Problem 2.2a

A First Course in String Theory

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2.2 Lorentz transformations for light-cone coordinates.

Consider coordinates \displaystyle{x^\mu = ( x^0, x^1, x^2, x^3 )} and the associated light-cone coordinates \displaystyle{x^\mu = ( x^+, x^-, x^2, x^3 )}. Write the following Lorentz transformations in terms of the light-cone coordinates.

(a) A boost with velocity parameter \displaystyle{\beta} in the \displaystyle{x^1} direction.

\displaystyle{ \begin{aligned} \begin{bmatrix} c t' \\ x' \\ y' \\ z' \end{bmatrix}  &= \begin{bmatrix} \gamma&-\beta \gamma&0&0\\ -\beta \gamma&\gamma&0&0\\ 0&0&1&0\\ 0&0&0&1\\ \end{bmatrix} \begin{bmatrix} c\,t \\ x \\ y \\ z \end{bmatrix} \\   \end{aligned}}

~~~

\displaystyle{ \begin{aligned} \begin{bmatrix} x^+ \\ x^- \end{bmatrix}  &= \frac{1}{\sqrt{2}}     \begin{bmatrix}        1 & 1 \\        1 & -1 \\     \end{bmatrix}     \begin{bmatrix} x^0 \\ x^1 \end{bmatrix}    \end{aligned}}

The matrix is its own inverse.

\displaystyle{ \begin{aligned} \begin{bmatrix} x^0 \\ x^1 \end{bmatrix}  &=  \frac{1}{\sqrt{2}}     \begin{bmatrix}        1 & 1 \\        1 & -1 \\     \end{bmatrix}  \begin{bmatrix} x^+ \\ x^- \end{bmatrix} \\   \end{aligned}}

\displaystyle{ \begin{aligned} \begin{bmatrix} (x^0)' \\ (x^1)' \end{bmatrix}  &=     \begin{bmatrix}        \frac{1}{\sqrt{2}}  & \frac{1}{\sqrt{2}}  \\        \frac{1}{\sqrt{2}}  & -\frac{1}{\sqrt{2}}  \\     \end{bmatrix}  \begin{bmatrix} (x^+)' \\ (x^-)' \end{bmatrix} \\   \end{aligned}}

Apply the result to the original transformation:

\displaystyle{ \begin{aligned} \begin{bmatrix} (x^0)' \\ (x^1)' \\ (x^2)' \\ (x^3)' \end{bmatrix}  &= \begin{bmatrix} \gamma&-\beta \gamma&0&0\\ -\beta \gamma&\gamma&0&0\\ 0&0&1&0\\ 0&0&0&1\\ \end{bmatrix} \begin{bmatrix} x^0 \\ x^1 \\ x^2 \\ x^3 \end{bmatrix} \\   \end{aligned}}

\displaystyle{ \begin{aligned}    \begin{bmatrix}        \frac{1}{\sqrt{2}}  & \frac{1}{\sqrt{2}} & 0 & 0 \\        \frac{1}{\sqrt{2}}  & -\frac{1}{\sqrt{2}} & 0 & 0 \\                         0  & 0 & 1 & 0 \\                         0  & 0 & 0 & 1 \\     \end{bmatrix}  \begin{bmatrix} (x^+)' \\ (x^-)' \\ y' \\ z' \end{bmatrix}  &= \begin{bmatrix} \gamma&-\beta \gamma&0&0\\ -\beta \gamma&\gamma&0&0\\ 0&0&1&0\\ 0&0&0&1\\ \end{bmatrix}     \begin{bmatrix}        \frac{1}{\sqrt{2}}  & \frac{1}{\sqrt{2}} & 0 & 0 \\        \frac{1}{\sqrt{2}}  & -\frac{1}{\sqrt{2}} & 0 & 0 \\                         0  & 0 & 1 & 0 \\                         0  & 0 & 0 & 1 \\     \end{bmatrix}  \begin{bmatrix} x^+ \\ x^- \\ y \\ z \end{bmatrix}  \end{aligned}}

\displaystyle{ \begin{aligned}    \begin{bmatrix}        1  & 1 \\        1  & -1 \\     \end{bmatrix}  \begin{bmatrix} (x^+)' \\ (x^-)' \end{bmatrix}  &= \begin{bmatrix}       \gamma & -\beta \gamma \\       -\beta \gamma &\gamma       \end{bmatrix}     \begin{bmatrix}        1  & 1 \\        1  & -1 \\     \end{bmatrix}  \begin{bmatrix} x^+ \\ x^- \end{bmatrix}  \end{aligned}}

\displaystyle{ \begin{aligned} \begin{bmatrix} (x^+)' \\ (x^-)' \end{bmatrix}  &= \begin{bmatrix}        \gamma (1-\beta) & 0 \\        0 & \gamma (1+\beta) \\     \end{bmatrix}  \begin{bmatrix} x^+ \\ x^- \end{bmatrix}  \end{aligned}}

— Me@2020-02-27 07:14:19 PM

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2020.02.27 Thursday (c) All rights reserved by ACHK

Superposition always exists, 2

Decoherence means that the different components in the superposition do not interact with each other, but it does not mean that the components separate to form different macroscopic realities.

