奥斯卡·克莱因Oskar Klein

奥斯卡·克莱因是瑞典第一位拉比Gottlieb 克莱因的小儿子，Gottlieb 克莱因原籍南喀尔巴阡。Gottlieb 克莱因在海德堡获得博士学位，并于1883年移居瑞典。他显然在年幼的儿子心中灌输了求知的兴趣，因为克莱因从小就对生物学颇为喜爱。大约15岁时，这种兴趣转向了化学，不久之后，即1910年，Svante Arrhenius似乎是在Gottlieb的授意下，邀请克莱因到他在诺贝尔研究所的实验室工作。在这里，他对溶解度产生了兴趣，并于1912年发表了他的第一篇论文，内容是关于氢氧化锌在碱中的溶解度。这一年正是他完成中学教育的那一年。然而，他直到1914年才参加大学考试。

Arrhenius想派克莱因去巴黎大学Jean-Baptiste Perrin的实验室工作，但这一计划因第一次世界大战的爆发而落空。I. Klein发现自己被卷入风暴之中，于1915年和1916年服兵役。服役结束后，战争仍在激烈进行，他回到Arrhenius身边工作。

他们现在的工作重心是研究醇在不同溶剂中的介电常数。在斯德哥尔摩的这段特殊逗留期间，他遇到了Hendrik A 汉斯·克拉莫斯，后者当时（1917年）是哥本哈根尼尔斯·玻尔的学生。在接下来的几年里，克拉莫斯和克莱因在斯德哥尔摩和哥本哈根多次会面，而哥本哈根将成为克莱因的下一个目的地。

1917年，克莱因获得了一笔出国留学奖学金，随后于1918年抵达哥本哈根。在接下来的两年里，他往返于斯德哥尔摩和哥本哈根之间，为尼尔斯·玻尔和Arrhenius工作，1919年夏天与克拉莫斯在哥本哈根度过，最终于1920年返回斯德哥尔摩。但这并不是他在哥本哈根经历的终结。事实上，这仅仅是个开始。

尼尔斯·玻尔于1920年前往斯德哥尔摩拜访克莱因，并说服他再次回到哥本哈根，在尼尔斯·玻尔的研究所工作。克莱因同意了，并开始了一段后来证明成果丰硕的合作关系，最终为他带来了第一份教职。

大约在这个时候，尼尔斯·玻尔正与Svein Rosseland研究原子与自由电子混合物的统计平衡。当时人们相信，电子与原子碰撞总会损失能量。然而，克莱因与Rosseland合作，提出了“第二类碰撞”，在这种碰撞中电子实际上获得了能量！

克莱因继续在“分子通道”的另一侧工作，将注意力转向离子。事实上，这引导他进行了学位论文研究，在其中他利用约西亚·威拉德·吉布斯的统计力学研究了强电解质中离子之间的力。其结果是对布朗运动的广义表述。他于1921年在斯德哥尔摩高等学校进行了博士论文答辩，答辩对手是Erik Ivar 埃里克·伊瓦尔·弗雷德霍姆，这位数学物理学家以其在积分方程和谱理论方面的工作最为著名。成功答辩后，克莱因回到哥本哈根，后来协助尼尔斯·玻尔前往哥廷根。

大约在这个时候，克莱因转向发表关于物理学的半通俗作品。他在这个新领域的首部作品是一篇哲学论文，反驳了瑞典哲学家对相对论的一项反对意见。毫不奇怪，正是在这个时候，他开始寻找工作。

1923年，克莱因与Gerda Agnete 海里格·冯·科赫结婚，并搬到密歇根州安娜堡，在密歇根大学任职，他获得这个职位很大程度上要感谢他尊敬的朋友尼尔斯·玻尔。他在安娜堡的第一项工作涉及反常埃里克·克里斯托弗·齐曼效应，这个问题源于当时没有人理解原子在磁场中的行为。经典齐曼效应简而言之被解释为磁场使谱线分裂。问题在于经典理论只能有效描述总电子自旋为零的原子。两者的差异可以从它们的哈密顿量中看出。对于正常齐曼效应，哈密顿量为：

