Problem 14.3b6

Quick Calculation 14.4b | A First Course in String Theory

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Massive level in the open superstring

Consider the first and second excited levels of the open superstring (\alpha' M^2 = 1 and \alpha' M^2 = 2). List the states in NS sector and the states in the R sector. Confirm that you get the same number of states.

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When \alpha' M^2 = 2, by Equation (14.54), the possible states are

\{ \alpha_{-2}^I, \alpha_{-1}^I \alpha_{-1}^J, d_{-1}^I d_{-1}^J \} | R_a \rangle, || ...

\{ \alpha_{-1}^I d_{-1}^J, d_{-2}^I \} | R_{\bar a} \rangle, || ...

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For \alpha_{-2}^I | R_a \rangle, the number of states is 8.

For \alpha_{-1}^I \alpha_{-1}^J | R_a \rangle, the number of states is \frac{8 \times 7}{2} + 8 = 36.

For d_{-1}^I d_{-1}^J | R_a \rangle, the number of states is \frac{8 \times 7}{2} = 28.

For \alpha_{-1}^I d_{-1}^J | R_{\bar a} \rangle, the number of states is 8 \times 8 = 64.

For d_{-2}^I | R_{\bar a} \rangle, the number of states is 8.

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However, since each of a and {\bar a} has 8 possible values, there is an additional multiple of 8.

The total number of states is 8 \left[ 8 + 36 + 28 + 64 + 8 \right].

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You can check this answer against Equation (14.67):

f_{NS} (x) = \frac{1}{\sqrt{x}} + 8 + 36 \sqrt{x} + 128 x + 402 x \sqrt{x} + 1152 x^2 + ...

— Me@2018.02.20 10:57 AM

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2018.02.20 Tuesday (c) All rights reserved by ACHK

The language of Change 1.2

Energy conservation, 6.2 | Energy 5.2

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time ~ change

energy ~ the ability of _keeping_ changing

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constant velocity ~ the amount of an object’s change of position, measured with respect to its observer’s unit of change, is constant

s = \Delta x

v = \frac{s}{\Delta t} = \frac{\Delta x}{\Delta t}

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kinetic energy ~ the amount of the ability of keeping changing an object’s position

\frac{1}{2} m v^2 ~ the square of (the amount of change of position, relative to the observer’s unit of change)

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Energy difference is _not_ exactly a measurement of the amount of change, time interval is.

— Me@2018-02-20 09:39:30 AM

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2018.02.20 Tuesday (c) All rights reserved by ACHK

潛行凶間 16

Inception 16

這段改編自 2010 年 8 月 13 日的對話。

d_2018_02_18__19_10_12_PM_

這幾個月來,我跟一位來自大學時代的朋友,討論一些較為深刻精采,一般人也不會討論的話題。怎料,《潛行凶間》竟然囊括那些深刻精采話題,之中的一大部分。

在我朋友的同意和鼓勵下,我把那些對話錄了音,再化成文字。那就彷彿是把那些意念,由氣體凝結成液體。怎料,該電影卻把那些意念,再由液體凝固成固體。

我把對話化成文字,莫非都是為了避免,同一堆說話,再講一次。同一堆說話講第二次,並不是一個快樂的過程。有了文字版後,當有其他朋友需要那些意念時,我就可以引述我的文章,從而開發下一個層次的話題。

但是,有了《潛行凶間》這電影後,凡是之中有的意念,我的相關文章,都可以一概置之不理;因為,我只要介紹該電影,給對那些意念,有興趣的朋友就可以。

電影內,有太多超凡的意念,不能一次過講得完。你們可以分開幾次來問我,例如:

1. 清醒夢

1.1 有些人在某些時候,在夢知道自己在發夢,卻又可以保持住,發夢的狀態。

1.2 那些人之中的部分人,在那些清醒夢時候的部分時候,甚至可以控制著,那些夢境的劇情演變。

你有沒有試過,在夢裡面,知道自己在發夢?

