- #1

leo.

- 96

- 5

## Homework Statement

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.

## Homework 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.

## The Attempt at a Solution

[/B]

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?