$H_{1} = \Large\frac{e}{2mc}\normalsize \bf{L} . \bf{B}$

对于反常齐曼效应，哈密顿量变为：

$H_{1} = \Large\frac{e}{2mc}\normalsize (\bf{L} + 2\bf{S}) . \bf{B}$

额外项来自具有自旋的物体的内禀偶极矩，其中$\bf{S}$是自旋角动量。在当时（1923年），这是一个相当棘手的问题，但克莱因并未止步于此。

他接着研究双原子分子与进动电子的相互作用，研究分子内部的角动量。次年，即1924年，他教授了一门电磁学课程，并讲授了引力场与电磁场联合场中的带电粒子。这是他关于统一场论的里程碑式工作的开端。

克莱因选择通过本质上将他的工作扩展到第五维来解决这个问题，尽管他早期的统一思想以量子物理为核心催化剂。他通过设定$p_{5}^{2} = m^{2}$来做到这一点。Brink [5]曾说克莱因的驱动力是：-

……希望有一种形式体系，把波方面和粒子方面都作为极限包含在内。

一段时间之后，克莱因越来越少主张量子物理能导致一幅统一图景，事实上他后来完全放弃了这一想法。然而，他确实看到了在五维中实现统一的可能性，这似乎在他最初的尝试中就已经存在。

此时，克莱因显然不知道西奥多·卡鲁扎的工作。西奥多·卡鲁扎在1919年寄给阿尔伯特·爱因斯坦一篇论文，提出将引力与詹姆斯·克拉克·麦克斯韦的光的理论统一起来。阿尔伯特·爱因斯坦起初对这篇论文不感兴趣，但后来意识到其中包含的高度原创的思想，并鼓励西奥多·卡鲁扎发表他的想法。事实上，这篇论文是由阿尔伯特·爱因斯坦本人于1921年12月8日提交的。

1925年，克莱因回到哥本哈根，染上了肝炎。他病了半年，不过1925年7月维尔纳·海森堡来看望过他，1926年1月埃尔温·薛定谔也来看望过他。大约就在这时，他终于能够重新工作。也正是在这时，他终于知道了西奥多·卡鲁扎的工作。沃尔夫冈·泡利把这项工作告诉了他和克莱因：-

……试图从沉船中抢救出我能抢救的东西。

克莱因对西奥多·卡鲁扎工作的改造与原作有一个重大区别：那个额外维度即第五维卷曲成一个球，其尺度约为马克斯·普朗克长度，即$10^{-33}$厘米。然而，重要的是要注意，这个额外维度虽然卷曲起来，本质上仍然是欧几里得的。基本上，第五坐标不可观测，但它是一个与电荷共轭的物理量。正如Kragh [4]所解释的，克莱因试图把电的原子性解释为一种量子定律。他还试图解释电子和质子。

克莱因假设第五维度是周期的，周期为$l = \hbar c \Large\frac{\sqrt {2k}} {e}$，其中$e$是电子的电荷，$k$是阿尔伯特·爱因斯坦的引力常数。该维度在马克斯·普朗克长度量级。

克莱因的结果于1926年秋发表在Nature上，引起了弗拉基米尔·福克、Leon Rosenfeld、路易·德布罗意和迪尔克·扬·斯特勒伊克等著名理论家的兴趣。不幸的是，尽管最初对统一抱有很大兴趣，大多数物理学家最终还是转向了更有希望、更可用实验检验的研究，把西奥多·卡鲁扎-克莱因理论留给将近半个世纪之后的另一代物理学家去探索。用克莱因自己的话说：-

保罗·狄拉克完全可以说，我的主要麻烦来自试图一次解决太多问题。

也正是在1926年，克莱因被任命为隆德大学的讲师（Docent），并在接下来的五年里成为尼尔斯·玻尔在通信和互补性方面最密切的合作者，而且显然对不确定性原理的发展有所贡献，正如维尔纳·海森堡所回忆的：-

经过几周并非没有压力的讨论，我们很快得出结论——尤其是由于克莱因的参与——我们实际上指的是同一回事，而不确定性关系只是更一般的互补原理的一个特例。

事实上，1926年对克莱因来说是一个辉煌的年份。除了终于从肝炎中康复并在隆德成为讲师之外，也正是在这一年，他做出了下一个伟大的理论突破。在一篇他确定原子跃迁概率的论文中（早于保罗·狄拉克），他引入了后来被称为克莱因-Gordon方程的初始形式。

