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整理人: charmer(2001-06-28 11:26:40), 站内信件
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世纪之交的理论物理学 (IX)
烈火战车
尽管爱德华。威腾(EDWARD WITTEN)现在还不到五十岁，但早已是跻身于当今数学名家之列了。所以他的个人小传，也与同
S。S。CHERN(陈省身)，M。ATIYAH，DONALDSON，S。T。YAU(丘成桐)，A。CONNES诸位当代几何(拓扑)学大师一起，被数学
界权威网站所收录(已附于文尾)。这当然与他获得90年的菲尔兹(FIELD)奖有关。但非常有趣的是，威腾毕竟是个物理学家！
更让人肃然的是，90年代中期有人就谁是家喻户晓的当代人物对美国人进行调查，威腾居然名列前五十强。这对于从事研究
的科学家而言，实属罕见。
下面的小传中提到了他获奖的一个原因，便是将丘成桐等人对广义相对论中正能定理的证明，进行了大大的简化。而事实上
现在回头来看威腾的工作，那只是他所有工作中的一个小部分而已。随后他在86年左右开创了拓扑场论的先河。这样的一门
分支，从数学的角度看，是用物理学中场论的方法，来对空间的拓扑结构进行研究；从物理的角度来看，是将空间的(特别是
非平庸的)拓扑结构，纳入场论的范围内来加以考虑，从而对一些依然存在的物理现象进行解释。所以无论是从数学的审美角
度上看，还是从物理的应用上来看，其工作完全可以和他对正能定理证明的简化相媲美。利用发展起来的量子拓扑场论，威腾
自己首先便对2+1维(两维空间，一维时间)量子引力进行了研究，证明了它完全是一个可解问题。随后95年左右，威腾和塞伯
格(SEIBERG)作了一个很漂亮的工作，现在这个工作被称为SEIBERG-WITTEN理论，它旨在对超对称中“电磁”对偶性进行研究。
这对于弦理论而言，具有非常重要的价值，这以后会再专门叙述。但同时这个理论在四维拓扑学中同时带来了突破性的进展。
它使得对四维拓扑不变量--DONALDSON结的计算量，大大地减少。所以去年在国际数学物理会议上(ICMP2000)，ATIYAH在总结
性的发言中，依然提起了他们的这项工作。
但是威腾的主要研究领域，应该说是在弦理论。因为弦理论的两次革命，都是直接和威腾的工作分不开的。威腾开始研究弦
时，前景并不被行家看好。我老板闲聊时侃过这样一件小事。八十年代初，钱德拉谢卡尔(Chandrasekhar)访问普林斯顿的时候，问威腾他
们正在作些什么。威腾告诉这位老前辈，说我们正在作一些弦方面的工作，然后钱德拉谢卡尔就语重心长地说，年青人，作
工作选课题时，可得慎重点儿！弄得威腾等人尴尬不已。
但弦随后便取得了突破性的进展。首先，原来的弦理论是一个玻色子理论，为了能将费米子也纳入到弦的框架里来，人们将
超对称也考虑了进来(将在后面对超对称另作简介)。这使得合理的时空维数由26维降到了十维。同时在这样的超弦框架里面，不象以前似乎可以建立起许多、
种不同的弦理论，人们发觉可以建立起来的自
恰的理论，只可能有五种！它们分别与不同的对称群相对应。这样方标志着超弦理论，初具轮廓。史瓦兹将其称为弦的第一次
革命。被他称为弦的第二次革命发生于95年前后，威腾等人发觉，即使是在这样五种不同的超弦理论中，他们并不是完全独立
的，它们被各种各样的对偶性(DUALITY)给联系了起来。如同电磁对偶一样，五种不同的弦理论，只是描述了同一事物的不同
侧面！那么这个“同一事物”又是什么？它为弦理论所预言，但至今依然没有答案。但无疑，它正是人们一直在梦寐以求的
“包括一切”(^_^)的大统一理论！这样的一个理论，现在被命名为M理论(或如SEN提议的U理论)。现在人们只知这样的一个理论
，应该是一个存在于十一维时空的理论，若把它比作一个三维立方体，那么你可以想象五种不同的十维弦理论，分布于立方体
不同的角或面，各能窥其一隅，但远远不够完整。
特别需要指出的是，这样的一个M理论，产生于弦理论，但不是弦理论！所以你可以说弦理论的第二次革命，“革的是自己的命”，
旧理论方兴未艾，新理论有待破壳而出，谁又能说接下来的二十年，不会迎来另一个激动人心的时代呢？。。。。。。
(待续)
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Born: 26 Aug 1951 in Baltimore, Maryland, USA
Edward Witten studied at Brandis University and received his B.A. in 1971. From there he went to Princeton receiving his M.A. in 1974 and his Ph.D. in 1976.
