Srednicki QFT chapter 67, LSZ formula

In summary, the LSZ formula for scalar fields can be equivalently expressed as a limit where the on-shell momentum is not fixed, allowing for a more convenient treatment of the correlation function.
  • #1
physicus
55
3

Homework Statement


I would like to know how to get from eq. (67.3) to (67.4) in Srednicki's book on QFT. The problem is the following:
Given the LSZ formula for scalar fields
[itex]\langle f|i \rangle = i \int d^{4}x_1e^{ik_1x_1}(\partial^{2}+m^{2})\ldots \langle 0|T\phi(x_1)\ldots|0\rangle[/itex]
This is supposed to be equivalent to:
[itex]\langle f|i \rangle = \lim_{k_i\to m^2} (-k_1^{2}+m^{2})\ldots \langle 0|T\widetilde{\phi}(k_1)\ldots|0\rangle[/itex]
where [itex]\widetilde{\phi}(k) = i \int d^4x e^{ikx} \phi(x)[/itex] an [itex]k^2=m^2[/itex] is not fixed.


Homework Equations


None


The Attempt at a Solution


Especially, I don't understand where the limes comes from. Here my attempt:
[itex]\langle f|i \rangle = i \int d^{4}x_1e^{ik_1x_1}(\partial^{2}+m^{2})\ldots \langle 0|T\phi(x_1)\ldots|0\rangle[/itex]
[itex]=\int d^{4}x_1e^{ik_1x_1}(\partial_1^{2}+m^{2})\ldots \langle 0|T \int\frac{d^4q_1}{(2\pi)^4}e^{-iq_1x_1}\widetilde{\phi}(q_1)\ldots|0\rangle[/itex]
[itex]=\int d^{4}x_1\int\frac{d^4q_1}{(2\pi)^4}e^{i(k_1-q_1)x_1}(-q_1^{2}+m^{2})\ldots \langle 0|T\widetilde{\phi}(q_1)\ldots|0\rangle[/itex]
[itex]=\int{d^4q_1}\delta^4(k_1-q_1)(-q_1^{2}+m^{2})\ldots \langle 0|T\widetilde{\phi}(q_1)\ldots|0\rangle[/itex]
[itex]=(-k_1^{2}+m^{2})\ldots \langle 0|T\widetilde{\phi}(k_1)\ldots|0\rangle[/itex]
[itex]=\ldots[/itex]

So, I am missing the limes in the last expression. Can it simply be introduced in the end since the on shell condition fixed [itex]k_1^{2}=m^{2}[/itex] before ?
Why isn't [itex]-k_1^{2}+m^{2}=0[/itex] true here? Is it because we are considering an interacting theory?

Very best, physicus
 
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  • #2
I'm not 100% certain, but here's my guess.

If you take [tex]k_i^2 = -m^2[/tex] through all of the steps you've shown, you get a problem here:

[tex]\langle f|i \rangle =\int{d^4q_1}\delta^4(k_1-q_1)(-q_1^{2}+m^{2})\ldots \langle 0|T\widetilde{\phi}(q_1)\ldots|0\rangle[/tex]

where as you integrate through all values of q_1, you get zero from the delta function unless q_1 = k_1, but when q_1 = k_1, then the term

[tex]-q_1^{2}+m^{2}[/tex]

still makes the integrand go to zero. In fact, the product of the delta and the above term at q_1 = k_1 is sort of like an indefinite form infinity*0 inside the integrand, so you sort of need to choose a prescription for dealing with it.. In your steps, you let the delta function take precedence, but then your last line must be identically zero for on-shell momenta. Srednicki works through this by letting the momenta be a little off-shell.

If you follow through the steps after 67.4, you'll see that Srednicki is also using this sort of limit thing to show how he's considering these effectively "singular terms" in the correlation function to be the only contributors to the scattering amplitude, so you let the momenta go on-shell eventually and take residues.
 

1. What is the LSZ formula in Srednicki QFT chapter 67?

The LSZ formula, also known as the Lehmann-Symanzik-Zimmermann formula, is a mathematical tool used in quantum field theory to calculate scattering amplitudes. It relates the S-matrix elements, which describe the scattering of particles, to the correlation functions of the underlying quantum field theory.

2. How is the LSZ formula derived in Srednicki QFT chapter 67?

The LSZ formula is derived using the path integral formalism in quantum field theory. It involves taking the Fourier transform of the correlation functions and then taking the limit as the external momenta approach infinity. This results in a relation between the S-matrix elements and the correlation functions.

3. What is the significance of the LSZ formula in Srednicki QFT chapter 67?

The LSZ formula is a very useful tool in quantum field theory as it allows for the calculation of scattering amplitudes, which are essential for understanding the behavior of particles in high-energy interactions. It also provides a connection between the fundamental concepts of quantum field theory, such as correlation functions and the S-matrix, allowing for a deeper understanding of the theory.

4. Are there any limitations to the LSZ formula in Srednicki QFT chapter 67?

While the LSZ formula is a powerful tool in quantum field theory, it does have some limitations. One limitation is that it assumes the particles involved in the scattering process are asymptotic states, meaning they are far apart in space and time. It also assumes that the scattering amplitudes are dominated by the lowest order Feynman diagrams, which may not always be the case.

5. Can the LSZ formula be applied to all quantum field theories discussed in Srednicki QFT chapter 67?

The LSZ formula can be applied to a wide range of quantum field theories, including scalar, fermion, and gauge theories. However, it is most commonly used in theories with a mass gap, meaning there is a nonzero minimum energy required to create a particle. It may not be as applicable in theories without a mass gap, such as conformal field theories.

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