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

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SUMMARY

The discussion centers on the potential omission of the term \(\alpha_{-1}\alpha_{1}\) in equation 12.130 of Zwiebach's text, which is linked to the previous equation 12.129. Participants confirm that upon evaluating the commutators in equations 12.125 and 12.126, and applying equation 12.128, the result falls short of 4L_0 by precisely the term \(\alpha_{-1}\alpha_{1}\). This indicates that the term is indeed necessary for the equation's accuracy.

PREREQUISITES
  • Understanding of string theory concepts, particularly commutators.
  • Familiarity with Zwiebach's "A First Course in String Theory".
  • Knowledge of equations 12.125, 12.126, 12.128, and 12.130.
  • Basic proficiency in mathematical notation used in theoretical physics.
NEXT STEPS
  • Review the derivation of equations 12.125 and 12.126 in Zwiebach's text.
  • Study the implications of commutators in string theory.
  • Examine the role of the term \(\alpha_{-1}\alpha_{1}\) in other equations within string theory.
  • Explore additional resources on the mathematical foundations of string theory.
USEFUL FOR

The discussion is beneficial for students and researchers in theoretical physics, particularly those focusing on string theory and its mathematical formulations.

ehrenfest
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Homework Statement


Shouldn't there be an

[tex]\alpha_{-1}\alpha_{1}[/tex]

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 [itex]4L_0[/itex] by exactly [itex]\alpha_{-1}\alpha_1[/itex].
 

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