(CPK:未。不過,我的姐姐試過。)

我中學時代,有一次清醒夢時,刻意留意一棵樹。

我發現,原來夢中的東西,不只有黑白色。我不單看到,那棵樹的葉是綠色的,而且,還可以見到,那些樹葉中的每一塊。

《潛行凶間》中的意念,你可以假想,有七成是真的。

又例如:

2. 多層夢

有時,明明已經起了床,關掉了鬧鐘;怎料,「明明已經起了床,關掉了鬧鐘」這劇情,只是另一個夢 — 我仍然未醒來。

3. 多重自我

3.1 每一個人,其實有超過一個自我。

3.2 而每一個自我,其實有超過一個層次的意識。

— Me@2018.02.18

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2018.02.18 Sunday (c) All rights reserved by ACHK

Creative constraints

Imagine you were asked to invent something new. It could be whatever you want, made from anything you choose, in any shape or size. That kind of creative freedom sounds so liberating, doesn’t it? Or … does it?

If you’re like most people you’d probably be paralyzed by this task. Why?

Brandon Rodriguez explains how creative constraints actually help drive discovery and innovation.

With each invention, the engineers demonstrated an essential habit of scientific thinking – that solutions must recognize the limitations of current technology in order to advance it.

Understanding constraints guides scientific progress, and what’s true in science is also true in many other fields.

Constraints aren’t the boundaries of creativity, but the foundation of it.

— The power of creative constraints

— Lesson by Brandon Rodriguez

— animation by CUB Animation

— TED-Ed

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We cannot change anything until we accept it. Condemnation does not liberate, it oppresses.

— Carl Jung

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2018.02.17 Saturday (c) All rights reserved by ACHK

Twelve-step program

A twelve-step program is a set of guiding principles outlining a course of action for recovery from addiction, compulsion, or other behavioral problems. Originally proposed by Alcoholics Anonymous (AA) as a method of recovery from alcoholism, the Twelve Steps were first published in the 1939 book Alcoholics Anonymous: The Story of How More Than One Hundred Men Have Recovered from Alcoholism. The method was adapted and became the foundation of other twelve-step programs.

As summarized by the American Psychological Association, the process involves the following:

– admitting that one _cannot_ control one’s alcoholism, addiction or compulsion;

– recognizing a higher power that can give strength;

– examining past errors with the help of a sponsor (experienced member);

– making amends for these errors;

– learning to live a new life with a new code of behavior;

– helping others who suffer from the same alcoholism, addictions or compulsions.

— Wikipedia on Twelve-step program

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We cannot change anything until we accept it. Condemnation does not liberate, it oppresses.

— Carl Jung

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2018.02.17 Saturday (c) All rights reserved by ACHK

深淵 2

與魔鬼戰鬥的人,應當小心自己不要成為魔鬼。當你遠遠凝視深淵時,深淵也在凝視你。

— 尼采

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勇者鬥惡龍,小心勇者變惡龍。

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As soon as men decide that all means are permitted to fight an evil, then their good becomes indistinguishable from the evil that they set out to destroy.

— Christopher Dawson

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2018.02.16 Friday ACHK

Problem 14.3b5

Quick Calculation 14.4b | A First Course in String Theory

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Massive level in the open superstring

Consider the first and second excited levels of the open superstring (\alpha' M^2 = 1 and \alpha' M^2 = 2). List the states in NS sector and the states in the R sector. Confirm that you get the same number of states.

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When \alpha' M^2 = 2, by Equation (14.37),

M^2 = \frac{1}{\alpha'} \left( - \frac{1}{2} + N^\perp \right)

N^\perp = \frac{5}{2}

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\{ b_{-1/2}^I b_{-1/2}^J b_{-1/2}^K b_{-1/2}^L b_{-1/2}^M,

b_{-3/2}^I b_{-1/2}^L b_{-1/2}^M,

b_{-5/2}^I,

\alpha_{-1}^I b_{-1/2}^J b_{-1/2}^K b_{-1/2}^M,

\alpha_{-1}^I b_{-3/2}^J,

\alpha_{-2}^I b_{-1/2}^J,

\alpha_{-1}^I \alpha_{-1}^J b_{-1/2}^K \} |NS \rangle \otimes |p^+, \vec p_T \rangle

