Monthly Archives: November 2019

Problem 13.5b

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A First Course in String Theory

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13.6 Unoriented closed strings

This problem is the closed string version of Problem 12.12. The closed string \displaystyle{X^{\mu} (\tau, \sigma)} with \displaystyle{\sigma \in [0, 2 \pi]} and fixed \displaystyle{\tau} is a parameterized closed curve in spacetime. The orientation of a string is the direction of the increasing \displaystyle{\sigma} on this curve.

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Introduce an orientation reversing twist operator \displaystyle{\Omega} such that

\displaystyle{\Omega X^I(\tau, \sigma) \Omega^{-1}} = X^I (\tau, 2 \pi - \sigma)

Moreover, declare that

\displaystyle{\Omega x_0^- \Omega^{-1} = x_0^-}

\displaystyle{\Omega p^+ \Omega^{-1} = p^+}

(b) Used the closed string oscillator expansion (13.24) to calculate

\displaystyle{\Omega x_0^I \Omega^{-1}}

\displaystyle{\Omega \alpha_0^I \Omega^{-1}}

\displaystyle{\Omega \alpha_n^I \Omega^{-1}}

\displaystyle{\Omega \bar \alpha_n^I \Omega^{-1}}

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Equation (13.24):

\displaystyle{X^{\mu} (\tau, \sigma) = x_0^\mu + \sqrt{2 \alpha'} \alpha_0^\mu \tau + i \sqrt{\frac{\alpha'}{2}} \sum_{n \ne 0} \frac{e^{-in\tau}}{n} (\alpha_n^\mu e^{i n \sigma} + \bar \alpha_n^\mu e^{-in \sigma})}

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\displaystyle{\begin{aligned} X^{\mu} (\tau, \sigma) &= x_0^\mu + \sqrt{2 \alpha'} \alpha_0^\mu \tau + i \sqrt{\frac{\alpha'}{2}} \sum_{n \ne 0} \frac{e^{-in\tau}}{n} (\alpha_n^\mu e^{i n \sigma} + \bar \alpha_n^\mu e^{-in \sigma}) \\ X^I (\tau, 2 \pi - \sigma) &= x_0^I + \sqrt{2 \alpha'} \alpha_0^I \tau + i \sqrt{\frac{\alpha'}{2}} \sum_{n \ne 0} \frac{e^{-in\tau}}{n} \left( \alpha_n^I e^{- in\sigma} + \bar \alpha_n^I e^{i n \sigma)} \right) \\ \end{aligned}}

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\displaystyle{\Omega X^I(\tau, \sigma) \Omega^{-1}} = X^I (\tau, 2 \pi - \sigma)

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By comparing \displaystyle{\Omega X^I(\tau, \sigma) \Omega^{-1}} with \displaystyle{X^I (\tau, 2 \pi - \sigma)}, we have:

\displaystyle{\begin{aligned} \Omega x_0^I \Omega^{-1} &= x_0^I \\ \Omega \alpha_0^I \Omega^{-1} &= \alpha_0^I \\ \Omega \alpha_n^I \Omega^{-1} &= \bar \alpha_n^I \\ \Omega \bar \alpha_n^I \Omega^{-1} &= \alpha_n^I \\ \end{aligned}}

— Me@2019-11-24 04:33:23 PM

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2019.11.24 Sunday (c) All rights reserved by ACHK

PhD, 3.8.1

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財政自由 1.3.1

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The secret to creativity is knowing how to hide your sources.

— Not Einstein

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In 1924, while working as a Reader (Professor without a chair) at the Physics Department of the University of Dhaka, Bose wrote a paper deriving Planck’s quantum radiation law without any reference to classical physics by using a novel way of counting states with identical particles. This paper was seminal in creating the very important field of quantum statistics. Though not accepted at once for publication, he sent the article directly to Albert Einstein in Germany. Einstein, recognising the importance of the paper, translated it into German himself and submitted it on Bose’s behalf to the prestigious Zeitschrift für Physik. As a result of this recognition, Bose was able to work for two years in European X-ray and crystallography laboratories, during which he worked with Louis de Broglie, Marie Curie, and Einstein.

— Wikipedia on Satyendra Nath Bose

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（問：根據你的講法，好像大部分情況下，都不應該讀研究院似的。）

在理想的情況下，你可能應該讀研究院。

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（問：那樣，你心目中的理想情況是什麼？）

假設你已經有財政自由，你就有可能，適合讀研究院；…

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讀研究院的最大作用是，獲取獨家資料和人脈。

做到物理學家的其中一個，近乎先決條件是，認識一些一流的物理學家，從而可以跟他們對話。換句話說，讀研究院的主要目的是，爭取跟一些物理神人，對話的機會。

（問：那金錢和時間成本奇高。間中約他們暢談可以嗎？）

那比較困難。

如果你不是他們原本的朋友、同事或學生的話，大概不會有足夠時間，分配給你。

（問：自己一個做研究，一定不可以嗎？）

那十分困難。

即使是獨行俠愛因思坦，他的獨行俠形象，也是假的。大概而言，那只是他的大眾形象、公關技巧。

實情是，他在學術上，有一個開放謙卑的態度，十分願意吸收他人的思想，無論對方當時的名氣是怎麼樣。

例如，他有一段時期會，參加維也納學團的學術討論聚會。

又例如，有來自印度一所大學的，一位尚未世界知名的物理學家，企圖發表一篇文章，但給學術期刊拒絕了。於是，他把那篇文章，寄給了愛因思坦。

雖然素未謀面(?)，愛因思坦仍然用心閱讀，發現該文有料到，十分有意思。不單如此，他更親自把文章由英文翻譯成德文。在他的引薦下，德國一著名物理期刊出版了該文。

那位印度物理學家，就是後來舉世聞名的玻色。

愛因思坦一生幾大曠世鉅著之一，玻色-愛因斯坦統計規律，就是緣起於他和玻色的這次合作。

— Me@2019-10-29 10:20:33 PM

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Bose adapted this lecture into a short article called Planck’s Law and the Hypothesis of Light Quanta and submitted it to the Philosophical Magazine. However, the referee’s report was negative, and the paper was rejected. Undaunted, he sent the manuscript to Albert Einstein requesting publication in the Zeitschrift für Physik. Einstein immediately agreed, personally translated the article from English into German (Bose had earlier translated Einstein’s article on the theory of General Relativity from German to English), and saw to it that it was published. Bose’s theory achieved respect when Einstein sent his own paper in support of Bose’s to Zeitschrift für Physik, asking that they be published together. The paper came out in 1924.

— Wikipedia on Bose–Einstein statistics

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2019.10.29 Tuesday (c) All rights reserved by ACHK