Here I would like to propose another possibility, namely that quantum theory will make more sense when regarded as part of a theory of spacetime. Furthermore, I claim that we can only see this from a category-theoretic perspective — in particular, one that de-emphasizes the primary role of the category of sets and functions.
— Quantum Quandaries: A Category-Theoretic Perspective, John C. Baez
2010.02.23 Tuesday ACHK
Since n-categories are purely algebraic structures but have a natural relationship to the study of n-dimensional spacetime, their study is sometimes called `higher-dimensional algebra’.
— Higher-Dimensional Algebra and Planck-Scale Physics, John C. Baez
2010.02.22 Monday ACHK
With regard to the issues related to the distinction between analytic and synthetic propositions, Kant and the logical positivists agreed about what “analytic” and “synthetic” meant. This would only be a terminological dispute. Instead, they disagreed about whether knowledge of mathematical and logical truths could be obtained merely through an examination of one’s own concepts. The logical positivists thought that it could be. Kant thought that it could not.
— Wikipedia on Analytic-synthetic distinction
2010.02.20 Saturday ACHK
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity and invertibility.
— Wikipedia on Group (mathematics)
2010.02.19 Friday ACHK
This is my first post in , using Google Chart API.
2010.02.18 Thursday (c) ACHK
A supersymmetry relating mesons and baryons was first proposed, in the context of hadronic physics, by H Miyazawa in 1966, but his work was ignored at the time. In the early 1970s, J. L. Gervais and B. Sakita (in 1971), Yu. A. Golfand and E.P. Likhtman (also in 1971), D.V. Volkov and V.P. Akulov (in 1972) and J. Wess and B. Zumino (in 1974) independently rediscovered supersymmetry, a radically new type of symmetry of spacetime and fundamental fields, which establishes a relationship between elementary particles of different quantum nature, bosons and fermions, and unifies spacetime and internal symmetries of the microscopic world.
— Wikipedia on Supersymmetry
2010.02.16 Tuesday ACHK
A part of A is a monomapping i with codomain A.
— Me
2010.02.15 Monday (c) All rights reserved by ACHK
The general theory of analytic continuation and its generalizations are known as sheaf theory.
— Wikipedia on Analytic continuation
2010.02.14 Sunday ACHK
The concept of a universal cover was first developed to define a natural domain for the analytic continuation of an analytic function. The idea of finding the maximal analytic continuation of a function in turn led to the development of the idea of Riemann surfaces.
— Wikipedia on Analytic continuation
2010.02.13 Saturday ACHK
別人明不明白高深的學問，很視乎你的包裝技巧。當然，不是任何高深的學問都可以這樣包裝。你不可以將 相對論 包裝成只有一兩句，令人很易理解又不會導致威力降低。
（安：可以包裝的，當然不會做成困擾。但是超凡的人，不會只是有一些容易包裝的高深學問，他們還會有一些類似 相對論 難度，但又不易包裝的學問。哪應怎麼辦呢？）
不說便行。既然是類似 相對論 難度的東西，通常即是凡人不需要的東西。那我不教他們，他們也沒有損失。
如果有一些超難又不易包裝，剛巧是一般人也需要的東西的話，整個社會自然會有一整個系統去輔助平民百姓去理解它。例如，在以前的世界，一般人是不需要懂邏輯的 … 起碼不需要太複雜的邏輯。但是，現代世界有了電腦，導致需要很多人寫程式。而寫程式所需要邏輯頭腦，比邏輯課時所需要的，還要複雜幾倍。所以，現代社會自然會有很多教育機構開設，教大眾寫程式。這又是一個自然定律。
