# 2007.03.26

— Me@2020-02-27 10:20:04 PM

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# 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}}

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\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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# 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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# 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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# 機遇創生論 1.3

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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

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」。

— Me@2020-02-24 10:52:08 PM

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# 1999.12.31

— Me@2020-02-23 03:34:08 PM

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# 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.

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:

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.

12.6 Type the command

(+ 1 1)

to test the system.

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

M-p

— Me@2020-02-22 06:25:47 PM

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# 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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# Pandemonium, 2.2

You get what you FIGHT for, nothing more, nothing less.

— Me < 2008

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# 反貼士搵笨大行動 1.1

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— Me@2020-02-19 05:33:09 PM

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# CSS, 2

— Me@2020-02-16 06:20:52 PM

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# 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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# 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.

— Hal Abelson

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# 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。

— 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

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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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— Me@2020-02-15 07:17:00 AM

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# 2006

— Me@2020-02-10 05:37:38 PM

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# 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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# 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

# 魔間傳奇

Hell Is Empty, and All the Devils Are Here

— The Tempest

— William Shakespeare

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2020.02.08 Saturday ACHK

# Ken Chan 時光機 3.4

（問：何出此言呢？）

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— Me@2020-02-08 11:36:41 AM

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