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Just like when a 100-soldier army’s marching gets interrupted, the decoherent soldiers do not form a single army anymore, because their actions become out of sync.

However, they do not become 100 armies either.

Instead, they form a group of 100 random people in the street.

Although now they are out of sync with each other, all original soldiers still exist, forming the (new) average result; all or most of them have become part of the environment.

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But it is an analogy only. It has an important distinction.

In quantum superposition, we discuss the relationships between different component states of the superposition. Those states exist not in physical space, but in a mathematical space.

In the army analogy, we discuss the relationships between the actions between different material items (solders in this case). Those material items exist in physical space.

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The unselected eigenstates do not cooperate with other particles to form macroscopic realities.

Although the spirit of the statement is correct, the statement itself is incorrect in multiple senses.

First, an eigenstate is a quantum state. It interferes with other eigenstates, not other particles.

Second, although the “unselected” eigenstates seem to disappear, they actually still exist; they entangles with the environment, which includes the apparatus and measurement devices of that experiment.

— Me@2013.01.01

— Me@2020-02-26 06:49:46 AM

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In “decoherence means that the different components do not interact with each other”, the meaning of “interact” is not defined yet.

The word should probably be “interfere”, instead of “interact”.

— Me@2020-02-25 10:44:23 PM

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interference ~ superposition with pattern

Decoherence means that the phase differences between different components in a superposition are not constants anymore. It does not mean that there is no superposition anymore.

Superposition is always there.

What disappears is the interference pattern, not the superposition.

— Me@2019-09-20 06:48:55 AM

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2020.02.26 Wednesday (c) All rights reserved by ACHK

Light, 2

Buddha 3 | 無額外論 6

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Whoever follows the principles is the Light.

— Me@2011.08.24

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The tricky part is that it is hard to know the principles, but feasible, as long as you are willing to pay the price.

— Me@2020-02-25 06:15:08 PM

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2020.02.25 Tuesday (c) All rights reserved by ACHK

機遇創生論 1.3

這段改編自 2010 年 4 月 18 日的對話。

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你可以花一生的時間,去令到自己受歡迎;但是,你的喪禮將會有多少人出席,還要視乎當日的天氣。

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萬事俱備,只欠東風時,就應該立刻離開,讓東風做事,不要阻礙東風。

亦即是話,一件事情的發展結果,受眾多因素影響。有些你控制得到,有些你控制不到。你既要盡力處理,你控制得到的部分,亦同時要盡力去忽略,你控制不到的部分。

千萬不要干涉,大自然的內政,因為,那只會弄巧反拙。

例如,如果你想受人歡迎,你可以做的,就是認真工作,善待自已,以誠待人等。最終會不會受歡迎,那不在你控制範圍。

試想想,如果你企圖保證,自己受歡迎的話,情形會怎樣?

情形就是,你會不斷地提醒他人,你自己的存在。那只會為他人,帶來極大的滋擾。那反而會令你討厭。

原理上,你就應該化成磁鐵,盡其在我地,增加自身的吸引力。重點是,你只可以吸引,而不能拉扯。

正如磁鐵一樣,它只能吸引磁性物質。其他東西,並不會受其吸引。

題外話:而更重要的問題是,為什麼要追求「受歡迎」呢?「受歡迎」的好處,真的大於壞處嗎?