克莱因-Gordon方程是第一个相对论性波动方程。该方程可以写成：

$\Large\frac{\partial^2 \psi}{\partial t^2}\normalsize=c^2 \nabla^2 \psi - \Large\frac{m^2 c^4}{\hbar^2}\normalsize\psi$

有趣的是，这个方程在弗洛伦斯·南丁格尔·大卫玻姆1951年的书Quantum Theory中完全按照所写的样子出现，但并未被称为克莱因-戈登方程。然而，汉斯·贝特和Jackiw的Intermediate Quantum Mechanics，最初写于1964年，确实将同一个方程称为克莱因-戈登方程。克莱因和Walter Gordon最终因此获得以他们名字命名方程的荣誉，尽管似乎花了超过四分之一世纪才获得这一荣誉。奇怪的是，埃尔温·薛定谔本人私下从他的原始波动方程发展出一个相对论波动方程，实际上这并不太难做到，并且在克莱因和Gordon之前就这样做了，尽管他从未发表结果。麻烦出现在方程未能得出氢原子正确的精细结构，以及一年后（1927年）沃尔夫冈·泡利引入自旋概念时。该方程结果与自旋不相容，因此仅适用于涉及无自旋粒子的计算。但尽管如此，它是量子理论中的一个重要点，并且连同他的统一理论，确保了克莱因的持久遗产，并巩固了1926年作为他生命中关键的一年。

1926年之后的几年里，克莱因转向教学并继续他的研究，尽管速度可能有所放缓。Brink [5]引用克莱因的一位朋友兼导师的话说：-

你现在将实现这些话：去教导人民。你伟大的教学才能一直是你最强的品质之一。我不认为发现新的自然定律和指出新方向是你的强项之一，尽管你在这个方向上总是表现出一定的雄心。

1927年，克莱因被任命为哥本哈根的讲师，但仍然继续他的研究，与帕斯库尔·约尔丹合作研究量子力学中的二次量子化。

在与帕斯库尔·约尔丹的合作中，他展示了量子场与量子统计之间的密切联系。已知二次量子化保证了光子遵循萨特延德拉·纳特·玻色-阿尔伯特·爱因斯坦统计，但克莱因表明二次量子化并不局限于自由粒子。他和帕斯库尔·约尔丹展示了可以对非相对论埃尔温·薛定谔方程进行量子化，为了纪念这项工作，他又获得了一个以他命名的数学工具，即帕斯库尔·约尔丹-克莱因矩阵。

在随后的几年里，他与正在哥本哈根进行长期研究访问的日本物理学家仁科芳雄合作，研究保罗·狄拉克电子的康普顿散射问题。尽管存在所谓的克莱因佯谬，即物理学家尚未完全理解正电子，他仍能说服物理学家相信保罗·狄拉克相对论性波动方程的合理性。他持续的工作包括热力学第二定律的量子力学和克莱因引理。

1930年，他受邀接替埃里克·伊瓦尔·弗雷德霍姆在斯德哥尔摩高等学校的职位，最终回到故乡城市担任该职位，直到1962年退休。

在20世纪30年代，克莱因帮助了许多因犹太血统而被德国和其他国家驱逐的难民物理学家。在他帮助的许多人中，有一位是沃尔特·戈登，他后来与克莱因一起成为我们刚才讨论的以他们命名的方程的受益者。1943年，克莱因还协助尼尔斯·玻尔从哥本哈根逃脱。

在20世纪30年代，克莱因还抽时间参加会议，其中尤其包括1938年华沙会议，他在会上就(almost) non-Abelian gauge theories发言。这次会议包括当时一些领先的理论家，如亚瑟·爱丁顿、尤金·维格纳等。正是在这次会议上，克莱因提出自旋为-1的粒子介导β衰变，并以类似于光子在电磁学中的作用方式参与弱相互作用。克莱因的假说是对统一场论的又一次尝试，这次是试图统一强、弱和电磁力。这项工作直到近二十年后才被注意，1957年由朱利安·施温格重新提出。

在20世纪40年代，克莱因研究了各种各样的课题，包括超导性（1945年与延斯·林哈德合作）、生物化学、通用$p$衰变、广义相对论和恒星演化。1947年后的某个时候，他和独立工作的乔瓦尼·普皮都意识到电子和μ介子都是“弱”粒子。

