Problem 14.1a

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}

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}

— Me@2015-07-26 08:43:22 AM
 
 
 
2015.07.26 Sunday (c) All rights reserved by ACHK