而「不要干涉大自然的內政」,並不代表,「種子論」是「被動」、「消極」的。或者說,那是有意義的「被動」和有策略的「消極」。

那並不是負面意思下的被動消極,原因是「種子論」的行動三部曲,其實也相當上進積極。

第一部,要盡你自己的能力,獲取最多的情報,有關哪裡的土壤,最有機會讓農作物茁壯成長。然後,在那些範圍播種。沒有種子,就不可能有果實。 

第二部,有種子,亦不一定會果實。所以,你要積極不做事,積極去等待。除了定期的灌溉和施肥等工序外,並沒有任何行動,令你可以保證某單一棵植物,一定開花結果。

第三部,而要訓練到自已,習慣在適當的地方和時間,去「積極不做事,積極去等待」的技巧是,先做其他東西,同時在其他土壤播種。

而這正正回應了第一部:正正是因為,在某一指定地方有成果的機會極細,你要在最多的土壤,灑下最多的機遇種子;而底線是,不要多到自己管理不到。

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一件事情的發展結果,受眾多因素影響。有些你控制得到,有些你控制不到。你既要盡力處理,你控制得到的部分,亦同時要盡力去忽略,你控制不到的部分。

While we are free to choose our actions, we are not free to choose the consequences of those actions. Consequences are governed by natural law.

— The 7 Habits of Highly Effective People

— Stephen R. Covey

那只是理論比喻。執行「種子論」的真正難度在於,有時,在事情完結前,很難去分清該事件中的哪些部分,是「可控部分」,而哪些部分,卻是「不可控部分」。

亦即是話,「謀事在人,成事在天」之中,哪些部分而要由「你」去「謀」,而哪些地方是由「天」去「成」,事前不一定知道。

所以,那不會有公式答案,只能靠自己或他人的學術知識和實際經驗。

Grant me the serenity to accept the things I cannot change,

the courage to change the things I can change,

and the wisdom to distinguish the two.

「經驗」的近義詞,是「錯誤」。它們不同義的地方,在於「經驗」可以累積,「錯誤」不一定可以累積。

「種子論」的重點,並不是給予你,一條必勝的方程式。

「種子論」的重點是,令你知道,無論如何,你也會遇到無數次的失敗;那是正常不過的事。「種子論」的重點是,令你不再害怕失敗,從而,你會勇於嘗試,敢於超大量地犯錯。

「種子論」的重點是,重複犯同一個錯誤本身,並不能提高成功的機會率。

「種子論」所要求的「不斷犯錯」是,每次也是「新的錯誤」。「新錯誤」的意思,並不只是相對於你來說是「新」,而是相對全人類來說,都是「新」。

別人犯過的錯誤,你也不可以再犯。

一千萬次的失敗,可以兌換一次的成功。

而一次的成功,卻可以兌換一千萬次的應用。

— Me@2020-02-24 10:52:08 PM

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2020.02.24 Monday (c) All rights reserved by ACHK

scmutils, 2.3.2

Scheme Mechanics Installation for GNU/Linux

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Steps:

1. The following steps are tested in Ubuntu 18.04. Prepare Ubuntu 18.04 if you can.

Note: Since the installation of the library scmutils requires the root access of your Linux system, please do NOT use it on your working computer. Instead, create an isolated virtual machine to use it.

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2. Go to the bottom of this post to click the category scmutils, so that you can see all the posts in this scmutils series.

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3. Go to the post titled “scmutils, 2.3” to download

scmutils-20160827-x86-64-gnu-linux

4. Although the official installation guide advises you to install “MIT/GNU Scheme system” before installing scmutils, you do NOT need to install “MIT/GNU Scheme system” at all.

5. Unzip the file scmutils-20160827-x86-64-gnu-linux.

In the following, if you need to copy any commands or programming codes, remember that any number on the left of the vertical green line is NOT part of the code.

6. Run the command

tar xzf scmutils-20160827-x86-64-gnu-linux.tar.gz

to further extract the file.

-x — extract files from an archive;
-f — specify the archive’s name;
-v — show a list of processed files.

