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differ diff --git a/source/know/concept/feynman-diagram/fullgf.png b/source/know/concept/feynman-diagram/fullgf.png deleted file mode 100644 index f376ef2..0000000 Binary files a/source/know/concept/feynman-diagram/fullgf.png and /dev/null differ diff --git a/source/know/concept/feynman-diagram/impurity.avif b/source/know/concept/feynman-diagram/impurity.avif new file mode 100644 index 0000000..1f78d10 Binary files /dev/null and b/source/know/concept/feynman-diagram/impurity.avif differ diff --git a/source/know/concept/feynman-diagram/impurity.png b/source/know/concept/feynman-diagram/impurity.png new file mode 100644 index 0000000..2e8fbf6 Binary files /dev/null and b/source/know/concept/feynman-diagram/impurity.png differ diff --git a/source/know/concept/feynman-diagram/index.md b/source/know/concept/feynman-diagram/index.md index c36e7df..ace8dbc 100644 --- a/source/know/concept/feynman-diagram/index.md +++ b/source/know/concept/feynman-diagram/index.md @@ -25,6 +25,7 @@ Below, we go through the most notable components of Feynman diagrams and how to translate them into a mathematical expression. + ## Real space The most common component is a **fermion line**, which represents @@ -37,9 +38,7 @@ Let the subscript $$I$$ refer to the and $$\mathcal{T}\{\}$$ denote the [time-ordered product](/know/concept/time-ordered-product/): - - - +{% include image.html file="fermion-light.png" width="60%" alt="Fermion line diagram" %} $$\begin{aligned} = i \hbar G_{s_2 s_1}^0(\vb{r}_2, t_2; \vb{r}_1, t_1) @@ -59,9 +58,7 @@ Less common is a **heavy fermion line**, representing a causal Green's function $$G$$ for the entire Hamiltonian $$\hat{H}$$, where the subscript $$H$$ refers to the [Heisenberg picture](/know/concept/heisenberg-picture/): - - - +{% include image.html file="fermion-heavy.png" width="60%" alt="Heavy fermion line diagram" %} $$\begin{aligned} = i \hbar G_{s_2 s_1}(\vb{r}_2, t_2; \vb{r}_1, t_1) @@ -75,9 +72,7 @@ which we assume to be instantaneous, i.e. time-independent hence it starts and ends at the same time, and no arrow is drawn: - - - +{% include image.html file="boson.png" width="60%" alt="Boson/interaction line diagram" %} $$\begin{aligned} = \frac{1}{i \hbar} W_{s_2 s_1}(\vb{r}_2, t_2; \vb{r}_1, t_1) @@ -99,9 +94,7 @@ $$\begin{aligned} One-body (time-dependent) operators $$\hat{V}$$ in $$\hat{H}_1$$ are instead represented by a special vertex: - - - +{% include image.html file="impurity.png" width="35%" alt="One-body perturbation (e.g. impurity) diagram" %} $$\begin{aligned} = \frac{1}{i \hbar} V_s(\vb{r}, t) @@ -148,6 +141,7 @@ so that a particle with a given spin propagates from vertex to vertex without getting flipped. + ## Fourier space If the system is time-independent and spatially uniform, @@ -177,9 +171,7 @@ Working in Fourier space allows us to simplify calculations. Consider the following diagram and the resulting expression, where $$\tilde{\vb{r}} = (\vb{r}, t)$$, and $$\tilde{\vb{k}} = (\vb{k}, \omega)$$: - - - +{% include image.html file="example.png" width="40%" alt="Example: fermion-fermion interaction" %} $$\begin{aligned} &= (i \hbar)^3 \sum_{s s'} \!\!\iint \dd{\tilde{\vb{r}}} \dd{\tilde{\vb{r}}'} @@ -274,6 +266,7 @@ then conservation removes all internal variables, so no integrals would be needed. + ## Imaginary time Feynman diagrams are also useful when working with diff --git a/source/know/concept/feynman-diagram/interaction.png b/source/know/concept/feynman-diagram/interaction.png deleted file mode 100644 index 3c2d093..0000000 Binary files a/source/know/concept/feynman-diagram/interaction.png and /dev/null differ diff --git a/source/know/concept/feynman-diagram/perturbation.png b/source/know/concept/feynman-diagram/perturbation.png deleted file mode 100644 index 2e8fbf6..0000000 Binary files a/source/know/concept/feynman-diagram/perturbation.png and /dev/null differ -- cgit v1.2.3