- #1
Fysicus
- 3
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Hi folks,
Been trying to fill some of the more formal gaps in my knowledge by tackling the more technical stuff in P&S Chapter 7. Their derivation of the LSZ formula is quite different to those of books like, say, Srednicki, as they basically Fourier transform the whole argument as I understand it to concentrate more interpreting the pole structure of a general correlation function.
My question is this - having Fourier transformed the n-point function with respect to one field co-ordinate:
[itex]\int d^4x e^{ip \cdot x} \left\langle \Omega \left| T{ \phi (x) \phi (z_{1}) \phi (z_{2}) ... } \right| \Omega \right\rangle[/itex]
And splitting the [itex]x^{0}[/itex] integral into three zones (I, II, III):
[itex] \int dx^{0} = \int^{\infty}_{T_{+}} dx^{0} + \int^{T_{+}}_{T_{-}} dx^{0} + \int^{T_{-}}_{\infty} dx^{0}[/itex]
They insert a complete set of states from the interacting theory, and choose to evaluate region I and do the momentum integrals that come with that, along with the position integrals from our Fourier Transform to arrive at the very ugly Eq. 7.36:
[itex]\int_{I} d^4x e^{ip \cdot x} \left\langle \Omega \left| T{ \phi (x) \phi (z_{1}) \phi (z_{2}) ... } \right| \Omega \right\rangle = \sum \frac{1}{2E_\textbf{p}(\lambda)} \frac{i e^{i(p^{0}-E_{\textbf{p}}+i\epsilon)T_{+}}}{p^{0}-E_{\textbf{p}}+i \epsilon} \left\langle \Omega \left| \phi (0) \right| \lambda_{0} \right\rangle \left\langle \lambda_{\textbf{p}} \left| T \phi (z_{1}) \phi (z_{2}) \ldots \right| \Omega \right\rangle [/itex]
Where the sum runs over the different mass states [itex] \lambda [/itex] and the 'I' on the integral just means [itex] dx^{0} [/itex] integrated over the region: [itex] \int^{\infty}_{T_{+}} dx^{0} [/itex].
At this point they state " the denominator is just that [as][itex] p^2 - m^2_{\lambda} [/itex] " . I can't see why this is the case - yes, the pole in [itex] p^{0} [/itex] is in the same location as one of the poles in [itex] p^2 - m^2_{\lambda} [/itex] , but since all the integrals are done I can't see any way to massage this into this form exactly (by that i mean playing with delta functions and the E).
In other words, how can this statement true? Are we to literally claim that:
Since, close to the pole, the exponential factor goes to 1, and the leftmost matrix element is the square root of the field strength renormalization factor [itex] \sqrt{Z} [/itex], do they just mean that this quantity has a pole with residue [itex] \sqrt{Z} [/itex] at [itex] p^{0} = E_{\textbf{p}} [/itex] much like the piece of the full propagator containing the mass singularities? Is this possibly what they mean by the next statement (7.37):
[itex]\int d^4x e^{ip \cdot x} \left\langle \Omega \left| T{ \phi (x) \phi (z_{1}) \phi (z_{2}) ... } \right| \Omega \right\rangle \sim \frac{i}{p^{2}-m^{2}+i\epsilon} \sqrt{Z} \left\langle \textbf{p} \left| T \phi (z_{1}) \ldots \right| \Omega \right\rangle [/itex]
as
[itex]p^{0} \rightarrow +E_{\textbf{p}}[/itex]
Had a look at the P&S questions reference thread: https://www.physicsforums.com/showthread.php?t=400073 , and it seems no-one has asked about this derivation before, so either I'm missing something, overthinking it, or any explanations might be useful to others for future reference. Cheers!
Been trying to fill some of the more formal gaps in my knowledge by tackling the more technical stuff in P&S Chapter 7. Their derivation of the LSZ formula is quite different to those of books like, say, Srednicki, as they basically Fourier transform the whole argument as I understand it to concentrate more interpreting the pole structure of a general correlation function.
