Problem 14.4a2

Closed string degeneracies | A First Course in String Theory

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(a) State the values of \alpha' M^2 and give the degeneracies for the first five mass levels of the closed bosonic string theory.

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p.288 Equation (13.48):

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

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p.290 “A basis vector | \lambda, \bar \lambda \rangle belongs to the state space _if and only if_ it satisfies the level-matching constraint”

N^\perp = \bar N^\perp

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\frac{1}{2} \alpha' M^2~| ~\text{Number of states}
-2~| ~1
0~| ~(D-2)^2

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\frac{1}{2} \alpha' M^2~| ~\text{Number of states}
2~| ...

{a_1^I}^\dagger {a_1^J}^\dagger \bar a_1^{K\dagger} \bar a_1^{L\dagger} | p^+, \vec p_T \rangle
{a_1^I}^\dagger {a_1^J}^\dagger \bar a_2^{K\dagger} | p^+, \vec p_T \rangle
{a_2^I}^\dagger \bar a_1^{K\dagger} \bar a_1^{L\dagger} | p^+, \vec p_T \rangle

Assuming the signs of the wave functions do not matter:

 | |~\text{Number of states}
{a_1^I}^\dagger {a_1^J}^\dagger \bar a_1^{K\dagger} \bar a_1^{L\dagger}~| \left[ \frac{(D-2)(D-1)}{2} \right]^2
{a_1^I}^\dagger {a_1^J}^\dagger \bar a_2^{K\dagger} | p^+, \vec p_T \rangle~|
{a_2^I}^\dagger \bar a_1^{K\dagger} \bar a_1^{L\dagger} | p^+, \vec p_T \rangle~|

p.291 How come the number of the components of the matrix represents the number of states?

p.292 For massless states, we have only one {a_1^{I}}^\dagger and \bar a_1^{J\dagger}, where a and \bar a cannot interchange. So I and J also cannot interchange. In other words, in Equation (13.69), there is no double count. All the states are independent.

But for non-massless states, this probably is not true anymore:

 | ~\text{Number of states}
{a_1^I}^\dagger {a_1^J}^\dagger \bar a_1^{K\dagger} \bar a_1^{L\dagger} | p^+, \vec p_T \rangle~| ~\left[ \frac{(D-2)(D-1)}{2} \right]^2
{a_1^I}^\dagger {a_1^J}^\dagger \bar a_2^{K\dagger} | p^+, \vec p_T \rangle~| ~\left[ \frac{(D-2)(D-1)}{2} \right](D-2)
{a_2^I}^\dagger \bar a_1^{K\dagger} \bar a_1^{L\dagger} | p^+, \vec p_T \rangle~| ~(D-2)\left[ \frac{(D-2)(D-1)}{2} \right]

— Me@2018-04-17 04:47:31 PM

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2018.04.17 Tuesday (c) All rights reserved by ACHK