# Problem 14.5a1

Counting states in heterotic $SO(32)$ string theory | A First Course in String Theory

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(a) Consider the left NS’ sector. Write the precise mass-squared formula with normal-ordered oscillators and the appropriate normal-ordering constant.

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$\displaystyle{\alpha' M_L^2 = \frac{1}{2} \sum_{n \ne 0} \bar \alpha_{-n}^I \bar \alpha_n^I + \frac{1}{2} \sum_{r \in \mathbf{Z} + \frac{1}{2}}r \lambda_{-r}^A \lambda_r^A}$

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What is normal-ordering?

Put all the creation operators on the left.

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What for?

p.251 “It is useful to work with normal-ordered operators since they act in a simple manner on the vacuum state. We cannot use operators that do not have a well defined action on the vacuum state.”

“The vacuum expectation value of a normal ordered product of creation and annihilation operators is zero. This is because, denoting the vacuum state by $|0\rangle$, the creation and annihilation operators satisfy”

$\displaystyle{\langle 0 | \hat{a}^\dagger = 0 \qquad \textrm{and} \qquad \hat{a} |0\rangle = 0}$

— Wikipedia on Normal order

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— This answer is my guess. —

$\displaystyle{\sum_{n \ne 0} \bar \alpha_{-n}^I \bar \alpha_n^I}$

$\displaystyle{= \sum_{n \in \mathbf{Z}^-} \bar \alpha_{-n}^I \bar \alpha_n^I + \sum_{n \in \mathbf{Z}^+} \bar \alpha_{-n}^I \bar \alpha_n^I}$

$\displaystyle{= \sum_{n \in \mathbf{Z}^+} \bar \alpha_{n}^I \bar \alpha_{-n}^I + \sum_{n \in \mathbf{Z}^+} \bar \alpha_{-n}^I \bar \alpha_n^I}$

$\displaystyle{= \sum_{n \in \mathbf{Z}^+} \left[ \bar \alpha_{n}^I \bar \alpha_{-n}^I - \bar \alpha_{-n}^I \bar \alpha_{n}^I + \bar \alpha_{-n}^I \bar \alpha_{n}^I \right] + \sum_{n \in \mathbf{Z}^+} \bar \alpha_{-n}^I \bar \alpha_n^I}$

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$\displaystyle{= \sum_{n \in \mathbf{Z}^+} \left[ \bar \alpha_{n}^I, \bar \alpha_{-n}^I \right] + \sum_{n \in \mathbf{Z}^+} \bar \alpha_{-n}^I \bar \alpha_{n}^I + \sum_{n \in \mathbf{Z}^+} \bar \alpha_{-n}^I \bar \alpha_n^I}$

$= \displaystyle{\sum_{n \in \mathbf{Z}^+} n \eta^{II} + 2 \sum_{n \in \mathbf{Z}^+} \bar \alpha_{-n}^I \bar \alpha_{n}^I}$

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c.f. p.251:

$\displaystyle{\sum_{n \ne 0} \bar \alpha_{-n}^I \bar \alpha_n^I}$

$\displaystyle{= \sum_{n \in \mathbf{Z}^+} n \eta^{II} + 2 \sum_{n \in \mathbf{Z}^+} \bar \alpha_{-n}^I \bar \alpha_{n}^I}$

$\displaystyle{= \frac{-1}{12} (D - 2) + 2 \sum_{n \in \mathbf{Z}^+} \bar \alpha_{-n}^I \bar \alpha_{n}^I}$

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Equation at Problem 14.5:

$\displaystyle{\alpha' M_L^2}$

$\displaystyle{= \frac{1}{2} \sum_{n \ne 0} \bar \alpha_{-n}^I \bar \alpha_n^I + \frac{1}{2} \sum_{r \in \mathbf{Z} + \frac{1}{2}}r \lambda_{-r}^A \lambda_r^A}$

$\displaystyle{= \frac{1}{2} \left[ \frac{-1}{12} (D - 2) + 2 \sum_{n \in \mathbf{Z}^+} \bar \alpha_{-n}^I \bar \alpha_{n}^I \right] + \frac{1}{2} \sum_{r \in \mathbf{Z} + \frac{1}{2}}r \lambda_{-r}^A \lambda_r^A}$

