Yang-Mills Equation of `U(1)` Gauge Theory

Nov 2 2017
--
-- This file has been auto-generated by egison-translator.
--

def N := 4

def g := [|[|-1, 0, 0, 0|], [|0, 1, 0, 0|], [|0, 0, 1, 0|], [|0, 0, 0, 1|]|]

def d X :=
  WedgeApplyExpr (ApplyExpr (VarExpr "flip") [VarExpr "\8706/\8706"]) [VectorExpr [VarExpr "t",VarExpr "x",VarExpr "y",VarExpr "z"],VarExpr "X"]

def hodge A :=
  let k := dfOrder A
   in withSymbols [i, j]
        sqrt (abs (M.det g_#_#)) *
        foldl
          (.)
          ((ε' N k)_(i_1)..._(i_N) . A..._(j_1)..._(j_k))
          (map 1#g~(i_%1)~(j_%1) (between 1 k))

def δ A :=
  let r := dfOrder A
   in (-1) ^ (N * r + 1) * hodge (d (hodge A))

def Δ A :=
  match dfrOrder A as integer with
    | #0 -> δ (d A)
    | #4 -> d (δ A)
    | _ -> d (δ A) + δ (d A)

def normalize2 A := withSymbols [t1, t2] A_t1_t2 - A_t2_t1

hodge (wedge [|1, 0, 0, 0|] [|0, 1, 0, 0|])

hodge (wedge [|0, 0, 1, 0|] [|0, 0, 0, 1|])

dfNormalize (d [|φ t x y z, Ax t x y z, Ay t x y z, Az t x y z|])

def F :=
  [|[|0, Ex t x y z, Ey t x y z, Ez t x y z|]
  , [|- Ex t x y z, 0, Bz t x y z, - By t x y z|]
  , [|- Ey t x y z, - Bz t x y z, 0, Bx t x y z|]
  , [|- Ez t x y z, By t x y z, - Bx t x y z, 0|]|]

hodge (d F)

δ F
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