Daily Archives: February 16, 2018

深淵 2

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與魔鬼戰鬥的人，應當小心自己不要成為魔鬼。當你遠遠凝視深淵時，深淵也在凝視你。

— 尼采

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勇者鬥惡龍，小心勇者變惡龍。

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As soon as men decide that all means are permitted to fight an evil, then their good becomes indistinguishable from the evil that they set out to destroy.

— Christopher Dawson

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2018.02.16 Friday ACHK

Problem 14.3b5

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Quick Calculation 14.4b | A First Course in String Theory

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Massive level in the open superstring

Consider the first and second excited levels of the open superstring (\alpha' M^2 = 1 and \alpha' M^2 = 2). List the states in NS sector and the states in the R sector. Confirm that you get the same number of states.

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When \alpha' M^2 = 2, by Equation (14.37),

M^2 = \frac{1}{\alpha'} \left( - \frac{1}{2} + N^\perp \right)

N^\perp = \frac{5}{2}

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\{ b_{-1/2}^I b_{-1/2}^J b_{-1/2}^K b_{-1/2}^L b_{-1/2}^M,

b_{-3/2}^I b_{-1/2}^L b_{-1/2}^M,

b_{-5/2}^I,

\alpha_{-1}^I b_{-1/2}^J b_{-1/2}^K b_{-1/2}^M,

\alpha_{-1}^I b_{-3/2}^J,

\alpha_{-2}^I b_{-1/2}^J,

\alpha_{-1}^I \alpha_{-1}^J b_{-1/2}^K \} |NS \rangle \otimes |p^+, \vec p_T \rangle

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There are \frac{8 \times 7 \times 6 \times 5 \times 4}{5!} = {8 \choose 5} = 56 states for

\{ b_{-1/2}^I b_{-1/2}^J b_{-1/2}^K b_{-1/2}^L b_{-1/2}^M |NS \rangle \otimes |p^+, \vec p_T \rangle

There are 8 \times \frac{8 \times 7}{2} = 224 states for

\{ b_{-3/2}^I b_{-1/2}^L b_{-1/2}^M \} |NS \rangle \otimes |p^+, \vec p_T \rangle

There are 8 states for

\{ b_{-5/2}^I \} |NS \rangle \otimes |p^+, \vec p_T \rangle

There are 8 \times {8 \choose 3} = 448 states for

\{ \alpha_{-1}^I b_{-1/2}^J b_{-1/2}^K b_{-1/2}^M \} |NS \rangle \otimes |p^+, \vec p_T \rangle

There are 8 \times 8 = 64 states for

\{ \alpha_{-1}^I b_{-3/2}^J \} |NS \rangle \otimes |p^+, \vec p_T \rangle

There are 8 \times 8 = 64 states for

\{ \alpha_{-2}^I b_{-1/2}^J \} |NS \rangle \otimes |p^+, \vec p_T \rangle

There are \left( \frac{8 \times 7}{2!} + 8 \right) \times 8 = 288 states for

\{ \alpha_{-1}^I \alpha_{-1}^J b_{-1/2}^K \} |NS \rangle \otimes |p^+, \vec p_T \rangle

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So the total number of states is 56 + 224 + 8 + 448 + 64 + 64 + 288 = 1152.

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You can check this answer against Equation (14.67):

f_{NS} (x) = \frac{1}{\sqrt{x}} + 8 + 36 \sqrt{x} + 128 x + 402 x \sqrt{x} + 1152 x^2 + ...

— Me@2018-02-16 03:22:13 PM

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2018.02.16 Friday (c) All rights reserved by ACHK