$\displaystyle{= \frac{-1}{24} (D - 2) + \sum_{n \in \mathbf{Z}^+} \bar \alpha_{-n}^I \bar \alpha_{n}^I + \frac{1}{2} \sum_{r \in \mathbf{Z} + \frac{1}{2}}r \lambda_{-r}^A \lambda_r^A}$

$\displaystyle{= \frac{-1}{8} + \sum_{n \in \mathbf{Z}^+} \bar \alpha_{-n}^I \bar \alpha_{n}^I + \frac{1}{2} \sum_{r \in \mathbf{Z} + \frac{1}{2}}r \lambda_{-r}^A \lambda_r^A}$

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$D = 10$

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$\displaystyle{\sum_{r \in \mathbf{Z} + \frac{1}{2}}r \lambda_{-r}^A \lambda_r^A}$

$\displaystyle{= \sum_{r = - \frac{1}{2}, - \frac{3}{2}, ...} r \lambda_{-r}^A \lambda_r^A + \sum_{r = \frac{1}{2}, \frac{3}{2}, ...} r \lambda_{-r}^A \lambda_r^A}$

$\displaystyle{= \sum_{r = \frac{1}{2}, \frac{3}{2}, ...} (-r) \lambda_{r}^A \lambda_{-r}^A + \sum_{r = \frac{1}{2}, \frac{3}{2}, ...} r \lambda_{-r}^A \lambda_r^A}$

$\displaystyle{= \sum_{r = \frac{1}{2}, \frac{3}{2}, ...} r \left[ (-1) \lambda_{r}^A \lambda_{-r}^A + \lambda_{-r}^A \lambda_r^A \right]}$

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$\displaystyle{= \sum_{r = \frac{1}{2}, \frac{3}{2}, ...} r \left[ (-1) \lambda_{r}^A \lambda_{-r}^A + \lambda_{-r}^A \lambda_r^A \right]}$

$\displaystyle{= \sum_{r = \frac{1}{2}, \frac{3}{2}, ...} r \left[ \lambda_{-r}^A, \lambda_r^A \right]}$

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Equation (14.29):

$\displaystyle{\left\{ b_r^I, b_s^J \right\} = \delta_{r+s, 0} \delta^{IJ}}$

$\displaystyle{b_r^I b_s^J = - b_s^I b_r^J + \delta_{r+s, 0} \delta^{IJ}}$

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$\displaystyle{\sum_{r \in \mathbf{Z} + \frac{1}{2}}r \lambda_{-r}^A \lambda_r^A}$

$\displaystyle{= \sum_{r = \frac{1}{2}, \frac{3}{2}, ...} r \left[ (-1) \lambda_{r}^A \lambda_{-r}^A + \lambda_{-r}^A \lambda_r^A \right]}$

$\displaystyle{= \sum_{r = \frac{1}{2}, \frac{3}{2}, ...} r \left[ (-1) \left( - \lambda_{-r}^A \lambda_r^A + \delta_{r-r, 0} \delta^{AA} \right) + \lambda_{-r}^A \lambda_r^A \right]}$

$\displaystyle{= \sum_{r = \frac{1}{2}, \frac{3}{2}, ...} r \left[ \lambda_{-r}^A \lambda_r^A + \lambda_{-r}^A \lambda_r^A - 1 \right]}$

$\displaystyle{= \sum_{r = \frac{1}{2}, \frac{3}{2}, ...} r \left[ 2 \lambda_{-r}^A \lambda_r^A - 1 \right]}$

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$\displaystyle{\sum_{r \in \mathbf{Z} + \frac{1}{2}}r \lambda_{-r}^A \lambda_r^A}$

$\displaystyle{= - \sum_{r = \frac{1}{2}, \frac{3}{2}, ...} r + \sum_{r = \frac{1}{2}, \frac{3}{2}, ...} r \left[ b_{-r}^A b_r^A + \lambda_{-r}^A \lambda_r^A \right]}$

