{
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  {
   "cell_type": "markdown",
   "id": "a4268ebd",
   "metadata": {},
   "source": [
    "# 電磁気学としての $U(1)$ ヤン＝ミルズ理論\n",
    "\n",
    "可換ゲージ場では、接続は時空上の1形式 $A$ であり、その曲率は電磁2形式\n",
    "\n",
    "$$F=dA.$$\n",
    "\n",
    "となります。マクスウェル方程式は、ビアンキ恒等式 $dF=0$ と源方程式\n",
    "$\\delta F=J$ で表されます。真空中では $J=0$ です。\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b6d09020",
   "metadata": {},
   "source": [
    "## ミンコフスキー時空と微分形式の演算子\n",
    "\n",
    "計量の符号数を $(-,+,+,+)$ とします。余微分はホッジスターと外微分から\n",
    "構成されるため、計量に由来するすべての符号を1か所で扱えます。\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "5f321928",
   "metadata": {},
   "outputs": [],
   "source": [
    "declare symbol t, x, y, z : MathValue\n",
    "\n",
    "def N : Integer := 4\n",
    "def coords : Vector MathValue := [| t, x, y, z |]\n",
    "def g : Matrix MathValue :=\n",
    "  [| [| -1, 0, 0, 0 |]\n",
    "   , [| 0, 1, 0, 0 |]\n",
    "   , [| 0, 0, 1, 0 |]\n",
    "   , [| 0, 0, 0, 1 |] |]\n",
    "\n",
    "def d (X : Tensor MathValue) : Tensor MathValue :=\n",
    "  !(flip ∂/∂) coords X\n",
    "\n",
    "def hodge (A : DiffForm MathValue) : DiffForm MathValue :=\n",
    "  let k := dfOrder A\n",
    "   in withSymbols [i, j]\n",
    "        sqrt (abs (M.det g_#_#)) *\n",
    "        foldl\n",
    "          (.)\n",
    "          ((ε' N k)_(i_1)..._(i_N) . A..._(j_1)..._(j_k))\n",
    "          (map (\\n -> g~(i_n)~(j_n)) [1..k])\n",
    "\n",
    "def δ (A : DiffForm MathValue) : DiffForm MathValue :=\n",
    "  let k := dfOrder A\n",
    "   in (-1) ^ (N * k + 1) * hodge (d (hodge A))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b512946c",
   "metadata": {},
   "source": [
    "基底に対する簡単な確認によって、ローレンツ計量における向きと符号規約を\n",
    "固定します：$\\star(dt\\wedge dx)=-dy\\wedge dz$。\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "e3bdf5d4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$\\begin{pmatrix} 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & -1 \\\\ 0 & 0 & 0 & 0 \\\\ \\end{pmatrix}$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "hodge (wedge [| 1, 0, 0, 0 |] [| 0, 1, 0, 0 |])\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "68aa9474",
   "metadata": {},
   "source": [
    "## ゲージポテンシャルと曲率\n",
    "\n",
    "任意の記号的な成分関数を用いて、\n",
    "$A=\\varphi\\,dt+A_x\\,dx+A_y\\,dy+A_z\\,dz$ とします。$dA$ を反対称化すると、\n",
    "電場と磁場を組み合わせた場の強さが現れます。\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "2736ecda",
   "metadata": {},
   "outputs": [],
   "source": [
    "def ϕ : MathValue := function (t, x, y, z)\n",
    "def Ax : MathValue := function (t, x, y, z)\n",
    "def Ay : MathValue := function (t, x, y, z)\n",
    "def Az : MathValue := function (t, x, y, z)\n",
    "\n",
    "def potential : DiffForm MathValue := [| ϕ, Ax, Ay, Az |]\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "326ec7e0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$\\begin{pmatrix} 0 & \\frac{-1}{2} \\frac{\\partial ϕ}{\\partial 2} + \\frac{1}{2} \\frac{\\partial Ax}{\\partial 1} & \\frac{-1}{2} \\frac{\\partial ϕ}{\\partial 3} + \\frac{1}{2} \\frac{\\partial Ay}{\\partial 1} & \\frac{-1}{2} \\frac{\\partial ϕ}{\\partial 4} + \\frac{1}{2} \\frac{\\partial Az}{\\partial 1} \\\\ \\frac{1}{2} \\frac{\\partial ϕ}{\\partial 2} + \\frac{-1}{2} \\frac{\\partial Ax}{\\partial 1} & 0 & \\frac{1}{2} \\frac{\\partial Ay}{\\partial 2} + \\frac{-1}{2} \\frac{\\partial Ax}{\\partial 3} & \\frac{1}{2} \\frac{\\partial Az}{\\partial 2} + \\frac{-1}{2} \\frac{\\partial Ax}{\\partial 4} \\\\ \\frac{1}{2} \\frac{\\partial ϕ}{\\partial 3} + \\frac{-1}{2} \\frac{\\partial Ay}{\\partial 1} & \\frac{-1}{2} \\frac{\\partial