在20世纪50年代和60年代，克莱因仍然活跃，1958年在第11届索尔维会议上发表演讲，1963年与Hannes Alfven共同发展了一种新的宇宙学模型，并在1964年发表于Astrophisica Norvegica的一篇论文中探讨了阿尔伯特·爱因斯坦的广义相对论。晚年，他还对哲学产生了浓厚兴趣，尤其是科学与宗教之间的类比。此外，他开始撰写几本通俗书籍，其中大部分已绝版。

克莱因在斯德哥尔摩去世，他是20世纪最杰出的理论物理学家之一。

Oskar Klein was was the youngest son of Sweden's first rabbi, Gottlieb Klein, who was originally from the Southern Carpathian. Gottlieb Klein received his doctorate from Heidelberg and moved to Sweden in 1883. He evidently instilled an interest in learning in his young son, as Oskar became quite fond of biology at an early age. This interest changed to chemistry around the age of 15 and soon after, in 1910, Svante Arrhenius, at what seems to be the behest of Gottlieb, invited Oskar to work in his laboratory at the Nobel Institute. Here he took up an interest in solubility and he published his first paper in 1912 on the solubility of zinc hydroxide in alkalis. This was the very same year that he finished his secondary education. He waited, however, until 1914 to take the University exam.

Arrhenius wanted to send Klein to work with Jean-Baptiste Perrin in his laboratory at the University of Paris but the plan was foiled by the outbreak of World War I. Klein found himself caught up in the tempest and saw military service in 1915 and 1916. After his service concluded, but with the war still raging, he returned to work with Arrhenius.

Their work now centred around studying dielectric constants of alcohols in various solvents. During this particular stay in Stockholm, he met Hendrik A Kramers, who, at the time (1917), was a student of Niels Bohr in Copenhagen. Kramers and Klein met several times during the next few years both in Stockholm and in Copenhagen, which was to be Klein's next destination.

In 1917 Klein received a fellowship to study abroad and, subsequently, arrived in Copenhagen in 1918. Over the course of the next two years he would travel between Stockholm and Copenhagen performing work for both Bohr and Arrhenius, spending the summer of 1919 with Kramers in Copenhagen, and finally returning to Stockholm in 1920. But that was not to be the end of his Copenhagen experience. In fact, it was merely the beginning.

Bohr traveled to Stockholm in 1920 to visit Klein and convinced him to return to Copenhagen once more to work at Bohr's Institute. Klein agreed and began what would prove to be quite a fruitful relationship that eventually would lead him to his first teaching position.

Around this time, Bohr was working with Svein Rosseland on the statistical equilibrium of a mixture of atomic and free electrons. At the time, it was believed that electrons colliding with atoms always lost energy. However, Klein, in conjunction with Rosseland, introduced "collisions of the second kind" where the electrons actually gained energy!

Klein continued his work on the other side of the 'molecular aisle' by turning his attention to ions. In fact, this led him to his thesis research in which he studied the forces between ions in strong electrolytes using Gibbs's statistical mechanics. The result was a generalized formulation of Brownian motion. He defended his doctorate in 1921 at Stockholm Högskola and was opposed by Erik Ivar Fredholm the mathematical physicist best known for his work on integral equations and spectral theory. After his successful defence, Klein returned to Copenhagen, later assisting Bohr on a trip to Göttingen.

Around this time Klein turned to publishing semi-popular writings on physics. His first work in this new arena was a philosophical paper that was a refutation of an objection to relativity theory by Swedish philosophers. Not surprisingly, it was around this time that he began to look for a job.

