Problem 14.1b

A First Course in String Theory

14.1 Counting bosonic states

~~~

Let $N(n, k) = {n + k - 1 \choose k - 1}$, the number of ways to put $n$ indistinguishable balls into $k$ boxes.

p.318 “For open bosonic strings $\alpha' M^2 = N^\perp - 1$, …”

When $\alpha' M^2 = 3$, $N^\perp = 4$, the cases are:

1. four $a_1^\dagger$‘s:

$N(4,24) = 17550$

2. one $a_2^\dagger$ and two $a_1^\dagger$‘s:

$24 \times N(2,24) = 24 \times 300$

3. two $a_2^\dagger$‘s:

$N(2,24) = 300$

4. one $a_3^\dagger$ and one $a_1^\dagger$:

$24 \times 24 = 576$

5. one $a_4^\dagger$:

$24$

Total number of possible states for $N^\perp = 4$ is 25650.

— Me@2015-08-13 12:05:57 PM