Documentation for Spinors

docs/make.jl

@@ -9,8 +9,8 @@ makedocs(sitename="LatticeGPU", modules=[LatticeGPU], doctest=true,
              "LatticeGPU.jl" => "index.md", 
              "Space-time" => "space.md",
              "Groups and algebras" => "groups.md",
-             "Fields" => "fields.md"
-             "Dirac" => "dirac.md"
+             "Fields" => "fields.md",
+             "Dirac" => "dirac.md",
              "Solvers" => "solvers.md"
-             ], 
+             ],
          repo = "https://igit.ific.uv.es/alramos/latticegpu.jl")

docs/src/dirac.md

@@ -28,24 +28,21 @@ Wilson-Dirac operator.
 The action of the Dirac operator `Dw!` is the following:
 
 ```math
-D\psi (\vec{x} = x_1,x_2,x_3,x_4) = (4 + m_0)psi(\vec{x})
+D_w\psi (\vec{x} = x_1,x_2,x_3,x_4) = (4 + m_0)\psi(\vec{x}) - 
 ```
 ```math
-    - \frac{1}{2}\sum_{\mu = 1}^4 \theta (\mu) (1-\gamma_\mu) U_\mu(\vec{x}) \psi(\vec{x} + \hat{\mu}) 
-```
-```math
-    + \theta^* (\mu) (1 + \gamma_\mu) U^{-1}_\mu(\vec{x} - \hat{\mu}) \psi(\vec{x} - \hat{\mu})
+    - \frac{1}{2}\sum_{\mu = 1}^4 \theta (\mu) (1-\gamma_\mu) U_\mu(\vec{x}) \psi(\vec{x} + \hat{\mu}) + \theta^* (\mu) (1 + \gamma_\mu) U^{-1}_\mu(\vec{x} - \hat{\mu}) \psi(\vec{x} - \hat{\mu})
 ```
 
 where $$m_0$$ and $$\theta$$ are respectively the values `.m0` and `.th` of [`DiracParam`](@ref).
-Note that $$|\theta(i)|=1$$ is not built into the code, so it should be imposed explicitly. 
+Note that $$|\theta(\mu)|=1$$ is not built into the code, so it should be imposed explicitly. 
 
 Additionally, if |`dpar.csw`| > 1.0E-10, the clover term is assumed to be stored in `ymws.csw`, which
-can be done via the [`Csw`](@ref) function. In this case we have the Sheikholeslami–Wohlert (SW) term
+can be done via the [`Csw!`](@ref) function. In this case we have the Sheikholeslami–Wohlert (SW) term
 in `Dw!`:
 
 ```math
-    \frac{i}{2}c_{sw} \sum_{\pi = 1}^6 F^{cl}_\pi \sigma_\pi \psi(\vec{x})
+    \delta D_w^{sw} = \frac{i}{2}c_{sw} \sum_{\pi = 1}^6 F^{cl}_\pi \sigma_\pi \psi(\vec{x})
 ```
 where the $$\sigma$$ matrices are those described in the `Spinors` module and the index $$\pi$$ runs
 as specified in `lp.plidx`.
@@ -54,7 +51,7 @@ If the boudary conditions, defined in `lp`, are either `BC_SF_ORBI,D` or `BC_SF_
 improvement term 
 
 ```math
-    (c_t -1) (\delta_{x_4,a} \psi(\vec{x}) + \delta_{x_4,T-a} \psi(\vec{x}))
+    \delta D_w^{SF} = (c_t -1) (\delta_{x_4,a} \psi(\vec{x}) + \delta_{x_4,T-a} \psi(\vec{x}))
 ```
 is added. Since the time-slice $$t=T$$ is not stored, this accounts to modifying the second
 and last time-slice. 
@@ -66,7 +63,7 @@ in the first time-slice is zero. To enforce this, we have the function
 SF_bndfix!
 ```
 
-The function [`Csw`](@ref) is used to store the clover in `dws.csw`. It is computed 
+The function [`Csw!`](@ref) is used to store the clover in `dws.csw`. It is computed 
 according to the expression
 
