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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] ?

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!

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# Correlation function poles in Peskin's derivation of LSZ formula

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