— Wikipedia on tar (computing)

Then two folders will be created: bin and scmutils.

7. Run the command

cd bin

to go into the folder.

You will see a file called mechanics.

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8.1 Run the command

mechanics

You will get the error

mechanics: command not found

8.2 Instead, you should run the command

./mechanics

to specify that the file mechanics is actually in the current folder.

You will get the error

./mechanics: line 16: exec: xterm: not found

It is because your Linux system has not the program xterm yet.

8.3 Run the following command to install it.

sudo apt-get install xterm

8.4 Run the command again:

./mechanics

There will be an xterm window popup, but with an error message inside:

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That means you should move the two folders, bin and scmutils, to the pre-defined locations.

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9.1 Run the command to move the folder scmutils to its pre-defined location:

mv scmutils /usr/local/

You will get the error

mv: cannot move 'scmutils' to '/usr/local/scmutils': 
Permission denied

9.2 Try again by

sudo mv scmutils /usr/local/

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10.1 Go inside the folder bin.

10.2 Move its content to the pre-defined location by this command:

sudo mv mechanics /usr/local/bin/

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11. Run the command

mechanics

Then you will see the Edwin window is opened. That means, in theory, you system has successfully installed the scmutils library. You can use it within the Edwin window if you like.

However, in practice, it is difficult, because it provides no syntax-highlighting. Also, you cannot use mouse in the Edwin window, so if you want to copy and paste a command or a series of commands, there will be no obvious way to achieve that.

So I suggest you to use the standard Emacs as the editor instead.

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12.1 If you do not know Emacs, learn its basics.

12.2 Also, learn how to open Emacs’ initialization file, which has the filename

.emacs

After opening the file, you will see that it is just a text file.

12.3 Go to the end of the .emacs file. Add the following code:

(defun mechanics()
  (interactive)
  (run-scheme
     "/usr/local/scmutils/mit-scheme/bin/scheme --library
     /usr/local/scmutils/mit-scheme/lib"))

12.4 Save the file. Close Emacs. Then re-open Emacs.

12.5 Within Emacs, type the command

M-x mechanics

M-x means that while the Alt key is pressed down, press also x. Then type the word mechanics.

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12.6 Type the command

(+ 1 1)

to test the system.

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13. If you want to access your last command without re-typing it, type the command

M-p

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— Me@2020-02-22 06:25:47 PM

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2020.02.22 Saturday (c) All rights reserved by ACHK

Unitarity, 2

Just as in the definition of a function in mathematics, two x values can map to the same y value, but one x value cannot map to two y values; the future is unique in classical physics.

In quantum mechanics, the uniqueness is not of the measurement results, but of the set of probabilities of the results.

— Me@2018-02-03 11:29:18 AM

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With respected to a known state, the future is unique in classical physics.

In quantum mechanics, the set of probabilities is encoded in the wave function.