 ```math
@@ -76,9 +73,13 @@ F_{\mu,\nu} = \frac{1}{8} (Q_{\mu \nu} - Q_{\nu \mu})
 where 
 ```math
 Q_{\mu\nu} = U_\mu(\vec{x})U_{\nu}(x+\mu)U_{\mu}^{-1}(\vec{x}+\nu)U_{\nu}(\vec{x}) +
-+ U_{\nu}^{-1}(\vec{x}-\nu) U_\mu (\vec{x}-\nu) U_{\nu}(\vec{x} +\mu - \nu) U^{-1}_{\mu}(\vec{x}) +
+U_{\nu}^{-1}(\vec{x}-\nu) U_\mu (\vec{x}-\nu) U_{\nu}(\vec{x} +\mu - \nu) U^{-1}_{\mu}(\vec{x}) +
+```
+```math
 + U^{-1}_{\mu}(x-\mu)U_\nu^{-1}(\vec{x} - \mu - \nu)U_\mu(\vec{x} - \mu - \nu)U_\nu^{-1}(x-\nu) +
-+ U_{\nu}(\vec{x})U_{\mu}^{-1}(\vec{x} + \nu - \mu)U^{-1}_{\nu}(\vec{x} - \mu)U_\mu(\vec{x}-\mu)
+```
+```math
++U_{\nu}(\vec{x})U_{\mu}^{-1}(\vec{x} + \nu - \mu)U^{-1}_{\nu}(\vec{x} - \mu)U_\mu(\vec{x}-\mu)
 
 ```
 
@@ -91,7 +92,8 @@ F[b,4,r] \to F_{31}(b,r) ,\quad F[b,5,r] \to F_{32}(b,r) ,\quad F[b,6,r] \to F_{
 ```
 where $$(b,r)$$ labels the lattice points as explained in the module `Space`
 
-The function [`pfrandomize!`](@ref), userfull for stochastic sources, is also present. 
+The function [`pfrandomize!`](@ref), userfull for stochastic sources, is also present. It
+randomizes a  fermion field either in all the space or in a specifit time-slice.
 
 The generic interface of these functions reads
 

docs/src/spinors.md

@@ -0,0 +1,86 @@
+# Spinors
+
+The module Spinors defines the necessary functions for the structure `Spinor{NS,G}`,
+which is a NS-tuple with values in G.
+
+The functions `norm`, `norm2`, `dot`, `*`, `/`, `/`, `+`, `-`, `imm` and `mimm`, 
+if defined for G, are extended to Spinor{NS,G} for general NS.
+
+For the 4d case where NS = 4 there are some specific functions to implement different
+operations with the gamma matrices. The convention for these matrices is
+
+
+```math
+\gamma _4 = \left( 
+    \begin{array}{cccc}
+        0 & 0 & -1 & 0\\
+        0 & 0 & 0 & -1\\
+        -1 & 0 & 0 & 0\\
+        0 & -1 & 0 & 0\\
+    \end{array}
+    \right)
+    \quad 
+    \gamma_1 = \left( 
+    \begin{array}{cccc}
+        0 & 0 & 0 & -i\\
+        0 & 0 & -i & 0\\
+        0 & i & 0 & 0\\
+        i & 0 & 0 & 0\\
+    \end{array}
+    \right)
+```
+```math 
+    \gamma _2 = \left( 
+    \begin{array}{cccc}
+        0 & 0 & 0 & -1\\
+        0 & 0 & 1 & 0\\
+        0 & 1 & 0 & 0\\
+        -1 & 0 & 0 & 0\\
+    \end{array}
+    \right)
+    \quad 
+    \gamma_3 = \left( 
+    \begin{array}{cccc}
+        0 & 0 & -i & 0\\
+        0 & 0 & 0 & i\\
+        i & 0 & 0 & 0\\
+        0 & -i & 0 & 0\\
+    \end{array}
+    \right)
+```
+
+
+The function [`dmul`](@ref) implements the multiplication over the $$\gamma$$ matrices
+
+```@docs
+dmul
+```
+
+The function [`pmul`](@ref) implements the $$ (1 \pm \gamma_N) $$ proyectors. The functions 
+[`gpmul`](@ref) and [`gdagpmul`](@ref) do the same and then multiply each element by `g`and 
+g^-1 repectively.
+
+```@docs
+pmul
+gpmul
+gdagpmul
+```
+
+## Some examples
+
+Here we just display some examples for these functions. We display it with `ComplexF64`
+instead of `SU3fund` or `SU2fund` for simplicity.
+
+
+```@setup exs
+import Pkg # hide
+Pkg.activate("/home/alberto/code/julia/LatticeGPU/") # hide
+using LatticeGPU # hide
+```
+```@repl exs
+spin = Spinor{4,Complex{Float64}}((1.0,im*0.5,2.3,0.0))
+println(spin)
+println(dmul(Gamma{4},spin))
+println(pmul(Pgamma{2,-1},spin))
+
+```