26 | July | 2015 | Physics Town

A First Course in String Theory

14.1 Counting bosonic states

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$n$ is the number of $a$ ‘s.

$k$ is the number of different $a$ ‘s.

For $a^{i_1} a^{i_2}$ , $n=2$ .

(Indices $i_1$ and $i_2$ are not powers. Instead, they are just upper indices for representing different $a$ ‘s.)

When all $a$ ‘s commute, $a^1 a^2$ and $a^2 a^1$ , for example, represent the same state. So we have to avoid double-counting, except for the same $a$ ‘s states, such as $a^3 a^3$ .

By direct counting, without using the formula, the number of products of the form $a^{i_1} a^{i_2}$ can be built is

$\frac{k(k-1)}{2} + k$
$= \frac{(k + 1)k}{2}$

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

By using the formula, the number of products of the form $a^{i_1} a^{i_2}$ can be built is

$N(2,k)$

$= \frac{(2+k-1)!}{2!(k-1)!}$

$= \frac{(k+1)k}{2}$

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Daily Archives: July 26, 2015

Problem 14.1a