2.10 A spacetime orbifold in two dimensions

A First Course in String Theory

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Consider a two-dimensional world with coordinates \displaystyle{x^0} and \displaystyle{x^1}.

A boost with velocity parameter \displaystyle{\beta} along the \displaystyle{x^1} axis is described by the first two equations in (2.36). We want to understand the two-dimensional space that emerges if we identify

\displaystyle{(x^0, x^1) \sim ({x'}^0, {x'}^1)}.

We are identifying spacetime points whose coordinates are related by a boost!

(a) Use the result of Problem 2.2, part (a), to recast (1) as

\displaystyle{(x^+, x^-) \sim \left( e^{-\lambda} x^+, e^{\lambda} x^- \right)}, where \displaystyle{e^\lambda \equiv \sqrt{\frac{1+\beta}{1-\beta}}}.

~~~

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

— Me@2021-05-04 10:48:28 PM

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2021.05.05 Wednesday (c) All rights reserved by ACHK

Logical arrow of time, 9.2

To confirm or disconfirm a prediction, you cannot check record; you can only observe the system evolving.

To confirm or disconfirm a retrodiction, you can only check record (or the logical consequence of that retrodiction); you cannot observe that past event directly.

— Me@2013-08-10 08:00 PM

— Me@2021-05-03 12:28 PM

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2021.05.03 Monday (c) All rights reserved by ACHK

相逢恨晚 2

尋覓 2.2.3.4 | Passion Test 2.2

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如果你覺得你女朋友,會阻礙你工作進度,或者是你時間負擔的話,你倆大概由始至終,就不應在一起;你倆根本不是,對方的另一半。

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第二,如果你喜歡一個人,她真的是你另一半的話:

初期相識時,甚至是第一次相見時,就已經有相逢恨晚、無所不談之感。 

We talked half the night, and in the middle of talk became lovers. — Bertrand Russell

你自然視她的意見,如珠如寶,尤其是有關工作上的;因為,你工作上的決定,直接影響著你倆,將來可否開花結果,化成夫妻。

中期相處時,就會心有靈犀。

心有靈犀者,心靈感應也。人的身體大腦,有如電腦硬件。人的思想靈魂,其實就是程式軟件。聊天等溝通,其實就是不斷地,複製自我於,對方的腦子之中。所以,相識不淺時,你倆其實就已經有了,對方的「自我」複本,在腦海心中。

深情者,即使不見也同在。

有感情 = 感受到相同的情境 — Me@2011.06.04

所以,你倆每天,毋須再 24 小時也,物理聊天相見。

每天再相見時,就只須談情說愛,毋須再談天說地;因為,你倆已能心靈感應,開聲前就大概知道,對方的心思。你再毋須事事也,問女何所思,問女何所憶。

後期相愛時,則會穿梭時空。

愛情三階段:

相識相逢恨晚,相處心有靈犀,相愛穿梭時空。

單身三境界:

青年工程師,中年物理學家,老年魔法師。

— Me@2021-04-05 06:14:46 PM

— Me@2021-05-01 04:46:18 PM

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2021.05.02 Sunday (c) All rights reserved by ACHK

Ex 1.21 The dumbbell, 2.2.1

Structure and Interpretation of Classical Mechanics

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b. Write the formal Lagrangian

\displaystyle{L(t;x_0, y_0, x_1, y_1, F; \dot x_0, \dot y_0, \dot x_1, \dot y_1, \dot F)}

such that Lagrange’s equations will yield the Newton’s equations you derived in part a.

