Computing the scattering amplitude from the S-matrix

In summary: Finally, we can use the definition of the S-matrix to write:\langle f | S | i \rangle = \langle f | T\left\{1 - i \dfrac{\lambda}{4}\int_{-\infty}^\infty \int d^3x \phi(x)\psi(x)\phi(x)\psi(x)dt\right\} | i \rangle = \langle f; \infty | i; -\infty\rangle_{\mathrm{Schrödinger
  • #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?
 
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  • #2


First, let's rewrite the interaction Hamiltonian in terms of creation and annihilation operators:
H_{\mathrm{int}} = \dfrac{\lambda}{4}\int d^3x \phi(x)^2 \psi(x)^2 = \dfrac{\lambda}{4}\int d^3x \left(a^\dagger(x)a^\dagger(x)a(x)a(x) + a^\dagger(x)a(x)a^\dagger(x)a(x)\right)

where a(x) and a^\dagger(x) are the annihilation and creation operators for the fields \phi and \psi respectively. Using Wick's theorem, we can simplify this to:
H_{\mathrm{int}} = \dfrac{\lambda}{4}\int d^3x \left(:\phi^2(x)\psi^2(x): + \phi(x)\psi(x)\phi(x)\psi(x) \right)

where :: indicates normal ordering. Now, using the definition of the S-matrix and Wick's contraction, we can write the O(\lambda) contribution to the scattering amplitude as:
\langle f | S | i \rangle = \lim_{t_{\pm}\to \pm\infty} \langle f | S(t_+,t_-) | i\rangle = \langle f | T\left\{e^{-i \int_{-\infty}^\infty H_{\mathrm{int}}(t)dt}\right\} | i \rangle

Expanding the time evolution operator to first order in \lambda, we have:
\langle f | S | i \rangle = \langle f | T\left\{1 - i \int_{-\infty}^\infty H_{\mathrm{int}}(t)dt\right\} | i \rangle

Using the expression for H_{\mathrm{int}}, we can write:
\langle f | S | i \rangle = \langle f | T\left\{1 - i \dfrac{\lambda}{4}\int_{-\infty}^\infty \int d^3x :\phi^2(x)\psi^2(x):dt\right\} | i \rangle

Now, we can use Wick's theorem to evaluate the contractions and obtain:
\langle f | S | i \rangle = \langle f | T\left\{1 - i \dfrac{\
 

Related to Computing the scattering amplitude from the S-matrix

1. What is the S-matrix?

The S-matrix is a mathematical tool used in quantum field theory to describe the scattering amplitudes of particles. It is a matrix that relates the initial and final states of a scattering process, and contains information about the probability of particles interacting and exchanging energy and momentum.

2. How is the scattering amplitude computed from the S-matrix?

The scattering amplitude is computed by taking the elements of the S-matrix that correspond to the initial and final states of the scattering process, and using them to calculate the probability amplitude for that process. This involves performing mathematical operations on the elements of the S-matrix, which can vary depending on the specific scattering process being studied.

3. What are the inputs and outputs of computing the scattering amplitude from the S-matrix?

The inputs for computing the scattering amplitude from the S-matrix are the initial and final states of the scattering process, as well as the energy and momenta of the particles involved. The output is a numerical value representing the probability amplitude for that process to occur.

4. How is the S-matrix used in experimental particle physics?

In experimental particle physics, the S-matrix is used to analyze the results of particle scattering experiments. By comparing the measured scattering amplitudes to the theoretical predictions calculated from the S-matrix, scientists can test and validate different theories and models of particle interactions.

5. Are there any limitations or challenges to computing the scattering amplitude from the S-matrix?

One limitation is that the S-matrix formalism is most useful for describing high energy scattering processes, and may not be as accurate at lower energies. Additionally, the calculations involved in computing the scattering amplitude from the S-matrix can be complex and time-consuming, requiring advanced mathematical techniques and computer simulations.

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