超弦理論及丘空間

David^_^b

CUHK Math Lecture - 超弦理論及丘空間

講者：梁廼聰教授
日期：2008 年 2 月 2 日 (星期六)
時間：上午 10:30 至 中午 12:00
地點：逸夫書院大講堂
語言：粵語

http://www.math.cuhk.edu.hk/publect/lecture17/lecture17.html

LLH

科學家通常說卡拉比(Calabi)-丘流形，或卡拉比-丘空間。


以下節錄自Wikipedia中的"Calabi–Yau manifold"

Calabi-Yau manifolds are a special class of manifolds used in some branches of mathematics (such as algebraic geometry) as well as in theoretical physics. For instance, in superstring theory the extra dimensions of spacetime are sometimes conjectured to take the form of a 6-dimensional Calabi-Yau manifold.
...
A Calabi-Yau manifold of complex dimension n is also called a Calabi-Yau n-fold. The mathematician Eugenio Calabi conjectured in 1957 that all such manifolds admit a Ricci-flat metric (one in each Kähler class), and this conjecture was proved by Shing-Tung Yau (丘成桐) in 1977 and became Yau's theorem.

[ 本帖最後由 LLH 於 2008-1-23 10:19 編輯 ]

David^_^b

1980年代起逐漸建立的超弦理論，嘗試統合相對論和量子力學，使用了1970年代末證明了的 Calabi-Yau 空間作為數學工具。

1.相對論 - 宏觀世界的四維空間

三維歐氐空間 R3 - 長度 = (x^2 + y^2 + z^2)^(1/2)
三維閔氐空間 R(3,1) - 長度 = (x^2 + y^2 + z^2 - c^2 t^2)^(1/2)

2.量子力學 - 微觀世界

薛丁格方程式 - E Y = T Y + V Y

3.超對稱 - 物極必反，化繁為簡

「超」 - 複數的共軛互換
例如：費米子與玻色子互換
推廣應用：強作用(較難計算)與弱作用(較易計算)互換

複數的共軛互換(x + yi -> x - yi) - 二元數 (1, i)
複數的不可交換延伸(乘法不符合交換律) - 四元數 (1, i, j, k)
四元數的推廣 - 八元數 (1, i, j, k, l, m, n, o)

4.廣義相對論 - 愛因斯坦場方程 - Ricci張量 = 0 (物質在宏觀世界的四維空間)

5.超弦理論 (加入物質在微觀世界的六維空間, 共十維)

6.丘成同証明 - 丘空間

7. M 理論 (及新興的 F 理論) - 統合十維空間的超弦理論及十維超重力



[ 本帖最後由 David^_^b 於 2008-2-11 00:06 編輯 ]

citystory

Yau-Shing Tong and Chern Shing Shen are my idols. Are there presentation or video I can download from web? CUHK Physics did a good job for posting public lecture in their website but seems not for Maths department.

I just finish reading of the book  Theodore Frankel's "The Geometry of Physics, An Introduction". What an excellent book which covers from manifold, geometry to Fibre Bundle and Chern form with most important the geometry and  physics insight. This is not like those "Tautology" books.

In advance, I am exploring reading of some introduction to supersymmetry, Calabi-Yau manifold and String theory. Appreciate if anyone could suggest some introductory book or lecture notes to read. So far I could only find Brian R Greene's "String Theory on Calabi-Yau manifolds" which is good but not too funny to read.

LLH

多謝推薦好書，有空會找來看看。

丘成桐教授將在3月28日於中大主講「數學新浪潮」，題目是「幾何神奇：妙用理、工、醫」。可能與天文無直接關係，但如果對幾何有興趣，不妨一聽。(注意：請先到中大網頁 http://www.math.cuhk.edu.hk/newwave/ 登記報名)