Zwiebach question with link to text

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SUMMARY

The discussion centers on the interpretation of equation 10.33 from "A First Course in String Theory." It confirms that this equation is presented as a general solution to the equation of motion for φ, specifically equation 10.14, rather than being derived. The term "normalization" refers to ensuring that the norm of φ_p equals 1. The "volume of space" factor V is clarified as a numerical factor intended to cancel out during volume integrals, not a Jacobian. Additionally, for Quick Calculation 10.3, it is advised to use equation (10.35) instead of (10.7) due to a typographical error in the latter.

PREREQUISITES
  • Understanding of differential equations, particularly in the context of physics.
  • Familiarity with normalization concepts in quantum mechanics.
  • Knowledge of volume integrals and their applications in theoretical physics.
  • Basic comprehension of string theory principles and terminology.
NEXT STEPS
  • Study the derivation and implications of equation 10.14 in "A First Course in String Theory."
  • Explore normalization techniques in quantum field theory (QFT).
  • Investigate the role of periodic boundary conditions in theoretical physics calculations.
  • Review the significance of Jacobians in volume integrals and their applications in physics.
USEFUL FOR

This discussion is beneficial for students and researchers in theoretical physics, particularly those studying string theory, quantum mechanics, and differential equations. It is especially relevant for individuals seeking clarity on normalization and solution techniques in quantum field theory.

ehrenfest
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Homework Statement


http://books.google.com/books?id=Xm...nDQ&sig=Eqnkxjj7B9gc4-nig1fgUkk7AqQ#PPA172,M1
I am very confused about equation 10.33.

This equation is not derived it is just given, correct?

What does normalization mean in this context?

By the "volume of space" factor V, does he really mean a Jacobian?

Also for Quick Calculation 10.3, how can E_p possibly get in the numerator when it is in the denominator of phi_p?

Homework Equations


The Attempt at a Solution

 
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ehrenfest said:
This equation (10.33) is not derived it is just given, correct?
Yes.

ehrenfest said:
What does normalization mean in this context?
It means that the norm (also called the length) of [itex]\phi_p[/itex] is 1.

ehrenfest said:
By the "volume of space" factor V, does he really mean a Jacobian?
No, it is just a numerical factor that is meant to cancel out somewhere down the line when a volume integral is performed. In fact, he points this out in the text just below equation (10.36) on page 173.

ehrenfest said:
Also for Quick Calculation 10.3, how can E_p possibly get in the numerator when it is in the denominator of phi_p?
This is straightforward, so I don't want to spoil your fun in figuring it out. Besides in the homework forum, I'm really only allowed to guide you, not tell you. So do as the author says and plug (10.33) into (10.7) [Note: actually, you had better plug it into (10.35) because even though they are meant to be the same, (10.7) has a typo in it or does in my copy of the book]. You will need to apply equation (2.65) page 24, to get the E factor correct.
 
ehrenfest said:

Homework Statement


http://books.google.com/books?id=Xm...nDQ&sig=Eqnkxjj7B9gc4-nig1fgUkk7AqQ#PPA172,M1
I am very confused about equation 10.33.

This equation is not derived it is just given, correct?

Actually, this is simply the most general solution to the equation of motion for phi (eq 10.14). As he wrot, a plane wave solution is of the form 10.16 so a general solution is a linear combination as in 10.33. At this point, there is nothing about qft of particle physics stuff...it's just solving a diff eq. The V factor is convenient if one works in a "box" with peridoic boundary conditions. It's put there to cancel later on in the calculation.
 

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