（安：這個想法很有趣。）
如果去到 2100 年，世界上大部人都需要 相對論 的話，中學公開試的課程自然會包括 相對論，亦自然會有很多補習社教人 相對論。你試想想，現在一個中學生的數學知識，已多過一個十七世紀數學家的數學知識。
— Me@2010.02.11
2010.02.12 Friday (c) All rights reserved by ACHK
（安：根據你的教學經驗 或者 你與其他人相處經驗，如果你講的東西，對方吸收不到，你會不會覺得很氣餒？）
大概都沒有什麼大不了… 當你學養足夠高時，自然大部分人不明白你的說話，這個是自然定律。（因為 ”超凡” 的意思就是 “超越凡人”，所以如果你是超凡的話，凡人自然不明白你。）
（安：但是，你的（教學）目的，就是要一般人也能明白你教的東西。）
那我就說一些沒有那麼高深的東西 …
或者那些高深的東西可以包裝成貌似淺白的東西，令人容易理解。
（安：這是一個常用的方法，將原本高深的東西包裝到很易理解、很大眾化。又或者用一種比喻的方法。）
但又不要減輕它的威力。
（安：但是這種方法的代價是，當中的 insight（洞察）會跌了很多倍。）
不一定。那要視乎你的教學技巧的高低。
其中一個技巧是，你將原本難以理解的事情，用 ”談戀愛” 來比喻。
例如，如果我說： ”有些事情，雖然適合做你的興趣，但是未必適合做你的事業。” 你未必明白我這句是什麼意思。
但是，如果我說： ”有些人適合做你的朋友，但是未必適合做你的情人；就算適合做你的情人，亦未必適合做你的太太。” 那就比較容易明白。
別人明不明白，很視乎你的包裝技巧。
當然，不是任何高深的學問都可以這樣包裝。你不可以將 相對論 包裝成只有一兩句，令人很易理解又不會導致威力降低。
— Me@2010.02.11
2010.02.12 Friday (c) All rights reserved by ACHK
Analytic continuations are unique in the following sense: if V is the connected domain of two analytic functions F1 and F2 such that U is contained in V and for all z in U
F1(z) = F2(z) = f(z),
then
F1 = F2
on all of V. This is because F1 – F2 is an analytic function which vanishes on the open, connected domain U of f and hence must vanish on its entire domain. This follows directly from the identity theorem for holomorphic functions.
— Wikipedia on Analytic continuation
2010.02.12 Friday ACHK
Two dimensional time 6.1
.
Physics
Special relativity describes spacetime as a manifold whose metric tensor has a negative eigenvalue. This corresponds to the existence of a “time-like” direction. A metric with multiple negative eigenvalues would correspondingly imply several timelike directions, i.e. multiple time dimensions, but there is no consensus regarding the relationship of these extra “times” to time as conventionally understood.
Philosophy
An Experiment with Time by J.W. Dunne (1927) describes an ontology in which there is an infinite hierarchy of conscious minds, each with its own dimension of time and able to view events in lower time dimensions from outside. His theory was often criticised as exhibiting an unnecessary infinite regress.
— Wikipedia on Multiple time dimensions
.
.
2010.02.11 Thursday ACHK
超凡的人（時常）會以為自己無困難的東西是理所當然地無困難的。他們以為其他人都和他們一樣，所以其他人並不需要特意向他們學習。有時，超凡的才能甚至成為了超凡的人潛意識的一部分。他們運用才能時，連想也不用想，導致他們運用了超凡的才能也不知道。
例如，我以前有一科的老師本身學科的知識學養很出眾。但是，很多時候，我問他問題時，他一開始答，就答（例如）第十二步的東西。他沒有講開頭的十一步，導致我很難理解他的答案。哪為什麼他不說之前的十一步？是不是他特意為難我呢？
不是。我估計原因是，他覺得頭十一步太明顯，不需要教。
（安：甚至他不知道有頭十一步。你的第十二步，其實是他的第一步。）
我以前教書時都有這個現象。所以教學經驗很重要。它可以令你知道其他人其實感受不到你以為很明顯的東西。教學天份和學術天份本身是兩樣東西。例如教物理，要教得好的話，是需要兩種天份合在一起，變成一種完整的 “教物理” 天份。我是教了一年書後，才可以有辦法知道你層次在哪裏、你在哪一步開始不明白、對你來說怎才算是 “一步”。然後，將那些步驟拆到足夠細小，適合你吸收為止。
（安：所以如果用這個準則在衡量老師的好壞，根據自然定律，好的老師是極少數。那是因為要一個人本身學養高，機會已經很小。而同時要他教學技巧好，機會就更小。）
我以前也向學生講過類似的說話。如果每一萬人有一個人物理好。而每一萬人有一個人教學好。那樣，教物理教得好的人，每一億人就只有一個。
— Me@2010.02.10
2010.02.10 Wednesday (c) All rights reserved by ACHK
The holographic principle states that the entropy of ordinary mass (not just black holes) is also proportional to surface area and not volume; that volume itself is illusory and the universe is really a hologram which is isomorphic to the information “inscribed” on the surface of its boundary.