In 1923, Oskar Klein married Gerda Agnete Koch and moved to Ann Arbor, Michigan to take up a post at the University of Michigan, a post he won with no small thanks to his venerable friend Niels Bohr. His first work in Ann Arbor dealt with the anomalous Zeeman effect which was a problem that arose out of the fact that no one at the time understood the behavior of atoms in a magnetic field. The classical Zeeman effect was explained, in a nutshell, as the splitting of spectral lines by the magnetic field. The problem was that the classical theory only effectively described atoms with a total electron spin of zero. The difference can be seen in the Hamiltonians of the two. For the normal Zeeman effect, the Hamiltonian reads:

$H_{1} = \Large\frac{e}{2mc}\normalsize \bf{L} . \bf{B}$

For the anomalous Zeeman effect, the Hamiltonian becomes:

$H_{1} = \Large\frac{e}{2mc}\normalsize (\bf{L} + 2\bf{S}) . \bf{B}$

The extra term arises from the intrinsic dipole moment of an object with spin, where $\bf{S}$ is the spin angular momentum. For the time (1923), this was a fairly large problem to tackle, but Klein did not stop there.

He went on to work on the interaction of diatomic molecules with precessing electrons, studying the angular momentum within the molecule itself. The following year, in 1924, he taught a course on electromagnetism and lectured on an electric particle in a combined gravitational and electromagnetic field. This was the beginning of his landmark work on a unified field theory.

Klein chose to solve the problem by essentially extending his work to a fifth dimension, though his early unification ideas centred around quantum physics as the catalyst. He did this by setting $p_{5}^{2} = m^{2}$. Brink [5] has said that Klein was driven by:-

... the wish to have a formalism which includes the wave aspect and the particle aspect as a limit.

After a time Klein argued less and less that quantum physics could lead to a unified picture, in fact he later abandoned the idea entirely. However, he did see the possibility of unification in five dimensions, which seems to have been present in his initial attempt.

At this time, Klein apparently was unaware of the work of Theodor Kaluza. Kaluza, in 1919, sent a paper to Albert Einstein proposing a unification of gravity with Maxwell's theory of light. Einstein initially was uninterested in the paper, but later realized the highly original ideas contained within it and encouraged Kaluza to publish his ideas. In fact the paper was communicated by Einstein himself on 8 December 1921.

In 1925, Klein returned to Copenhagen and contracted hepatitis. He was ill for half a year, though he was visited by Heisenberg in July of 1925 and Schrödinger in January of 1926. This was around the time he was finally able to return to work. It was at this time that he finally became aware of Kaluza's work. Wolfgang Pauli communicated this work to him and Klein:-

... tried to rescue what I could from the shipwreck.

Klein's adaptation of Kaluza's work had a major difference from the original in that the extra or fifth dimension was curled up into a ball that was on the order of the Planck length, $10^{-33}$ cm. It is important to note, however, that the extra dimension, though curled up, was still Euclidean in nature. Basically, the fifth coordinate was not observable but was a physical quantity that was conjugate to the electrical charge. As Kragh [4] explains, Klein attempted to explain the atomicity of electricity as a quantum law. He also attempted to account for the electron and the proton.

Klein assumed the fifth dimension to be periodic with a period $l = \hbar c \Large\frac{\sqrt {2k}} {e}$ where $e$ was the charge of the electron and $k$ was Einstein's constant of gravitation. The dimension was on the order of the Planck length.

Klein's results were published in Nature in the autumn of 1926 and generated interest from such eminent theorists as Vladimir Fock, Leon Rosenfeld, Louis de Broglie, and Dirk Struik. Unfortunately, despite a lot of initial interest in unification, most physicists eventually went on to more promising and experimentally testable research leaving Kaluza-Klein theory to be explored by another generation of physicists nearly half a century later. In Klein's own words:-

Dirac may well say that my main trouble came from trying to solve too many problems at a time.

It was also in 1926 that Klein was appointed as docent at Lund University and became, for the next five years, Bohr's closest collaborator both on correspondence and complimentarity, and apparently contributed to the development of the uncertainty principle, as Heisenberg recalled:-

After several weeks of discussion, which were not devoid of stress, we soon concluded, not least thanks to Oskar Klein's participation, that we really meant the same, and that the uncertainty relations were just a special case of the more general complementarity principle.

In fact, 1926 was a banner year for Klein. In addition to finally recovering from the hepatitis and becoming docent at Lund, it was in this same year that he made his next great theoretical breakthrough. In a paper in which he determined the atomic transition probabilities (prior to Dirac), he introduced the initial form of what would become known as the Klein-Gordon equation.

