# Wick's Theorm to Feynman Diagrams?

1. Nov 27, 2009

### pudilz

Is this right? To calculate a scattering amplitude for a certain scattering event, you take the term in the wick's expansion of the S-Matrix that suits your situation best. Then you draw a Feynman Diagram that corresponds to the term in the wick's expansion, then just use the Feynman Rules to compute the amplitude. Let's just say for example, you have two free ,spin zero, quantum field theory's given by their Lagrangian Densities.

$$L_{real} = \partial_{\mu}\varphi\partial^{\mu}\varphi -m^{2}\varphi^{2}$$
$$L_{complex} = \partial_{\mu}\psi\partial^{\mu}\psi^{\dag} - M^{2}\psi^{\dag}\psi$$

And Lets just say for pure example the interaction Hamiltonian between these two fields is:

$$H_{int} = g\int d^{3}x\psi^{\dag}\psi\varphi$$

Where $$\psi^{\dag}$$ is the Hermitian Conjugate. From that we can get the S-Matrix:

$$S = Texp(-ig\int d^{4}x \psi^{\dag}\psi\varphi)$$

Where $$T$$ is the Time-Ordering operator. For ease I'll only go to order $$g^{2}$$ for the S-Matrix expansion. So the S-Matrix will be:

$$S = (-ig)^{2}\int d^{4}xd^{4}yT[\psi^{\dag}(x)\psi(x)\varphi(x)\psi^{\dag}(y)\psi(y)\varphi(y)]$$

Using Wick's Theorem, we can calculate

$$T[\psi^{\dag}(x)\psi(x)\varphi(x)\psi^{\dag}(y)\psi(y)\varphi(y)]$$

To Be 3 different terms:

$$N[\psi^{\dag}(x)\psi(x)\psi^{\dag}(y)\psi(y)]$$$$\varphi(x)\varphi(y)$$ + $$N[\psi^{\dag}(x)\varphi(x)\psi(y)\varphi(y)]$$$$\psi(x)\psi^{\dag}(y)$$ + $$N[\psi(x)\varphi(x)\psi^{\dag}(y)\varphi(y)]$$$$\psi^{\dag}(x)\psi(y)$$

Where $$\varphi(x)\varphi(y)$$ is contraction/propagator for the $$\varphi$$ particle to go from $$x$$ to $$y$$, likewise for the other two. $$N$$ is the Normal Ordering Operator.

Now lets say we have some sort of scattering event between the $$\varphi$$ and the $$\psi^{\dag}\psi$$ particles. To get the Amplitude we choose one of the terms in the Wick's expansions that satisfies the initial state the best, draw out the Feynman Diagram being in 1-1 correspondence with the term selected, and then use the Feynman Rules to compute the Amplitude.

Is this right, or somewhat on track? Or I'm I just hopelessly lost and confused?

p.s
I can absolutely assure you this is NOT any type of homework assignment.

Last edited: Nov 27, 2009