DOC

doc/discretization.md

@@ -12,7 +12,7 @@ The action is
   \,,
 ```
 where $`\hat{\nabla}_j Q = \partial_j Q - i[\mathcal{A}_j, Q]`$
-and $\Omega = \epsilon\tau_3 + i\Delta$, $\Delta^\dagger = \Delta$.
+and $`\Omega = \epsilon\tau_3 + i\Delta$, $\Delta^\dagger = \Delta`$.
 We'd like the discretized functional to be also gauge invariant.
 
 We subdivide the space to cells, centered on a rectangular lattice
@@ -21,15 +21,14 @@ and $h$ are the lattice spacings. See the figure below:
 
 ![Lattice discretization](img/discretization.svg "FIG 1. Lattice discretization.")
 
-We choose $`Q(j) = Q(\vec{r}_j)`$ to be values of $`Q`$ at the lattice sites. We define the Wilson link matrices
-$`U_{ij} =U_{ji}^{-1} =U(\vec{r}_{i},\vec{r}_{j}) = \mathrm{Pexp}[i\int_{L(\vec{r}_i,\vec{r}_j)}d\vec{r}'\,\cdot\vec{\mathcal{A}}(\vec{r}')]`$
-where $`L(\vec{r}_i,\vec{r}_j)`$ is straight line from $`\vec{r}_j`$ to
-$`\vec{r}_i`$ and $`\mathrm{Pexp}`$ is the path-ordered integral.
-The neighbor cells of $`j`$ are $`i\in{}\mathrm{neigh}(j)`$
-ie. $`i=(j_x\pm1,j_y,j_z)$, $i=(j_x,j_y\pm1,j_z)`$,
-$`i=(j_x,j_y,j_z\pm1)`$. Since the gauge field $`\vec{\mathcal{A}}`$ is
-fixed, the link matrices can be computed cheaply, and there's no need
-to expand them.
+We choose $`Q(j) = Q(\vec{r}_j)`$ to be values of $`Q`$ at the lattice
+sites. We define the Wilson link matrices $`U_{ij} =U_{ji}^{-1}
+=U(\vec{r}_{i},\vec{r}_{j}) =
+\mathrm{Pexp}[i\int_{L(\vec{r}_i,\vec{r}_j)}d\vec{r}'\,\cdot\vec{\mathcal{A}}(\vec{r}')]`$
+where $`L(\vec{r}_i,\vec{r}_j)`$ is straight line from $`\vec{r}_j`$
+to $`\vec{r}_i`$ and $`\mathrm{Pexp}`$ is the path-ordered integral.
+Since the gauge field $`\vec{\mathcal{A}}`$ is fixed, the link
+matrices can be computed ahead of time.
 
 We want a discretized theory that is invariant under the discrete gauge transformation
 ```math