— Me@2020-02-21 06:25:52 PM

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2020.02.22 Saturday (c) All rights reserved by ACHK

反貼士搵笨大行動 1.1

在中四升中五的那個暑假,有一位楊姓同學送了,當時某補習社一個講座的入場劵給同學們,包括我。(該補習社,現已不存在。)物理科的部分,由 Ken Chan 主講。

他的第一句是:「我正職唔係日校物理老師,我係做研究o既。」

然後,他自問:「嘩,阿 sir 好犀利呀!」

自答:「唔一定o既,可能係我俾人研究呢。」

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另一句金句是:「後生又唔代表無經驗o既。」

他宣稱他在當屆的會考物理中,選擇題中,貼中了多少題,而長題目中,又貼中了多少題。選擇題的命中數目,我不太記得;大概是 45 題中的 39 題。而長題目的貼中率,則是 5 題中有 4 題。

聽過了他的講座後,我感到他與別不同,有如半人半神。之後的那個星期,我立刻去補習社報讀他的課程。每星期上他的課,大概是該星期最快樂的時光,雖然他的課堂時間,定在星期六的早上八時半。那個年代,仍是一個真人授課的時代。

他在授課中,亦間中,但不常,提及他的試題預測命中率。

我是在中五初之前的那個月(八月),開始上 Ken Chan 的課。我不記得是當時還是後來,他在星期六的下午 5 時半,有開另一班;課程和進度,和上午 8 時半那班,大致相同。

幾個月(可能少於三個月)後,我有一次改了上,下午的那一課,遇到暑期介紹我去聽講座的那位楊姓同學。我發覺他在那時才開始補,所以他要借我九月以來的講義。他以外,他還介紹了我日校的另一位同學,去上那個補習班。那另一位同學姓馬。

幾個月後,考了物理科後的一晚,我和馬同學於電話傾談。他說他在見到試題時,十分生氣,因為不如傳說中的所謂「5 題貼中 4 題」。(根本就一題也沒有貼中,除非「貼中」的定義,就只是「同一課題」而已;例如,貼「力學」題目,又真是出了「力學」題目。)

我則不會有那樣的期望。我回答他說:「如果他真的可以貼中題目,他在平時又何須那麼用心去授課?他又何須在聖誕假和農曆年假中,再要不斷的補課呢?」

— Me@2020-02-19 05:33:09 PM

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2020.02.19 Wednesday (c) All rights reserved by ACHK

Problem 2.1b

A First Course in String Theory

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2.1 Exercises with units

(b) Explain the meaning of the unit K (degree kelvin) used for measuring temperatures, and explain its relation to the basic length, mass, and time units.

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{\displaystyle {\frac {1}{T}}=\left({\frac {\partial S}{\partial U}}\right)_{V,N}},

where \displaystyle{U} is the internal energy.
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The units of \displaystyle{k_B T} and \displaystyle{E} are the same.

\displaystyle{[k_B T] = [E]}

In other words, the Boltzmann constant \displaystyle{k_B} translates the temperature unit \displaystyle{K} to the language of energy unit \displaystyle{J}.

However, although the temperature unit \displaystyle{K} and the energy unit \displaystyle{J} have the relation

\displaystyle{k_B K = J},

just \displaystyle{k_B T} would not give the correct value of energy \displaystyle{E}, not to mention that we have not yet specified of which the energy \displaystyle{E} is.

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For an ideal gas,

\displaystyle{pV=Nk_B T}

and the average translational kinetic energy is

{\displaystyle {\frac {1}{2}}m{\overline {v^{2}}}={\frac {3}{2}}k_BT}

for 3 degrees of freedom. In 3D space, if there are only translational motions, there are only 3 degrees of freedom.

In other words, just the value of {k_B T} itself gives no physical meaning. Instead, {\tfrac{1}{2}k_B T} can be interpreted as the average translational kinetic energy of the particles in an one dimensional space. Equivalently, \displaystyle{\tfrac{3}{2}k_BT} gives that in our three dimensional space.

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Another main difference is that although energy is an extensive property, temperature is an intensive property.

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We define the temperature unit Kelvin \displaystyle{K} by requiring the water triple point temperature,

\displaystyle{T_{tp} \equiv 273.16K}

Once this value is fixed, the Boltzmann constant \displaystyle{k_B} value can be estimated by using, for example, the ideal gas law

\displaystyle{pV = N k_B T},

because \displaystyle{k_B} always comes with \displaystyle{T}.

— Me@2020-02-16 11:14:24 AM

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2020.02.16 Sunday (c) All rights reserved by ACHK

Unix

Linux is headache. It gives you too many choices.

— Me@2011.07.14

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But Linux is necessary.

— Me@2020-01-23 08:53:29 PM

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Applicants must also have extensive knowledge of Unix, although they should have sufficiently good programming taste to not consider this an achievement.