— Wikipedia on Holographic principle
2010.02.09 Tuesday ACHK
Magician
“搞 gag”（弄笑話）在大部分情況下是學不到的。哪小部分情況下怎樣學呢？我不知道。
還有，最出奇的地方是，到你學到時，那些 gag 就已經不再是 gag 了。如果你弄一個 system（系統）教人如何搞 gag，其實就已經不能再搞 gag 了。搞 gag 的意思是對觀眾講一些好 unexpected（突然）但又好有趣的 point（點）。但是當一大班人學到如何搞 gag，而可以 mechanically generate （機械式製造）出來的話，那就已經不是 gag 了，因為那已不再 unexpected。
以下的內容，好似和以上的內容自相矛盾，其實沒有。
搞 gag 是有方法的。我知道一些方法。但是，我不是因為知道那些方法，所以識搞 gag。相反，我是識搞 gag，然後觀察自己在想什麼：”哦 … 原來我用了這方法。” 即是說，我不是學回來的。
第二，我知道這些方法，但我不會說出來。魔術精采的原因是，魔術師不會公開自己的魔術方法。搞 gag 的難度和搞魔術差不多：你知道方法後，就不再精采了。
第三，即使有一個原本不懂搞 gag 的人，透過用我所知的方法，訓練到自己識搞 gag，他也不容易得到 搞 gag 時候所需的神髓。一舉手、一投足，會比先天懂搞 gag 的人差一點。
— Me@2010.02.07
2010.02.08 Monday (c) All rights reserved by ACHK
Shannon’s efforts to find a way to quantify the information contained in, for example, an e-mail message, led him unexpectedly to a formula with the same form as Boltzmann’s. Bekenstein summarizes that “Thermodynamic entropy and Shannon entropy are conceptually equivalent: the number of arrangements that are counted by Boltzmann entropy reflects the amount of Shannon information one would need to implement any particular arrangement…” of matter and energy.
— Wikipedia on Holographic principle
2010.02.08 Monday ACHK
Theodore A. (Ted) Jacobson (born November 27, 1954) is an American theoretical physicist. He is known for his work on the connection between gravity and thermodynamics. In particular, in 1995 Jacobson proved that the Einstein field equations describing relativistic gravity can be derived from thermodynamic considerations.
— Wikipedia on Theodore Jacobson
2010.02.07 Sunday ACHK
This theory combines the thermodynamic approach to gravity with ‘t Hooft’s holographic principle. If proven correct, gravity is not a fundamental force, but an emergent phenomenon which arises from the statistical behaviour of microscopic degrees of freedom encoded on a holographic screen.
— Wikipedia on Gravity as an entropic force
2010.02.06 Saturday ACHK
《漫畫榮格》
間書的一個極端是一句也不間。那我們就不知哪些是重要句子。
間書的另一個極端是句句間。那我們也不知哪些是重要句子。
— Me@2003-2004
2007.11.18 Sunday 2010.02.05 Friday (c) CHK2 ACHK
Physics Town 
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