~~~

[guess]


(define (U-constraint q0 q1 F l)
  (* (/ F (* 2 l))
     (- (square (- q1 q0))
        (square l))))

(define ((extract-particle pieces) local i)
  (let* ((indices (apply up (iota pieces (* i pieces))))
         (extract (lambda (tuple)
                    (vector-map (lambda (i)
                                  (ref tuple i))
                                indices))))
    (up (time local)
        (extract (coordinate local))
        (extract (velocity local)))))

(define q-rect
  (up (literal-function 'x_0)
      (literal-function 'y_0)
      (literal-function 'x_1)
      (literal-function 'y_1)
      (literal-function 'F)))

(show-expression q-rect)


(show-expression (q-rect 't))


(show-expression (Gamma q-rect))


(show-expression ((Gamma q-rect) 'w))


(show-expression ((Gamma q-rect) 't))


(show-expression (time ((Gamma q-rect) 't)))


(show-expression (coordinate ((Gamma q-rect) 't)))

[guess]

— based on /sicmutils/sicm-exercises

— Me@2021-04-27 05:03:59 PM

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2021.04.28 Wednesday (c) All rights reserved by ACHK

Alfred Tarski 4

Godel 20

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[guess]

A system can be both complete and consistent because in that system, a metatheorem cannot be translated into a theorem, so a paradox cannot be made.

[guess]

— Me@2017-04-10 10:40:38 AM

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2021.04.27 Tuesday (c) All rights reserved by ACHK

深淵 3

愚善 3.2 | Good intentions 2 | 地獄之路 2.2

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魔鬼,也是由天使變成的。

不企圖做天使,就不會變成魔鬼。

— Me@2016-08-06 03:02:36 PM

— Me@2021-04-26 06:26:35 PM

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2021.04.26 Monday (c) All rights reserved by ACHK

一萬個小時 3.3

這段改編自 2010 年 4 月 24 日的對話。

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我現在的狀態是,在某些範疇中,有能力教人之餘,還有能力教人如何教。亦即是話,如果教員甲的教學質素,好過教員乙的話,我可以解釋到,甲比乙好在哪裡。

(安:這一點對我好有用。)

但那不能成為課程,因為,它只需一個下午,就可以大概完整地,講給你聽。

(安:你的講法是,有些知識,你只需要幾個小時,就可以傳授給我,不成課程,因為時間所需太短;而另一些知識,我就需要花「幾年加幾年」,才可以領會得到,不成課程,因為時間所需太長。)

現在,有三個概念混淆在一起。我們先釐清一下。

1. 幾乎

任何學問,或者才藝,如果要訓練自己,純熟到足以用來維生賺錢,或者娛樂滿足,需要的時間幅度是,一日幾小時,持續一兩年,才會開竅;然後,再持續一兩年,才會知道自己,在該門學問才藝,為何會開竅,開了什麼竅;又再過一兩年,才會有能力教人,如何在該門學問才藝開竅。

2. 過程中的一個重要環節是,會領悟到一些洞見。

3. 雖然要「自己領悟」第二點中的洞見,來自第一點中的「幾年加幾年」過程,但是,如果只是「吸收別人已知的」知識本身,絶不需要花「幾年加幾年」。有時,甚至只是,花數個小時就可以。

4. 1. 但是,「知道」本身,並不代表自己有足夠功力去「運用」,而「運用」則是,第一點的水平。不過,如果中途有第三點,即是別人情報資料上的提點,你就可以走少很多冤往路。那樣,原本是(例如)十年的過程,可以壓縮到(例如)三年。 

— Me@2021-04-25 09:29:14 PM

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2021.04.26 Monday (c) All rights reserved by ACHK

