Equation 12.130 on Zwiebach Page 225: Is an Alpha Missing?

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Homework Statement


Shouldn't there be an

\alpha_{-1}\alpha_{1}

added to equation 12.130 on account of equation 12.129, or is that 0 for some reason?

Homework Equations


The Attempt at a Solution

 
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If you work out the sum of the commutators of the terms in parentheses in equations (12.125) and (12.126), and use equation (12.128), you will find it deficient from 4L_0 by exactly \alpha_{-1}\alpha_1.
 

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