jordan had advertised the fact that he was a Nazi in the most vociferous of ways, this statement by Heisenberg could only mean that he wanted all Nazis reinstated. Jordan also asked Niels Bohr to support his reinstatement but Bohr was less kind in that he simply replied by sending Jordan a list of his friends and relatives murdered by the Nazis. In 1947 Jordan was offered temporary professorships at Freiburg and at Hamburg. He chose the latter and in 1953 the position became permanent and Jordan was allowed to supervise Ph.D. students. Wolfgang Kundt writes [24]:-

The first twenty years of my scientific life, 1951-71, were strongly shaped by Pascual Jordan, the dominant lecturer of my classes in theoretical physics at Hamburg University, my supervisor for the diploma, Ph.D., and habilitation, later as the head of the weekly 'Relativitätstheorieseminar', and as the scientific father who cared equally for the mental and physical wealth of his scholars.

Jordan retired from Hamburg in 1971 but before this he had had a political career being a Christian-Democratic MP in the Bundestag from 1957 to 1961. In doing so he had ignored Pauli's advice for when he spoke up for Jordan to be reinstated, he advised him not to enter politics. When in the Bundestag he frequently called for a return to the 'true front spirit'. He continued to believe in militarisation, advocated Germany being allowed to develop a nuclear bomb, remained a staunch anti-Communist, and, in 1965, argued that Germany should not recognise the Oder-Neisse line as the border between Germany and Poland. He continued to argue strongly for Germany to believe more in itself, writing in 1973:-

... what the German people most urgently need today is the overcoming of its national inferiority complex.

He objected to the fact that when he spoke out in favour of a strong Germany, he was called a fascist by those on the political Left.

Let us end our biography by quoting from [21]:-

That Jordan never won a Nobel Prize in physics is a puzzle. Some blame his inability to give elegant lectures because of a stutter; some blame his pro-Nazi politics or his support, after World War II, for a German nuclear weapons program; some blame the fact that Born misplaced Jordan's 1925 manuscript in which Fermi-Dirac statistics were first presented, thus depriving the modest Jordan of his rightful claim to priority over Pauli. But the fact remains that his contributions to the development of modern quantum theory were as fundamental and far-reaching as those of many whose achievements were recognized with a Nobel Prize. It was Jordan, more than anyone else, who developed a mathematically elegant formulation of matrix mechanics. It was Jordan who went on to consolidate matrix mechanics with Dirac's alternative operator calculus and Erwin Schrödinger's wave-mechanical formulation in the comprehensive formalism known as statistical transformation theory. It was Jordan who did more than anyone other than Dirac to inaugurate the program of quantum field theory, in ways such as developing the second quantization approach and being the first to discover the problem of divergences in quantum field theory. And it was Jordan who, along with von Neumann and Eugene Wigner, was developing more abstract algebraic frameworks for quantum mechanics. Not without reason has Jordan been described as "the unsung hero among the creators of quantum mechanics".