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Computing the scattering amplitude from the S-matrix

  1. May 15, 2017 #1
    1. The problem statement, all variables and given/known data
    Consider two real scalar fields [itex]\phi,\psi[/itex] with masses [itex]m[/itex] and [itex]\mu[/itex] respectively interacting via the Hamiltonian [tex]\mathcal{H}_{\mathrm{int}}(x)=\dfrac{\lambda}{4}\phi^2(x)\psi^2(x).[/tex]

    Using the definition of the S-matrix and Wick's contraction find the [itex]O(\lambda)[/itex] contribution to the scattering amplitude for the process [itex]\phi(p_1)+\psi(p_2)\to \phi(p_1')+\psi(p_2')[/itex], being [itex]p_i[/itex] the initial momenta and [itex]p_i'[/itex] the final momenta.

    2. Relevant equations
    First I would consider the definition of the S-matrix from the book: [tex] \langle f | S | i \rangle_{\mathrm{Heisenberg}}= \langle f; \infty | i; -\infty\rangle_{\mathrm{Schrödinger}},[/tex] secondly I believe Wick's theorem would be needed: [tex]T\{\phi_1\cdots\phi_n\}=:\phi_1\cdots\phi_n + \text{all contractions}:[/tex]

    Finaly I believe Dyson's series can help here [tex] U(t,t_0)= T\exp \left[ i \int _{t_0}^t H_I(t')dt'\right][/tex] being [itex]U(t,t_0)[/itex] the interaction picture time evolution operator and [itex]H_I[/itex] the interaction Hamiltonian at the interaction picture.

    3. The attempt at a solution

    I wasn't able to do much really. My first guess was to use Dyson's series to expand the time evolution operator of the interaction picture to first order in [itex]\lambda[/itex], that is, being [itex]H_I[/itex] the interaction Hamiltonian in the interaction picture defined by [tex]H_I (t) = e^{iH_0(t-t_0)} H_\mathrm{int} e^{-i H_0(t-t_0)}[/tex] we have Dyson's series

    [tex]U(t,t_0)= T\exp\left[-i \int_{t_0}^t H_I(t') dt'\right] \approx T\left\{1 - i \int_{t_0}^t H_I(t')dt' - \dfrac{1}{2}\int_{t_0}^t \int_{t_0}^t H_I(t')H_I(t'')dt'dt'' \right\},[/tex]

    now [itex]H_I[/itex] is just the same as [itex]H_{\mathrm{int}}[/itex] changing the fields by their interaction picture counterparts, which can be expanded in terms of creation and annihilation operators, so that
    [tex]H_I(t)= \dfrac{\lambda}{4}\int d^3x \phi(x)^2 \psi(x)^2[/tex]

    but I don't know how this can be used. The definition of the S-matrix as I understand works as follows: given [itex]S(t,t_0)[/itex] the time evolution operator associated to the full Hamiltonian [itex]H= H_0+H_\mathrm{int}[/itex] the S-matrix is defined as
    [tex]\langle f | S | i \rangle = \lim_{t_{\pm}\to \pm\infty} \langle f | S(t_+,t_-) | i\rangle[/tex]

    so we pick the state [itex]|i\rangle[/itex] at [itex]t=-\infty[/itex] and evolve it to [itex]t=+\infty[/itex] according to the full Hamiltonian, and project onto the final state.

    The issue is that [itex]S(t,t_0)= e^{i H_0(t-t_0)}U(t,t_0)[/itex] so I can't use [itex]U[/itex] directly.

    By the problem statement I guess I can't use Feynman rules derived from the LSZ formula, just the definition of the S-matrix.

    I'm really stuck with this. How can I proceed?
  2. jcsd
  3. May 21, 2017 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
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