2.9 Lightlike compactification, d, 2

A First Course in String Theory

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Represent your answer to part (c) in a spacetime diagram. Show two points related by the identification (2) and the space and time axes for the Lorentz frame S’ in which the compactification is standard.

~~~

[guess]

In case you would like to have a fundamental domain:

[guess]

— Me@2021-04-21 10:35:13 PM

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2021.04.24 Saturday (c) All rights reserved by ACHK

Modification of the cumulative concept of time

The past is part of the present, in the sense that part of the present is the same as the past, or is based on something of the past. For example, “I was 10 years old in the past” is the necessary condition for “I am 20 years old in the present”.

However, not all of the past is still part of the present, because some of the past is already lost. In other words, some of the past is not stored in the present.

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You need physical macroscopic law (induction) in order to get the cumulative property of time.

Microscopically, due to the unitarity of the Schrödinger equation, any past quantum state has a one-one correspondence to a quantum state at any particular future moment. So any future quantum state already has all the information needed to deduce its past state. In this sense, there is no time microscopically.

Time is not cumulative microscopically, because the past is not part of the future.

We can also say that time is 100% cumulative microscopically, because the past is all of the future; instead of being part of any future quantum state, the past quantum state already has all the (equivalent) information of a quantum state at any particular future moment.

— Me@2013-08-10 07:45 PM

— Me@2021-04-21 04:24 PM

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2021.04.21 Wednesday (c) All rights reserved by ACHK

相聚一刻

光速飛行 1.2.3 | 尋覓 2.2.3.3 | Passion Test 2

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我是一個奇怪的人。我比一般人益智和勵志。我不會因為愛情問題,而無心向學,放慢工作。我既不會因為單身一人,而茶飯不思,亦不會因為有心上人,而心神恍惚。

(我只會因自身的問題,而頹廢怠慢,無所事事。)

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如果你的所謂「愛情」,會打擊你的學業或者事業的話,你的「愛情」,大概是假的,因其不會長久,更遑論永久。

如果你覺得的女朋友,會阻礙你的工作進度,或者是你的時間負擔的話,你倆大概由始至終,就不應在一起;你倆根本不是,對方的另一半。

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第一,如果你喜歡一個人,她真的是你的另一半的話,你會願意花無限的時間,在她身上。

你不會覺得她是,你時間上的負擔。相反,你其他時間的所有努力,尤其是工作上的辛勞,都只為賺取,與她相處的一分一秒。

穿越人海 只為與你相擁

— 尹初七

— Me@2021-04-11 07:10:08 AM

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2021.04.20 Tuesday (c) All rights reserved by ACHK

Ex 1.21 The dumbbell, 3

Structure and Interpretation of Classical Mechanics

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c. Make a change of coordinates to a coordinate system with center of mass coordinates x_{cm}, y_{cm}, angle \theta, distance between the particles c, and tension force F. Write the Lagrangian in these coordinates, and write the Lagrange equations.

~~~

[guess]

\displaystyle{ \begin{aligned}   y_{cm} &= \frac{m_0 y_0 + m_1 y_1}{m_0 + m_1} \\   x_{cm} &= \frac{m_0 x_0 + m_1 x_1}{m_0 + m_1} \\   \end{aligned}}

\displaystyle{ \begin{aligned}   y_{cm} &= \frac{m_0 y_0 + m_1 (y_0 + c(t) \sin \theta)}{m_0 + m_1} \\   x_{cm} &= \frac{m_0 x_0 + m_1 (x_0 + c(t) \cos \theta)}{m_0 + m_1} \\   \end{aligned}}

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Let \displaystyle{M = m_0 + m_1}.

\displaystyle{ \begin{aligned}   y_{cm} &= \frac{M y_0 + m_1 c(t) \sin \theta}{M} \\   x_{cm} &= \frac{M x_0 + m_1 c(t) \cos \theta}{M} \\   \end{aligned}}

\displaystyle{ \begin{aligned}   y_0 &= y_{cm} - \frac{m_1}{M} c(t) \sin \theta \\   x_0 &= x_{cm} - \frac{m_1}{M} c(t) \cos \theta \\   \end{aligned}}

\displaystyle{ \begin{aligned}   y_1 &= y_{cm} + \frac{m_0}{M} c(t) \sin \theta \\   x_1 &= x_{cm} + \frac{m_0}{M} c(t) \cos \theta \\   \end{aligned}}

