Learning Feynman Diagrams: Matrices & Dirac Adjoints

[Immortalis]

I am trying to learn about Feynman diagrams and am confused by some of the notation. In particular, what do matrices, in particular the Dirac adjoint, have to do with Feynman propogators and why are they in imaginary terms?

 

[samalkhaiat]

I am trying to learn about Feynman diagrams and am confused by some of the notation. In particular, what do matrices, in particular the Dirac adjoint, have to do with Feynman propogators and why are they in imaginary terms?

Feynman diagrams are terms in the perturbative expansion of the scattering operator
$$S = \sum_{n = 0}^{\infty} \frac{(-i)^{n}}{n!} \int \cdots \int d^{4}x_{1} \cdots d^{4}x_{n} \ T \big\{ \mathcal{H}(x_{1}) \cdots \mathcal{H}_{I}(x_{n}) \big\} ,$$
where $T$ time-orders the product of operators, and $\mathcal{H}_{I}$ is the interaction Hamiltonian. For QED,
$$\mathcal{H}_{I}(x) = - \mathcal{L}_{int.}(x) = -e N \big\{ \bar{\psi}(x) \gamma^{\mu}\psi (x) A_{\mu}(x) \big\} ,$$ where $\mathcal{L}_{int.}$ is the interaction Lagrangian, and $N\{\cdots\}$ denotes the normal ordering product. For two scalar operators at $t_{1}\neq t_{2}$, Wick’s theorem relates the T-product to the Normal order product $N\{\cdots\}$ and the Feynman propagator $\Delta_{F}$:
$$T \big\{ \phi^{\dagger}(x_{1}) \phi (x_{2}) \big\} = N \big\{ \phi^{\dagger}(x_{1}) \phi (x_{2}) \big\} + i \Delta_{F}(x_{1}-x_{2}) ,$$ with
$$i \Delta_{F}(x_{1}-x_{2}) = \langle 0 | T \big\{ \phi^{\dagger}(x_{1}) \phi (x_{2}) \big\} | 0 \rangle .$$
Since you are learning about Feynman diagrams, make sure that you learn and understand all the under-lined words and phrases.

 

[Immortalis]

Feynman diagrams are terms in the perturbative expansion of the scattering operator
$$S = \sum_{n = 0}^{\infty} \frac{(-i)^{n}}{n!} \int \cdots \int d^{4}x_{1} \cdots d^{4}x_{n} \ T \big\{ \mathcal{H}(x_{1}) \cdots \mathcal{H}_{I}(x_{n}) \big\} ,$$
where $T$ time-orders the product of operators, and $\mathcal{H}_{I}$ is the interaction Hamiltonian. For QED,
$$\mathcal{H}_{I}(x) = - \mathcal{L}_{int.}(x) = -e N \big\{ \bar{\psi}(x) \gamma^{\mu}\psi (x) A_{\mu}(x) \big\} ,$$ where $\mathcal{L}_{int.}$ is the interaction Lagrangian, and $N\{\cdots\}$ denotes the normal ordering product. For two scalar operators at $t_{1}\neq t_{2}$, Wick’s theorem relates the T-product to the Normal order product $N\{\cdots\}$ and the Feynman propagator $\Delta_{F}$:
$$T \big\{ \phi^{\dagger}(x_{1}) \phi (x_{2}) \big\} = N \big\{ \phi^{\dagger}(x_{1}) \phi (x_{2}) \big\} + i \Delta_{F}(x_{1}-x_{2}) ,$$ with
$$i \Delta_{F}(x_{1}-x_{2}) = \langle 0 | T \big\{ \phi^{\dagger}(x_{1}) \phi (x_{2}) \big\} | 0 \rangle .$$
Since you are learning about Feynman diagrams, make sure that you learn and understand all the under-lined words and phrases.

As I am currently a high school senior with basic experience in physics and calculus, and much of this terminology flies over my head, would you know any good places where I can break this down so I can understand it?

 

[samalkhaiat]

As I am currently a high school senior with basic experience in physics and calculus, and much of this terminology flies over my head, would you know any good places where I can break this down so I can understand it?

Well, calculations based on Feynman diagrams are post-grad stuff, normally covered in quantum field theory courses. So, you should for now just trust the descriptive explanation of the diagrams. I believe Frank Close wrote a nice little book about elementary particles and theire interaction, which can be appropraite to your level.

 

[sandy stone]

I think Feynman himself touches on his diagrams towards the end of his book QED: the Strange Theory of Light and Matter. It's a pretty good introduction.