$\displaystyle{= - \frac{1}{2} \sum_{r = 1, 3, ...} r + \sum_{r = \frac{1}{2}, \frac{3}{2}, ...} r \left[ b_{-r}^A b_r^A + \lambda_{-r}^A \lambda_r^A \right]}$

$\displaystyle{= \left[ - \frac{1}{24} + \sum_{r = \frac{1}{2}, \frac{3}{2}, ...} r \left( b_{-r}^A b_r^A + \lambda_{-r}^A \lambda_r^A \right) \right]}$

— This answer is my guess. —

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— Me@2018-08-06 10:23:48 PM

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# CSS

— Me@2018-04-05 06:09:36 PM

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# Cat

— Me@2018-02-19 02:33:42 PM

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# The Sixth Sense, 2

It is difficult only for the others.

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It is also true when you are stupid.

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2018.01.26 Friday ACHK

# Teachers, 2

The most important teachers in my life, 2

6. 程兆海 (pure maths)

7. Christopher Nolan (film director)

8. Lubos Motl (physicist)

# 如何拯救眾生 2

How to improve the world indefinitely?

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There is law of conservation of energy.
There is law of conservation of mass.
But, there is no law of conservation of ideas.

— Me

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If you have an apple and I have an apple, and we exchange apples, we both still only have one apple. But if you have an idea and I have an idea, and we exchange ideas, we each now have two ideas.

— George Bernard Shaw

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. . . it is the action of the thinking power called an idea, which an individual may exclusively possess as long as he keeps it to himself; but the moment it is divulged, it forces itself into the possession of every one, and the receiver cannot dispossess himself of it. Its peculiar character, too, is that no one possesses the less, because every other possesses the whole of it. He who receives an idea from me, receives instruction himself without lessening mine; as he who lights his taper at mine, receives light without darkening me.

That ideas should freely spread from one to another over the globe, for the moral and mutual instruction of man, and improvement of his condition, seems to have been peculiarly and benevolently designed by nature, …

— Thomas Jefferson

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# 如何拯救眾生 1

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The meeting of two personalities is like the contact of two chemical substances: if there is any reaction, both are transformed. — Carl Jung

Generalized Newton’s Third Law: You cannot change others without being changed. — John Wheeler

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You are an A person. You befriend with an F person. The result is that your friend becomes a better person, you become a worse person. In theory, you become a C person. Your friend becomes a C person. In practice, since being a bad person is much easier than being a good person, you two will become both F people. — Me

Be slow in making friends, slower in changing. — Benjamin Franklin

You have to know how to accept rejection and reject acceptance. — Ray Bradbury

If you hire all A people, he said, they’ll also hire A people. But if you hire B people, they’ll hire the C people and then it’s all over. — Joel Spolsky

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So that means you cannot help anyone weaker than you?

NO.

Instead, you should help others. But you have to help others in a smart way. — Me

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How?

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. . . it is the action of the thinking power called an idea, which an individual may exclusively possess as long as he keeps it to himself; but the moment it is divulged, it forces itself into the possession of every one, and the receiver cannot dispossess himself of it. Its peculiar character, too, is that no one possesses the less, because every other possesses the whole of it. He who receives an idea from me, receives instruction himself without lessening mine; as he who lights his taper at mine, receives light without darkening me.

— Thomas Jefferson

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# Amazing Gags 7

Chapter 7 Painting

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The secret source of Humor itself is not joy but sorrow.

– Mark Twain

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– Translation by Me

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2008.10.17 Friday (c) CHK2

# 3 Storyline

3 Storyline

3.1 In the beginning,
…3.1.1 in a sense, we are all time travellers

3.2 Now is the past of the future, the future of the past

3.3 You are nowhere but now-here

3.4 Living in now-here

3.5 Time messenger
…3.5.1 how to send messages to your past self
…………接收來自未來的訊息
…3.5.3 deja vu 似曾相識
…3.5.4 how to send messages to your future self
…3.5.5 regrets
…3.5.6 have no regrets
…3.5.7 palm
…3.5.8 preserve time