— MIT job advertisement

— Hal Abelson

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2020.02.15 Saturday (c) All rights reserved by ACHK

大腦物理性損毀不是比喻

99.85%的人不會,終身被匪欺騙教育,大腦已經物理性損毀了,假如說匪的謊言灌輸有20層,那麼絕大多數人遭鐵拳之後的反思,最多只能突破一兩層,達到“我愛國所以要說話”的程度,離理解台人港人藏人維人還非常遠,至少還需要十幾個層次的突破。所以覺醒者極少極少,而且沒有力量。很悲哀,但事實如此。

所以匪的滅亡,恐怕不會是由於覺醒後的民眾起來反抗,而是傳統的王朝滅亡現象,天災加人禍造成經濟崩潰,引發統治崩潰,帶來的不是社會進步,而是社會危機。

大腦物理性損毀不是比喻,而是 literally 如此,人腦的神經細胞本身不構成思想,神經細胞之間的連接方式才是,從嬰兒出生到不斷學習成長,就是逐步構建神經細胞連接到過程,連接完成之後,是不可逆的,所以經匪教育的人,99.85%都是終身殘疾,剩下那點點漏網之魚,只是教育失敗而已。

肉體也是一樣,從受精卵、胚胎開始,細胞都是幹細胞,然後分化成各種完全不同的組織細胞,成長為肉體,這個分化、成長、發育的過程是非常可塑的,外界條件可以對肉體發育造成很大影響,但是一旦發育完成,還有可能退回去變成幹細胞,再重新發育一次,變成一個不一樣的肉體嗎?沒有可能的。

所以您國成年人99.85%都是無救的,不要對他們抱任何希望,期待發生什麼什麼事件能夠改變他們的思維,從這個意義上看,他們都是已經死亡的人。唯一的希望恐怕在還沒有受匪教育或受教育不多的小孩子,以及還沒有出生的下一代,只有趕緊跑路,把他們帶離地獄,才有一點點挽救的希望。

所以逃離匪區最大的收穫是什麼,不是自己能賺錢,能自由,而是孩子可以不用再經歷我們曾經經歷的那些毫無意義的痛苦、摧殘、扭曲,孩子可以有機會做個正常人。我們自己每個人都已經多多少少被毀滅了一部分,但是能救孩子,下一代還有新的希望,還有什麼比這更重要的呢。

— 李穎

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People do not change. They die, replaced by other people.

Most people do not change. They die, replaced by a new generation of people.

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You cannot change the people around you, but you can change the people you choose to be around.

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2020.02.15 Saturday ACHK

機遇創生論 1.2

這段改編自 2010 年 4 月 18 日的對話。

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還有,「人生攻略理論」令人聯想到很多東西,而大部分也不是,我們那個合體大理論的內容。

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「緣份攻略」都不行,因為感覺有點怪。

(安:那就不如叫做「緣份理論」。)

「理論」很空泛。不應把「理論」,視為名字的一部分。

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不如就直接收做「機械人生論」啦,因為那個大統一理論,其實就即是「人生攻略」。

這個大統一理論的成員,包括(但不止於):

精簡圖:

種子論
反白論
完備知識論

自由決定論

它們可以大統一的成因,在於它們除了各個自成一國外,還可以合體理解和應用。

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詳細圖:

種子論 可遇不可求

你只能控制因,不能控制果。事情結果,除了受你的因以外,還受千百萬樣其他因素影響。所以,人生的大目標之中,很多也只有極細的成功機會率。

第一方面,如果你很努力做事,事情仍不能如你所願的話,不要覺得很奇怪。

第二方面,做事時,不要過份用力之外,亦不要在不適當的地方用力。

比喻說,你向心儀示愛後,就只能等待她的答覆,讓她決定,接不接受你的愛意。

示愛以後,並沒有任何東西你可以做,從而控制到她去喜歡你,因為,那是她的自由。

你唯一可以做的是,遠在示愛之前,就積極提升自己的外貌和人格等,從而提高,她喜歡你的機會率。

在適當的地方用力以後,要懂得「放手」,讓事情自己發展。

把手緊握 什麼都沒有
把手放開 你得到一切

萬事俱備,只欠東風時,就應該立刻離開,讓東風做事,不要阻礙東風。

亦即是話,一件事情的發展結果,受眾多因素影響。有些你控制得到,有些你控制不到。你既要盡力處理,你控制得到的部分,亦同時要盡力去忽略,你控制不到的部分。

千萬不要干涉,大自然的內政,因為,那只會弄巧反拙。

— Me@2020-02-15 07:17:00 AM

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2020.02.15 Saturday (c) All rights reserved by ACHK

scmutils, 2.3

Within the MIT Scheme environment, it is not the original command line (bash) anymore. I can neither repeat the last command by just pressing the up key once, nor select the last command by mouse in order to copy it.