My question is this - having Fourier transformed the n-point function with respect to one field co-ordinate:
[itex]\int d^4x e^{ip \cdot x} \left\langle \Omega \left| T{ \phi (x) \phi (z_{1}) \phi (z_{2}) ... } \right| \Omega \right\rangle[/itex]
And splitting the [itex]x^{0}[/itex] integral into three zones (I, II, III):
[itex] \int dx^{0} = \int^{\infty}_{T_{+}} dx^{0} + \int^{T_{+}}_{T_{-}} dx^{0} + \int^{T_{-}}_{\infty} dx^{0}[/itex]
They insert a complete set of states from the interacting theory, and choose to evaluate region I and do the momentum integrals that come with that, along with the position integrals from our Fourier Transform to arrive at the very ugly Eq. 7.36:
[itex]\int_{I} d^4x e^{ip \cdot x} \left\langle \Omega \left| T{ \phi (x) \phi (z_{1}) \phi (z_{2}) ... } \right| \Omega \right\rangle = \sum \frac{1}{2E_\textbf{p}(\lambda)} \frac{i e^{i(p^{0}-E_{\textbf{p}}+i\epsilon)T_{+}}}{p^{0}-E_{\textbf{p}}+i \epsilon} \left\langle \Omega \left| \phi (0) \right| \lambda_{0} \right\rangle \left\langle \lambda_{\textbf{p}} \left| T \phi (z_{1}) \phi (z_{2}) \ldots \right| \Omega \right\rangle [/itex]
Where the sum runs over the different mass states [itex] \lambda [/itex] and the 'I' on the integral just means [itex] dx^{0} [/itex] integrated over the region: [itex] \int^{\infty}_{T_{+}} dx^{0} [/itex].
At this point they state " the denominator is just that [as][itex] p^2 - m^2_{\lambda} [/itex] " . I can't see why this is the case - yes, the pole in [itex] p^{0} [/itex] is in the same location as one of the poles in [itex] p^2 - m^2_{\lambda} [/itex] , but since all the integrals are done I can't see any way to massage this into this form exactly (by that i mean playing with delta functions and the E).
In other words, how can this statement true? Are we to literally claim that:
[itex]
\frac{1}{2E_\textbf{p}} \frac{1}{p^{0}-E_{\textbf{p}}+i\epsilon} = \frac{1}{p^{2}-m_{\lambda}^{2}}[/itex] ?
\frac{1}{2E_\textbf{p}} \frac{1}{p^{0}-E_{\textbf{p}}+i\epsilon} = \frac{1}{p^{2}-m_{\lambda}^{2}}[/itex] ?
Since, close to the pole, the exponential factor goes to 1, and the leftmost matrix element is the square root of the field strength renormalization factor [itex] \sqrt{Z} [/itex], do they just mean that this quantity has a pole with residue [itex] \sqrt{Z} [/itex] at [itex] p^{0} = E_{\textbf{p}} [/itex] much like the piece of the full propagator containing the mass singularities? Is this possibly what they mean by the next statement (7.37):
[itex]\int d^4x e^{ip \cdot x} \left\langle \Omega \left| T{ \phi (x) \phi (z_{1}) \phi (z_{2}) ... } \right| \Omega \right\rangle \sim \frac{i}{p^{2}-m^{2}+i\epsilon} \sqrt{Z} \left\langle \textbf{p} \left| T \phi (z_{1}) \ldots \right| \Omega \right\rangle [/itex]
as
[itex]p^{0} \rightarrow +E_{\textbf{p}}[/itex]
Had a look at the P&S questions reference thread: https://www.physicsforums.com/showthread.php?t=400073 , and it seems no-one has asked about this derivation before, so either I'm missing something, overthinking it, or any explanations might be useful to others for future reference. Cheers!