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The Lagrangian

\displaystyle{ \begin{aligned}   L &= \frac{1}{2} m_0 (\dot x_0^2 + \dot y_0^2) + \frac{1}{2} m_1 (\dot x_1^2 + \dot y_1^2) + \lambda \left[ (x_1(t) - x_0(t))^2 + (y_1(t) - y_0(t))^2 - l^2 \right] \\     &= \frac{1}{2} m_0 (\dot x_0^2 + \dot y_0^2) + \frac{1}{2} m_1 (\dot x_1^2 + \dot y_1^2)   - \frac{F}{2l} \left[ (c(t))^2 - l^2 \right] \\   \end{aligned}}

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\displaystyle{ \begin{aligned}   \dot y_0 &= \dot y_{cm} - \frac{m_1}{M} \dot c(t) \sin \theta - \frac{m_1}{M} c(t) \dot \theta \cos \theta\\  \dot x_0 &= \dot x_{cm} - \frac{m_1}{M} \dot c(t) \cos \theta - \frac{m_1}{M} c(t) \dot \theta \sin \theta \\   \end{aligned}}

[guess]

— Me@2021-04-17 05:40:46 PM

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2021.04.18 Sunday (c) All rights reserved by ACHK

Causal diamonds in time travel

Quantum mechanics is a set of rules that allows an observer to predict, explain, and/or verify observations (and especially their mutual relationships) that he has access to.

No observer can detect inconsistencies within the causal diamonds. However, inconsistencies between “stories” as told by different observers with different causal diamonds are allowed (and mildly encouraged) in general (as long as there is no observer who could incorporate all the data needed to see an inconsistency).

— Raphael Bousso is right about firewalls

— Lubos Motl

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There is no “god’s eye view” in physics.

— Me@2021-04-17 03:12:58 PM

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Macroscopic time is with respect to an observer. Actually, physics is with respect to an observer.

In the real universe, any observer’s observations must be consistent. When two observers compare their observations, their results must be consistent, because the comparison itself is an observation of an observer.

Time travel in the absolute sense is logically impossible. Let’s assume that it is logically possible.

If a time travel story follows the principle of “an observer’s observations must be consistent”, each character in that story must see a consistent timeline, even if different characters’ timelines may be inconsistent. That is fine as long as such inconsistent observers never meet to compare their results.

If two of such observers choose to meet to compare their results, the action to “meet to compare” itself will render the results consistent. It is similar to the resolution of the twin paradox in special relativity.

There is no “god’s eye view” in physics. Every physical event must be described with respect to an observer. Every physical event, even if the event is “to compare observation results”, must be described with respect to an observer.

— Me@2017-05-10 07:45:36 AM

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2021.04.17 Saturday (c) All rights reserved by ACHK

黎明 3

講到做唔到,唔好講。

好辛苦做咗啲嘢無人知,唔好做。

— 黎明

— Me@2016-05-16 05:58:16 PM

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2021.04.17 Saturday ACHK

To realize is to realize, 1.2

For example, when you ask “how to be a polite person”, you become a polite person.

In other words, when you realize that you should be a polite person, that polite person is realized at that moment.

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The ultimate self-fulfilling prophecies:

1. free will or not

2. god or no god

3. afterlife or not

4. future spouse exists or not

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Why self-fulfilling?

1. free will or not

If we knew the exact physical laws and all the data of the present state of the universe, we would be able to predict, with 100% accuracy, the state of the universe at any future moment.

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In theory, the exact physical laws exist, whether we know them or not. Also, in theory, all the details of the present state of the universe exist, whether we know them or not.

So in theory, there is no free will, because the future is already fixed, by the physical laws.

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However, even if we knew the exact physical laws, it would be still logically impossible to get all the data of the present state of the whole universe, because it is logically impossible for any observer to observe itself directly. For example, no camera can take a picture of itself directly.

So “in practice”, which is actually “in theory”, there is free will, because logically, no one can predict, with 100% accuracy, your future actions.

— Me@2021-04-16 04:45:47 PM

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2021.04.16 Friday (c) All rights reserved by ACHK