So I think I have to use an older version of scmutils.

However, this method is not easy to implement, because the author’s website does not provide an older version. Luckily, I have found an old version in my computer. You can download it here:

scmutils-20160827-x86-64-gnu-linux

Note: Since the installation of the library scmutils requires the root access of your Linux system, please do NOT use it on your working computer. Instead, create an isolated virtual machine to use it.

— Me@2020-02-09 10:27:32 PM

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2020.02.09 Sunday (c) All rights reserved by ACHK

Black hole information paradox, 4

So we seem to have a direct contradiction between [QM and unitarity] and [GR and causality]. Both of these principles, unitarity and causality, cannot be exactly correct because a contradiction arises from their explosive mixture.

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As most quantum mechanicians have known from the very beginning, it is the unitarity, a principle of quantum mechanics, that wins in the battle and remains universally valid.

On the other hand, causality becomes an approximate principle that is only valid in the limit of infinitely large causal diamonds. In the presence of black holes, the internal causal structure is modified by quantum phenomena and the information can “tunnel” out of the black hole.

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It shouldn’t be so surprising that unitarity survives completely while causality doesn’t. After all, the basic postulates of quantum mechanics, including unitarity, the probabilistic interpretation of the amplitudes, and the linearity of the operators representing observables, seem to be universally necessary to describe physics of any system that agrees with the basic insights of the quantum revolution.

On the other hand, geometry has been downgraded into an effective, approximate, emergent aspect of reality. The metric tensor is just one among many fields in our effective field theories including gravity. In string theory, there are, in some sense, infinitely many such fields besides the metric tensor – the whole “stringy tower”. The metric tensor doesn’t have to exist as a good degree of freedom at the Planck scale or in other extreme conditions. We know many other fields that are only good enough at low energies – e.g. the pion field.

— Black hole information puzzle

— Lubos Motl

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2020.02.09 Sunday ACHK

Ken Chan 時光機 3.4

如果可以用時光機,把那些讀書策略傳送到 16 歲時的自己,而又只能夠長話短說的話,我會提醒自己,

不要企圖去,違反「自然定律」。 

比喻說,如果你企圖把二公升的水,倒進只有一公升的杯中,悲劇注定會發生。

同理,如果有一件工作,正常人要花十小時,才能圓滿完成,而我企圖去,僅僅用五小時的話,質素必然會奇差。

一切要符合邏輯和自然定律。

第一定律是:

相對於當年香港中學會考(中五公開試)的課程時間長度和深淺程度而言,一個平均智力的人,只要運用一些簡單的時間管理系統,都可以在各科之中,獲得上佳成績。

會考課程由中四開始,至中五開考,大約有一年零七個月的時間。

但是,當時我詢問 Ken Chan 讀書方法,或者時間管理技巧時,情境並不是那樣的。

(問:何出此言呢?)

我在中四時,大部分時間,都用來擔憂和內疚,真正用來讀書時間奇少。所以,我是在中五開始時,才真正(企圖)開始準備會考的。而最慘的是,那時,我仍然花了不少的時間,去擔憂和內疚。

當時我詢問 Ken Chan 讀書方法,或者時間管理技巧時,已經是中五的中段時期;距離應考,只剩幾個月的時間,並沒有一年零七個月的時間。所以,心態上,我不應想像,仍然有什麼方法,可以令我可以在所有科目,都奪取 A 級成績。合理的問題應該是:

如何善用那剩下的幾個月,把它發揮最大的功效?

有多少科目,仍然有機會奪 A?

有哪些科目,只能退而求其次,奪 B 了事?

又有哪些科目,其次也不行,只能期望守住 C?

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第二定律是:

假設某一科奪 A 需要的分數是 100 分,而對應需要研習時間 1000 小時。

如果奪 B 需要的分數是 80 分的話,對應需要研習時間,必定遠低於 800 小時;因為凡事該科課程中,最艱深和最費時的題目,你都可以選擇放棄。

— Me@2020-02-08 11:36:41 AM

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2020.02.08 Saturday (c) All rights reserved by ACHK