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There are \frac{8 \times 7 \times 6 \times 5 \times 4}{5!} = {8 \choose 5} = 56 states for

\{ b_{-1/2}^I b_{-1/2}^J b_{-1/2}^K b_{-1/2}^L b_{-1/2}^M |NS \rangle \otimes |p^+, \vec p_T \rangle

There are 8 \times \frac{8 \times 7}{2} = 224 states for

\{ b_{-3/2}^I b_{-1/2}^L b_{-1/2}^M \} |NS \rangle \otimes |p^+, \vec p_T \rangle

There are 8 states for

\{ b_{-5/2}^I \} |NS \rangle \otimes |p^+, \vec p_T \rangle

There are 8 \times {8 \choose 3} = 448 states for

\{ \alpha_{-1}^I b_{-1/2}^J b_{-1/2}^K b_{-1/2}^M \} |NS \rangle \otimes |p^+, \vec p_T \rangle

There are 8 \times 8 = 64 states for

\{ \alpha_{-1}^I b_{-3/2}^J \} |NS \rangle \otimes |p^+, \vec p_T \rangle

There are 8 \times 8 = 64 states for

\{ \alpha_{-2}^I b_{-1/2}^J \} |NS \rangle \otimes |p^+, \vec p_T \rangle

There are \left( \frac{8 \times 7}{2!} + 8 \right) \times 8 = 288 states for

\{ \alpha_{-1}^I \alpha_{-1}^J b_{-1/2}^K \} |NS \rangle \otimes |p^+, \vec p_T \rangle

So the total number of states is 56 + 224 + 8 + 448 + 64 + 64 + 288 = 1152.

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You can check this answer against Equation (14.67):

f_{NS} (x) = \frac{1}{\sqrt{x}} + 8 + 36 \sqrt{x} + 128 x + 402 x \sqrt{x} + 1152 x^2 + ...

— Me@2018-02-16 03:22:13 PM

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2018.02.16 Friday (c) All rights reserved by ACHK

The language of Change

Energy conservation, 6 | Energy 5

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time ~ change

energy ~ the ability of causing change

Assuming

1. a system of one single particle

2. has only kinetic energy

3. and that kinetic energy is conserved.

conservation of energy ~ an object’s potential amount of change of position, measured with respect to its observer’s unit of change, is constant

s = \Delta x

v = \frac{s}{\Delta t} = \frac{\Delta x}{\Delta t}

— Me@2018-02-15 02:21:20 PM

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Note:

The above argument has a bug:

If the mass m is constant, the kinetic energy E_K should be proportional to velocity squared v^2, instead of velocity v.

E_K = \frac{1}{2} m v^2

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However, the above argument is still technically correct:

When E_K is constant, v^2 is constant. In turn, the magnitude of v also remains unchanged.

— Me@2018-02-19 09:37:24 PM

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2018.02.15 Thursday (c) All rights reserved by ACHK

機遇再生論 1.6

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所以,「同情地理解」,亦可稱為「意念淘金術」。

機遇再生論,可以同情地理解為,有以下的意思:

(而這個意思,亦在「機遇再生論」的原文中,用作其理據。)

假設,你現在手中,有一副樸克牌,存在於某一個排列 A 。洗牌一次之後,排列仍然是 A 的機會極微。

一副完整的撲克牌,共有 N = 52! \approx 8.07 \times 10^{67} 個,可能的排列。亦即是話,洗牌後仍然是排列 A 的機會率,只有 \frac{1}{N}

由於分母 N 太大(相當於 8 之後,還有 67 個位),洗牌後,理應變成另外一個排列 B 。

P(A) = \frac{1}{N}

P(\text{not} A) = 1 - \frac{1}{N}

洗了一次牌後,發覺排列是 B 不是 A 後,我們可以再問,如果再洗一次牌,「是 A」和「不是 A」的機會,分別是多少?

