Wick's Theorm to Feynman Diagrams?

[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 Let's 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 let's 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.

 

[azntoon]

I'm just trying to get a better understanding of how to calculate scattering amplitudes. Yes, this is correct. You have provided a good explanation of how to calculate the scattering amplitude for the given scattering event.

 

[Entropia]

Yes, you are on the right track! Wick's Theorem and Feynman Diagrams are important tools in quantum field theory for calculating scattering amplitudes.

Wick's Theorem allows us to break down a time-ordered product of fields into a sum of normal-ordered products, which are much easier to work with. This is because the normal-ordered products have all the annihilation operators to the right and creation operators to the left, making it easier to apply the Feynman Rules.

The Feynman Rules allow us to assign a specific diagram to each term in the normal-ordered product, where the lines represent particles and the vertices represent interactions. By using these rules, we can calculate the amplitude for a specific scattering event by summing over all possible diagrams that correspond to that event.

So, in summary, you are correct in saying that to calculate a scattering amplitude, we use Wick's Theorem to break down the time-ordered product into normal-ordered products and then use Feynman Diagrams and Rules to compute the amplitude. Keep up the good work in your studies of quantum field theory!