顺手一贴：Simons，杨振宁和纤维丛
2011-10-15 06:23阅读：
顺手一贴：Simons，杨振宁和纤维丛
近日在“新雨丝”看到杨振宁先生的一篇纪念文章［1］， 谈到“七十年代我了解了规范场［Gauge
Theory］与数学家的纤维丛［Fiber bundle］的密切关系，了解了美妙的陈氏级［Chern
class］”以及“如果韦伊［Weil］1948年的猜想是对的，那么陈先生的开创性的陈氏级等数学工作的重要性就要旁及物理世界的最最基本结构了”。
这让我想起了前一段时间看到的一个采访录像，James Simons and C.N. Yang: Stony Brook
Masters Series[2] － 但对采访的最初了解还是从Woit的博客看到的［3］。
Jim Simons（1938－）一生传奇，学数学出身，早年毕业于MIT（本科）和UC
Berkeley（博士）。他一度在大学教过书，后来辞职去创立并管理一个“对冲基金”（Renaissance
Technologies），从中赚了大钱；现在退了休，集中精力做慈善［4］。
去年十二月，Simons作为杰出校友回MIT有个演讲，“
Mathematics,
Common Sense, and Good Luck: My Life and
Careers”。行者以前曾仔细读过江才健写的“杨振宁传”。所以，从此Simons的经历，行者立即想到他应该就是七十年代在纽约州大石溪分校（State
University of New York at Stony
Brook）的那个数学系主任Simons。果然，Woit的记录和那段采访，验证了行者的猜想。
They describe how Yang went to Simons to try and find
out about fiber bundles and what they might have to do with gauge
theory. Simons started by referring Yang to Steenrod’s The Topology
of Fibre Bundles, which Yang couldn’t
make any sense of (Simons admits he never made it all the way
through the book himself). This did in the end lead Simons and Yang
to some real understanding of how vector potentials in gauge theory
and connections on bundles were the same thing, with monopoles
examples of topologically non-trivial bundles. Simons lectured at
Stony Brook in 1975 on this, and a paper later that year by Wu and
Yang included what became known as the “Wu-Yang dictionary”
relating terminology in gauge theory and geometry. Singer learned
about this soon thereafter when he visited Stony Brook, and went on
to spread the news to Oxford, MIT and elsewhere （Woit）.
这一点，在杨振宁先生自己的“论文自选评论集”［5］里也有涉及：
It gradually became clear that field theorists must
learn the mathematical concept of fibre bundles. In early 1975, I
invited Jim Simons to give my colleages and me a series of luncheon
lectures on differential forms and fibre bundles. He kindly
accepted the invitation, and we learned about the Stoke's theorem,
the de Rham theorem, and so on. What we learned allowed us to
understand the mathematical meaning of the Bohm-Aharonov experiment
and the Dirac quantization rule of electric and magnetic monopoles.
T.T.Wu and I later also understood the profound and very general
Chern-Weil theorem. We appreciated that gauge fields have global
geometrical connections (not to be confused with physicists' global
phase factors) that are naturally formulated in terms of fibre
bundle concepts.
后来杨振宁先生由衷地写道：
七十年代我了解了规范场与数学家的纤维丛的密切关系，了解了美妙的陈氏级，写了一首诗《赞陈氏级》：“天衣岂无缝，匠心剪接成。浑然归一体，广邃妙绝伦。造化爱几何，四力纤维能。千古寸心事，欧高黎嘉陈。”
因为一直关注杨李两位大师，记于此备存。
（完）
NOTE
1，“
菩萨、量子数与陈氏级”， 杨振宁（原编者按
2011年10月28日是国际数学大师陈省身先生（1911-2004）百年诞辰，他创立的南开大学陈省身数学研究所及美国国家数学研究所（MSRI）届时将联合举办纪念会议。陈先生的学生和好友、诺贝尔物理学奖得主杨振宁先生为此撰写了一篇纪念文章，并授权《物理》杂志及《中国青年报》同时刊发。）
2, James Simons and C.N. Yang:
Stony Brook Masters
Series。
3,
Interview With Simons and Yang, Peter Woit (Not Even
Wrong)
4，On December 9, 2010, MIT and the Dean of Science welcomed Dr. Jim
Simons back to campus to give the School of Science Dean’s
Colloquium entitled, “Mathematics, Common Sense, and Good Luck: My
Life and Careers.” Simons talked about his careers first as a
mathematician, then as founder and CEO of a successful hedge fund,
and now as a philanthropist （Excerpt from Science @ MIT, Spring
2011）.
5,
Selected papers, 1945-1980, with commentary
By Chen Ning Yang (Google Book)
附图：“Selected papers, 1945-1980, with commentary”（局部）