The Klein-Gordon equation was the first relativistic wave equation. The equation can be written:

$\Large\frac{\partial^2 \psi}{\partial t^2}\normalsize=c^2 \nabla^2 \psi - \Large\frac{m^2 c^4}{\hbar^2}\normalsize\psi$

It is interesting to note that this equation appeared exactly as it has been written in David Bohm's 1951 book Quantum Theory but was not called the Klein-Gordon equation. However, Bethe and Jackiw's Intermediate Quantum Mechanics, originally written in 1964, does refer to the same equation as the Klein-Gordon equation. Klein and Walter Gordon were thus eventually honoured with having the equation named after them, though it seems to have taken over a quarter of a century to receive the honour. Oddly enough, Schrödinger himself privately developed a relativistic wave equation from his original wave equation, which, in reality, was not that difficult to do, and did so prior to Klein and Gordon, though he never published his results. The trouble came when the equation did not result in the correct fine structure of the hydrogen atom and when Pauli introduced the concept of spin a year later (1927). The equation turned out to be incompatible with spin and, as a result, is only useful for calculations involving spinless particles. But, nonetheless, it was an important point in quantum theory and, along with his unification theory, was to ensure a lasting legacy for Klein and cemented 1926 as a pivotal year in his life.

In the years following 1926, Klein turned to teaching and continued his research, though possibly at a reduced pace. Brink [5] quotes a friend and mentor to Klein as having said:-

You will now fulfill the words: go and teach the people. Your great pedagogical talents always were one of your strongest qualities. I am not of the opinion that finding new laws of nature and indicating new directions is one of your great strengths, although you always have developed a certain ambition in this direction.

In 1927, Klein was appointed Lektor in Copenhagen but nonetheless continued his research working with Pascual Jordan on the second quantization in quantum mechanics.

In his work with Jordan, he demonstrated the close connection between quantum fields and quantum statistics. It was known that second quantization guarantees that photons obey Bose-Einstein statistics, but Klein showed that second quantization is not confined to free particles only. He and Jordan showed that one can quantize the non-relativistic Schrödinger equation and, in honour of this work, he was the recipient of yet another named mathematical tool, the Jordan-Klein matrices.

In subsequent years he collaborated with the Japanese physicist Yoshio Nishina who was in Copenhagen on an extended research visit and worked on the problem of Compton scattering of a Dirac electron. Despite the so-called Klein paradox, that being that the positron was not completely understood by physicists, he was able to convince physicists of the soundness of Dirac's relativistic wave equation. His continued work included the quantum mechanics of the second law of thermodynamics and Klein's lemma.

In 1930, he was offered Fredholm's position at Stockholm Högskala and he finally returned to his native city to take up a post that he held until his retirement in 1962.

During the 1930s, Klein helped many refugee physicists who were expelled from Germany and other nations largely due to their Jewish heritage. Of the many he helped, one included Walter Gordon who would later join Klein in being the beneficiaries of the named equation we have just discussed. In 1943, Klein also aided in Bohr's escape from Copenhagen.

During the 1930s Klein also found time to attend conferences, not the least of which included the 1938 Warsaw Conference where he spoke on (almost) non-Abelian gauge theories. This conference included some of the leading theorists of the day including Sir Arthur Eddington, Eugene Wigner, and others. It was at this conference that Klein suggested that a spin -1 particle mediated beta decay and played a role in weak interactions in a similar manner to the photon in electromagnetism. Klein's hypothesis was yet another crack at a unified field theory, this time in attempt to unify the strong, weak, and electromagnetic forces. The work was not noticed until nearly twenty years later when it was resurrected by Julian Schwinger in 1957.

In the 1940s Klein worked on a wide variety of subjects including superconductivity (with Jens Lindhard in 1945), biochemistry, universal $p$-decay, general relativity, and stellar evolution. Sometime after 1947 he, and independently Giovanni Puppi, realized that both the electron and the μ-meson were "weak" particles.

In the 1950s and 1960s Klein remained active, addressing the 11th Solvay Conference in 1958, developing a new model for cosmology in conjunction with Hannes Alfven in 1963, and tackling Einstein's General Relativity in a paper published in Astrophisica Norvegica in 1964. During his later years, he also became very interested in philosophy and especially in analogies between science and religion. In addition, he took to writing a few popular books, most of which are out of print.

Oskar Klein died in Stockholm, one of the finest theoretical physicists of the twentieth century.