17 | April | 2018 | Physics Town

Closed string degeneracies | A First Course in String Theory

(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):

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

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

$N^\perp = \bar N^\perp$

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

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

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

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

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

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Problem 14.4a2