Srednicki's Lehmann-Kallen propagator derivation doubt

[msid]

Homework Statement

Problem with the ordering of integrals in the derivation of the Lehmann-Kaller form of the exact propagator in Srednicki's book.

We start with the definition of the exact propagator in terms of the 2-point correlation function and introduce the complete set of momentum eigenstates and then define a certain spectral density in terms of a delta function. But the spectral density is also a function of 'k', so we cannot take the spectral density outside the integral over 'k'. Since that is not possible, the subsequent manipulations fail too.

Homework Equations

In Srednicki's book :
Equation 13.11 and 13.12

If that is incorrect, the use of 13.15 to get 13.16 is not possible.

The Attempt at a Solution

I don't see how it is possibe to derive the equation without that interchange.

I'd appreciate any clarifications on this issue. Am I missing some trivial thing?

 

[gabbagabbahey]

I don't have Srednicki's text, could you post the relevant equations?

 

[msid]

Luckily, the author has a free draft version of the textbook on his website. The relevant contents are the same.

http://www.physics.ucsb.edu/~mark/qft.html
Pg 107-108
Eqns 13.11, 13.12 and the use of 13.15 to get 13.16

 

[szg138]

I am new to the Physics forum, so I appologise if I doing something wrong by digging up an old thread. I too got stuck at this proof for a while. Now I think I have figured out the correction to the proof. I thought I will share it with others.

There seems to be a mistake in the proof. But the final result is correct.
Instead of rearranging the integrals as Srednicki has done in equation 13.12, retain ρ inside the momentum integral.
1) Now, convince yourself that ρ is independent of the spatial direction of k.
2) Do the time ordering similar to the one in 13.13 and 13.14 but now retaining ρ inside the k integral.
3) The author has used equation 13.15 which is proved in chapter 8. However this is not applicable anymore because ρ is present. However one can obtain an identity similar to the identity 13.15. In this, each integral has an additional function F(|\vec{k}|) inside. Proof is on the lines of the proof of 13.15. Try it.
4) Inserting the new identity you arrive at 13.16

Hope this helps. BTW let me know if there is flaw in my new proof. :)

 

[szg138]

Ignore my previous post. Equation 13.12 is correct.

Heres the explanation:
rho(s) is invariant under Lorentz transformation because, we have summed over all the possibilities for n.
Since rho is invariant under LorentzTransformations, it can depend only on k^2=k_\mu \eta^{\mu\nu} k_\nu=M^2. But then k^2 is effectively equal to s due to the dirac delta.

Since rho is independent of k, it can be taken out of the integral!