After completing his doctorate, Witten went to Harvard where he was postdoctoral fellow during session 1976-77 and then a Junior Fellow from 1977 to 1980. In September 1980 Witten was appointed professor of Physics at Princeton. He was awarded a MacArthur Fellowship in 1982 and remained as professor of Physics at Princeton until 1987 when he was appointed as a Professor in the School of Natural Sciences at the Institute for Advanced Study.
Basically Witten is a mathematical physicist and he has a wealth of important publications which are properly in physics. However, as Atiyah writes in [3]:-
Although he is definitely a physicist (as his list of publications clearly shows) his command of mathematics is rivalled by few mathematicians, and his ability to interpret physical ideas in mathematical form is quite unique. Time and again he has surprised the mathematical community by his brilliant application of physical insight leading to new and deep mathematical theorems.
Speaking at the American Mathematical Society Centennial Symposium in 1988, Witten explained the relation between geometry and theoretical physics:-
It used to be that when one thought of geometry in physics, one thought chiefly of classical physics - and in particular general relativity - rather than quantum physics. ... Of course, quantum physics had from the beginning a marked influence in many areas of mathematics - functional analysis and representation theory, to mention just two. ... Several important influences have brought about a change in this situation. One of the principal influences was the recognition - clearly established by the middle 1970s - of the central role of nonabelian gauge theory in elementary particle physics. The other main influence came from the emerging study of supersymmetry and string theory.
In his study of these areas of theoretical physics, Witten has achieved a level of mathematics which has led him to be awarded the highest honour that a mathematician can receive, namely a Fields Medal. He received the medal at the International Congress of Mathematicians which was held in Kyoto, Japan in 1990. The Proceedings of the Congress contains two articles describing Witten's mathematical work which led to the award. The main tribute is the article [3] by Atiyah, but Atiyah could not be in Kyoto to deliver the address so the address at the Congress was delivered by Faddeev [5] who quotes freely from Atiyah [3].
The first major contribution which led to Witten's Fields Medal was his simpler proof of the positive mass conjecture which had led to a Fields Medal for Yau in 1982. Gawedzki and Soulé describe this work by Witten, which appeared in 1981, in [9]:-
The proof ... employed in a subtle way the idea of supersymmetry. This became the centrepiece of many of Witten's subsequent works...
One of Witten's subsequent works was a paper which Atiyah singles out for special mention in [3], namely Supersymmetry and Morse theory which appeared in the Journal of differential geometry in 1984. Atiyah writes that this paper is:-
... obligatory reading for geometers interested in understanding modern quantum field theory. It also contains a brilliant proof of the classic Morse inequalities, relating critical points to homology. ... Witten explains that "supersymmetric quantum mechanics" is just Hodge-de Rham theory. The real aim of the paper is however to prepare the ground for supersymmetric quantum field theory as the Hodge-de Rham theory of infinite dimensional manifolds. It is a measure of Witten's mastery of the field that he has been able to make intelligent and skilful use of this difficult point of view in much of his subsequent work.
Since this highly influential paper, the ideas in it have become of central importance in the study of differential geometry. Further new ideas of fundamental importance were introduced by Witten and described in [9]:-
Witten subsequently gave a string interpretation of the elliptic genus and provided arguments for its rigidity ... Another piece of new mathematics stemmed from Witten's papers on global gravitational anomalies. ... In recent years, Witten focused his attention on topological quantum field theories. These correspond to Lagrangians ... formally giving manifold invariants. Witten described these in terms of the invariants of Donaldson and Floer (extending the earlier ideas of Atiyah) and generalised the Jones knot polynomial ...
The authors of [9] sum up Witten's contributions to mathematics:-
Although mostly not in the form of completed proofs, Witten's ideas have triggered major mathematical developments by the force of their vision and their conceptual clarity, his main discoveries soon becoming theorems. His Fields Medal at the 1990 International Congress of Mathematicians acknowledged the growing impact of his work on contemporary mathematics.
Atiyah, in [3], expresses the same ideas in the following way:-
... he has made a profound impact on contemporary mathematics. In his hands physics is once again providing a rich source of inspiration and insight in mathematics. Of course physical insight does not always lead to immediately rigorous mathematical proofs but it frequently leads one in the right direction, and technically correct proofs can then hopefully be found. This is the case with Witten's work. So far the insight has never let him down and rigorous proofs, of the standard we mathematicians rightly expect, have always been forthcoming.
Article by: J J O'Connor and E F Robertson
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Witten.html
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