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}}}.

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\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