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由於,機會率只是與未知的事情有關,或者說,已知的事件,發生的機會率必為 1;所以,如果發生了第一次洗牌,而你又知道其結果的情況下,問「如果再洗一次牌,『是 A』和『不是 A』的機會,分別是多少」,第二次洗牌各個可能結果,發生的機會率,與第一次洗牌的結果無關。

第二次洗牌結果為組合 A 的機會率,仍然是

P(A) = \frac{1}{N}

不是組合 A 的機會率,仍然是

P(\text{not} A) = 1 - \frac{1}{N}

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(問:那樣,為什麼要問多一次呢?)

我是想釐清,我真正想問的是,並不是這個問題,而是另一個:

如果在第一次洗牌之前,亦即是話,一次牌都未洗的話,問:

「如果洗牌兩次,起碼一次洗到原本排列 A 的機會率是多少?」

把該事件標示為 A_2

A_2 = 兩次洗牌的結果,起碼一次洗到原本排列 A

再把該事件的機會率,標示為 P(A_2)

由於 P(A_2) 相對麻煩,我們可以先行運算其「互補事件」的機會率。

A_2 的互補事件為「不是 A_2」:

不是 A_2

= 兩次洗牌的結果,不是起碼一次洗到原本排列 A

= 兩次洗牌的結果,都不是排列 A

其機會率為

P(\text{not} A_2) = (1 - \frac{1}{N})^2

那樣,我們就可推斷,

P(A_2)
= 1 - P(\text{not} A_2)
= 1 - (1 - \frac{1}{N})^2

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同理,在一次牌都未洗的時候,問:

如果洗牌 m 次,起碼一次洗到原本排列 A 的機會率是多少?

答案將會是

P(A_m)= 1 - (1 - \frac{1}{N})^m

留意,N = 52! \approx 8.07 \times 10^{67},非常之大,導致 (1 - \frac{1}{N}) 極端接近 1。在一般情況,m 的數值還是正常時, P(A_m) 會仍然極端接近 0。

例如,你將會連續洗一千萬次牌(m = 10,000,000),起碼有一次,回到原本排列 A 的機會是:

P(A_m)
= 1 - (1 - \frac{1}{N})^m
= 1 - (1 - \frac{1}{52!})^{10,000,000}

你用一般手提計算機的話,它會給你 0。你用電腦的話,它會給你

1.239799930857148592 \times 10^{-61}

— Me@2018-01-25 12:38:39 PM

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2018.02.13 Tuesday (c) All rights reserved by ACHK

Riemann Surfaces

Imaginary Numbers Are Real [Part 1: Introduction]

Imaginary Numbers Are Real [Part 2: A Little History]

Imaginary Numbers Are Real [Part 3: Cardan’s Problem]

Imaginary Numbers Are Real [Part 4: Bombelli’s Solution]

Imaginary Numbers Are Real [Part 5: Numbers are Two Dimensional]

Imaginary Numbers Are Real [Part 6: The Complex Plane]

Imaginary Numbers Are Real [Part 7: Complex Multiplication]

Imaginary Numbers Are Real [Part 8: Math Wizardry]

Imaginary Numbers Are Real [Part 9: Closure]

Imaginary Numbers Are Real [Part 10: Complex Functions]

Imaginary Numbers Are Real [Part 11: Wandering in 4 Dimensions]

Imaginary Numbers Are Real [Part 12: Riemann’s Solution]

Imaginary Numbers Are Real [Part 13: Riemann Surfaces]

— Welch Labs

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In case the original videos are lost, please use the Internet Archive link:

https://web.archive.org/web/20170714105446/https://www.youtube.com/watch?v=T647CGsuOVU

— Me@2018-02-12 02:14:51 PM

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2018.02.12 Monday (c) All rights reserved by ACHK

Eyes on me

dziungles 14 days ago

Hey, this is really cool to see natural eyesight topic on the hacker news.
I practice this for more than 10 years. Each day I work with computer for ~10 h., drive a car and do other things, and never wear glasses, even though the traditional ophtalmologic measures clearly indicate that I need strong glasses and I shouldn’t see even the biggest letter on the Snellen chart, but I see not only the biggest, but sometimes even the 20/20.

