Peskin Schröder Chapter 7.1 Field Strength Renormalization

In summary, the conversation discusses questions about the derivation of the two point function for the interacting case in chapter 7.1 of "An Introduction to Quantum Field Theory" by Peskin and Schröder. The third line in the equation is derived by setting the state |\lambda_p\rangle to be a momentum eigenstate and replacing the momentum operator P with the eigenvalue p. The change e^{-iPx} \rightarrow e^{-ipx} is due to the momentum operator acting on the vacuum, which is assumed to be Lorentz invariant.
  • #1
Phileas.Fogg
32
0
Hello,
I read chapter 7.1 of "An Introduction to Quantum Field Theory" by Peskin and Schröder and have two questions.

They derive the two point function for the interacting case.
On page 213 they manipulate the matrix element, after insertion of the complete set of eigenstates.

[tex] <\Omega | \Phi (x) | \lambda_p > [/tex]
[tex] = < \Omega | e^{iPx} \Phi (0) e^{- iPx} | \lambda_p > [/tex]
[tex] = < \Omega | \Phi (0) | \lambda_p > e^{- ipx} \end{array} [/tex]

with [tex] E_p = p^0[/tex]

1. Could anyone explain to me, how they get the third line above (also see P & S: equation (7.4) ) ?

2. Later on page 215 they make a Fourier Transform of the spectral decomposition. I don't know, how they derive equation 7.9

[tex] \int d^4 x e^{ipx} < \Omega | T \Phi (x) \Phi(0) | \Omega> = \frac{iZ}{p^2 - m^2 + i \epsilon} + \int_{~4m^2}^{\infty} \frac{d M^2}{2 \pi} \rho(M^2) \frac{i}{p^2 - M^2 + i \epsilon} [/tex]

Regards,
Mr. Fogg
 
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  • #2
Phileas.Fogg said:
Hello,
[tex] <\Omega | \Phi (x) | \lambda_p > [/tex]
[tex] = < \Omega | e^{iPx} \Phi (0) e^{- iPx} | \lambda_p > [/tex]
[tex] = < \Omega | \Phi (0) | \lambda_p > e^{- ipx} \end{array} [/tex]

with [tex] E_p = p^0[/tex]

1. Could anyone explain to me, how they get the third line above (also see P & S: equation (7.4) ) ?


The momentum operator acting to the left on the vacuum gives zero since the vacuum is assumed to be Lorentz invariant .

-JB
 
  • #3
And where does the change [tex] e^{-iPx} \rightarrow e^{-ipx} [/tex] come from?
 
  • #4
Don't have the book with me, but my best guess would be that they set the state [itex]|\lambda_p\rangle[/itex] to be a momentum eigenstate. This means the momentum operator P acts on this state and is replaced by the momentum eigenvalue p (which is ofcourse a 4-vector).

You can also check that when the operator [itex]e^P[/itex] acts on the state the P is also replaced by the eigenvalue p turning it into [itex]e^p[/itex].
 

1. What is the purpose of renormalization in field theory?

Renormalization is a technique used in quantum field theory to remove infinities that arise in calculations of particle interactions. These infinities are a result of treating point-like particles as if they have no size, which leads to unphysical results. By renormalizing, we can extract meaningful and finite predictions from the theory.

2. Why is field strength renormalization important in Quantum Electrodynamics (QED)?

In QED, field strength renormalization is crucial for obtaining finite and physically meaningful predictions. Without it, calculations would lead to infinite values for the electric charge and mass of the electron, making the theory useless. By renormalizing the field strength, we can obtain finite values for these quantities and make accurate predictions.

3. How does field strength renormalization affect the coupling constant in QED?

In QED, the coupling constant is a measure of the strength of the interaction between particles. Field strength renormalization affects the coupling constant by removing the infinities that arise in its calculation, resulting in a finite and physically meaningful value. This allows us to make accurate predictions for the strength of particle interactions.

4. Is field strength renormalization applicable to all quantum field theories?

Yes, field strength renormalization is a general technique that can be applied to any quantum field theory. It is not specific to QED and has been successfully used in other theories, such as Quantum Chromodynamics (QCD) and the Standard Model of particle physics.

5. What are some challenges in implementing field strength renormalization?

One major challenge in implementing field strength renormalization is the complexity of the calculations involved. The process can be quite time-consuming and requires advanced mathematical techniques. Additionally, there is always the risk of making errors in the calculations, which can lead to incorrect predictions. Another challenge is determining the appropriate scale at which to renormalize the field strength, as this can affect the final results and requires careful consideration.

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