3.6 Now-here-I
…3.6.1 Foresight: prequel
…3.6.3 You are all of yourselves
…3.6.4 Now-here-I: the holographic universe
…3.6.6 Nothing less
…3.6.7 No persistence needed
…3.6.8 Eliminate the passage of time 與時並進
…3.6.9 Event realism 事件實在論
…3.6.10 Lost in time

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# Chrono Trigger 3

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Masato Kato confirmed that Cross featured a central theme of parallel worlds, as well as the fate of Schala, which he was previously unable to expound upon in Chrono Trigger. Concerning the ending sequences showing Kid searching for someone in a modern city, he hoped to make players realize that alternate futures and possibilities may exist in their own lives, and that this realization would “not … stop with the game”.

— Wikipedia on Chrono Cross

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2011.05.02 Monday ACHK

# Ra’s al Ghul

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Planned cast

Bale said he would return if Nolan did. Aaron Eckhart had expressed his enthusiasm to return for a sequel if asked, although he later stated Nolan verified that his character, Harvey Dent, is dead. Nolan confirmed that the ensemble cast from the first two films will be returning. Tom Hardy and Anne Hathaway have been cast as Bane and Selina Kyle. Juno Temple has been cast in a role described as a “street-smart Gotham girl”. Josh Pence was cast as a young Ra’s al Ghul, with his scenes to take place thirty years before the events of Batman Begins. Nestor Carbonell is in negotiations to reprise his role as Mayor Anthony Garcia. Joseph Gordon-Levitt has been cast as John Blake, “a Gotham City beat cop,” along with Marion Cotillard, who will be portraying Miranda Tate, “a Wayne Enterprises board member eager to help a still-grieving Bruce Wayne resume his father’s philanthropic endeavors for Gotham”.

— Wikipedia on The Dark Knight Rises

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2011.04.30 Saturday ACHK

# Teachers 4

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I’m about to become a teacher. How can I be a good one?

The best teachers I remember from school had three things in common:

(1) They had high standards. Like three year olds testing their parents, students will test teachers to see if they can get away with low-quality work or bad behavior. They won’t respect the teachers who don’t call them on it.

(2) They liked us. Like dogs, kids can tell very accurately whether or not someone wishes them well. I think a lot of our teachers either never liked kids much, or got burned out and started not to like them. It’s hard to be a good teacher once that happens. I can’t think of one teacher in all the schools I went to who managed to be good despite disliking students.

(3) They were interested in the subject. Most of the public school teachers I had weren’t really interested in what they taught. Enthusiasm is contagious, and so is boredom.

— Paul Graham

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2011.04.05 Tuesday ACHK

# Socialising

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Socialising on the internet is to socialising, what reality TV is to reality.

— Aaron Sorkin

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2011.04.04 Monday ACHK

# PayPal

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No, because I think we didn’t know what we were doing. I think the hallmark of a really good entrepreneur is that you’re not really going to build one specific company. The goal — at least the way I think about entrepreneurship — is you realize one day that you can’t really work for anyone else. You have to start your own thing.

— Max Levchin

— Founders at Work

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2011.03.31 Thursday ACHK

# Great Idea 4

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If so, this is good news indeed. Good ideas always tend to win eventually. The problem is, it can take a very long time. It took decades for relativity to be accepted, and the greater part of a century to establish that central planning didn’t work. So even a small increase in the rate at which good ideas win would be a momentous change — big enough, probably, to justify a name like the “new economy.”

— Paul Graham

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2011.03.29 Tuesday ACHK

# Blog 4

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In an attempt to avoid that trap, and to clarify my own thinking, I’ve decided to write for a very limited audience: me, and people who might be similar to me.

— Paul Buchheit

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2011.03.22 Tuesday ACHK

# Point of view

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“Point of view” is that quintessentially human solution to information overload, an intuitive process of reducing things to an essential relevant and manageable minimum.

— Paul Saffo

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2011.03.21 Monday ACHK

# Write 2

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Don’t write the essay readers expect; one learns nothing from what one expects. And don’t write the way they taught you to in school.

— Paul Graham

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2011.03.19 Saturday ACHK