Ay}{\\partial 2} + \\frac{1}{2} \\frac{\\partial Ax}{\\partial 3} & 0 & \\frac{1}{2} \\frac{\\partial Az}{\\partial 3} + \\frac{-1}{2} \\frac{\\partial Ay}{\\partial 4} \\\\ \\frac{1}{2} \\frac{\\partial ϕ}{\\partial 4} + \\frac{-1}{2} \\frac{\\partial Az}{\\partial 1} & \\frac{-1}{2} \\frac{\\partial Az}{\\partial 2} + \\frac{1}{2} \\frac{\\partial Ax}{\\partial 4} & \\frac{-1}{2} \\frac{\\partial Az}{\\partial 3} + \\frac{1}{2} \\frac{\\partial Ay}{\\partial 4} & 0 \\\\ \\end{pmatrix}$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "dfNormalize (d potential)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "067854b4",
   "metadata": {},
   "source": [
    "## マクスウェルテンソル\n",
    "\n",
    "場の方程式を見慣れた変数で表示するため、記号的な電場と磁場の成分から\n",
    "反対称テンソル $F_{\\mu\\nu}$ を定義します。Egisonは微分2形式を\n",
    "$\\tfrac12F_{\\mu\\nu}dx^\\mu\\wedge dx^\\nu$ として保持するため、完全な\n",
    "反対称成分行列には2分の1の係数が明示的に現れます。\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "5a7c770b",
   "metadata": {},
   "outputs": [],
   "source": [
    "def Ex : MathValue := function (t, x, y, z)\n",
    "def Ey : MathValue := function (t, x, y, z)\n",
    "def Ez : MathValue := function (t, x, y, z)\n",
    "def Bx : MathValue := function (t, x, y, z)\n",
    "def By : MathValue := function (t, x, y, z)\n",
    "def Bz : MathValue := function (t, x, y, z)\n",
    "\n",
    "def F : DiffForm MathValue :=\n",
    "  (1 / 2) *\n",
    "    [| [| 0, Ex, Ey, Ez |]\n",
    "     , [| -Ex, 0, -Bz, By |]\n",
    "     , [| -Ey, Bz, 0, -Bx |]\n",
    "     , [| -Ez, -By, Bx, 0 |] |]\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "add2b46f",
   "metadata": {},
   "source": [
    "双対化したビアンキ式には、$\\nabla\\cdot B=0$ と\n",
    "$\\nabla\\times E=-\\partial_tB$ がまとめて表現されています。\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "cb58c180",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$\\begin{pmatrix} \\frac{\\partial Bz}{\\partial 4} + \\frac{\\partial By}{\\partial 3} + \\frac{\\partial Bx}{\\partial 2} \\\\ \\frac{\\partial Ez}{\\partial 3} - \\frac{\\partial Ey}{\\partial 4} + \\frac{\\partial Bx}{\\partial 1} \\\\ -\\frac{\\partial Ez}{\\partial 2} + \\frac{\\partial Ex}{\\partial 4} + \\frac{\\partial By}{\\partial 1} \\\\ \\frac{\\partial Ey}{\\partial 2} - \\frac{\\partial Ex}{\\partial 3} + \\frac{\\partial Bz}{\\partial 1}\\\\ \\end{pmatrix}$"
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    }
   ],
   "source": [
    "hodge (d F)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bc32663e",
   "metadata": {},
   "source": [
    "余微分にはガウスの法則とアンペール＝マクスウェルの法則がまとめて\n",
    "表現されています。\n",
    "このベクトルを電流1形式と等しいと置くことで、$\\delta F=J$ が得られます。\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "fe97bce5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$\\begin{pmatrix} -\\frac{\\partial Ez}{\\partial 4} - \\frac{\\partial Ey}{\\partial 3} - \\frac{\\partial Ex}{\\partial 2} \\\\ -\\frac{\\partial Ex}{\\partial 1} + \\frac{\\partial Bz}{\\partial 3} - \\frac{\\partial By}{\\partial 4} \\\\ -\\frac{\\partial Ey}{\\partial 1} - \\frac{\\partial Bz}{\\partial 2} + \\frac{\\partial Bx}{\\partial 4} \\\\ -\\frac{\\partial Ez}{\\partial 1} + \\frac{\\partial By}{\\partial 2} - \\frac{\\partial Bx}{\\partial 3}\\\\ \\end{pmatrix}$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "δ F\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "583f31ad",
   "metadata": {},
   "source": [
    "ここで、成分に2分の1を付ける規約は `dfNormalize (d potential)` と\n",
    "一致します。これらの微分の組み合わせが、斉次マクスウェル方程式と\n",
    "源を含むマクスウェル方程式です。これは可換な $U(1)$ ヤン＝ミルズ系であり、\n",
    "非線形項 $A\\wedge A$ は消えます。\n"
   ]
  }
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