Doctors can’t explain this, and only congrats me on my achievement. Of course, the eyesight is not perfect. I see clearly in the daytime, but in the nighttime or low light conditions it becomes much harder to distinguish faces.

The best book I found so far is “Relearning to See” by Thomas R. Quackenbush. The originator of this theory was William Bates.

Actually, there is no clear unified theory on how to achieve this. Everyone interprets it differently and the results are inconsistent. There is also a lot of criticism from the medical establishment.

Natural eyesight improvement really works. And the unified theory, in a form of an app, or a good book, maybe including findings from neuroplasticity, would be a great gift for humanity.

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bsder 14 days ago

Natural eyesight improvement really works. And the unified theory, in a form of an app, or a good book, maybe including findings from neuroplasticity, would be a great gift for humanity.

It SORT OF works in stable and predictable situations.

What seems to be happening is that your brain, in all of its neuroplastic glory, is learning to make better inferences from the broken information it receives.

The issue is that this works as long as the inferences are correct. That’s fine when you are reading a newspaper, using the computer, etc. as the situation is stable and predictable.

The problem is that when you are suddenly confronted by a situation where the inferences are NOT correct–such as a nighttime emergency situation while driving. Now you are relying on the “uninferenced” data coming in from your eyes and that data is subpar with all the resultant problems.

The best solution is both: fix the data coming in with corrective lenses for unpredictable situations, and train your brain to make better inferences so you can deal with predictable situations better.

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dziungles 14 days ago

Before, I also had this theory in my mind for a year or so.
But for example, right now I’m in Thailand, traveling here for the first time. Everything is new, unfamiliar and unpredictable, daytime, nightime. I have no problems seeing things, everything is almost perfectly clear and sharp. I needed some time to adapt to a smaller screen of my laptop (I was using 24 inch before), but now I’m doing fine.

Actually, the more you look, the better you see. Like in Aaron Swartz blog post, if you want to retrain your weak legs, you need to walk more. Same with eyes.

https://web.archive.org/web/20180204094207/https://news.ycombinator.com/item?id=16194580

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2018.02.09 Friday ACHK

Logical arrow of time, 6.2

Source of time asymmetry in macroscopic physical systems

Second law of thermodynamics

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Physics is not about reality, but about what one can say about reality.

— Bohr

— paraphrased

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Physics should deduce what an observer would observe,

not what it really is, for that would be impossible.

— Me@2018-02-02 12:15:38 AM

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1. Physics is about what an observer can observe about reality.

2. Whatever an observer can observe is a consistent history.

observer ~ a consistent story

observing ~ gathering a consistent story from the quantum reality

3. Physics [relativity and quantum mechanics] is also about the consistency of results of any two observers _when_, but not before, they compare those results, observational or experimental.

4. That consistency is guaranteed because the comparison of results itself can be regarded as a physical event, which can be observed by a third observer, aka a meta observer.

Since whenever an observer can observe is consistent, the meta-observer would see that the two observers have consistent observational results.

5. Either original observers is one of the possible meta-observers, since it certainly would be witnessing the comparison process of the observation data.

— Me@2018-02-02 10:25:05 PM

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2018.02.03 Saturday (c) All rights reserved by ACHK

War and Peace

Is peace better than war?

Peace is acceptance of the status quo

You cannot answer the question of whether peace is better than war without reference to the status quo. Preserve peace is another way of saying accept the status quo. Accepting the status quo is OK only if the status quo is acceptable. Sometimes, it is not.

Diplomacy is meaningless unless it is backed by potential use of force.

Nations negotiate diplomatically to prevent war. Were there no threat of war behind international negotiations, each side could just say “No” to the other forever

Objecting to all wars is not an option for anyone unless the objector is protected by someone who does not object to all wars. No one should be willing to risk his life for such people.

https://web.archive.org/web/20180130120844/https://johntreed.com/blogs/john-t-reed-s-blog-about-military-matters/68514755-is-peace-better-than-war

— Is peace better than war?

— John Reed

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2018.01.30 Tuesday ACHK