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   "source": [
    "# ミンコフスキー時空のホッジスター\n",
    "\n",
    "電磁気学は、時空上の微分形式を用いると自然に記述できます。\n",
    "計量の符号が $(-,+,+,+)$ の場合、ホッジスターは2形式を2形式へ写しますが、\n",
    "時間的な基底要素によって符号が生じます。\n",
    "\n",
    "$$\\star(dt\\wedge dx)=-dy\\wedge dz,\\qquad\n",
    "  \\star(dy\\wedge dz)=dt\\wedge dx.$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cde9f17e",
   "metadata": {},
   "source": [
    "## ローレンツ計量\n",
    "\n",
    "計量の行列式は負なので、体積密度には $\\sqrt{|\\det g|}$ を用います。\n",
    "時間方向の添字を上げると、結果に現れる追加の負符号が生じます。\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "b789a048",
   "metadata": {},
   "outputs": [],
   "source": [
    "declare symbol t, x, y, z : MathValue\n",
    "\n",
    "def N : Integer := 4\n",
    "def params : 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 dt : DiffForm MathValue := [| 1, 0, 0, 0 |]\n",
    "def dx : DiffForm MathValue := [| 0, 1, 0, 0 |]\n",
    "def dy : DiffForm MathValue := [| 0, 0, 1, 0 |]\n",
    "def dz : DiffForm MathValue := [| 0, 0, 0, 1 |]\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a5cce8ed",
   "metadata": {},
   "source": [
    "## 計量に依存する双対性\n",
    "\n",
    "定義にはユークリッド空間と同じ縮約を用います。変わるのは次元と計量だけです。\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "d05c698f",
   "metadata": {},
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   "source": [
    "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"
   ]
  },
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   "cell_type": "markdown",
   "id": "8f0e75a0",
   "metadata": {},
   "source": [
    "まず、時間方向を含む2形式の双対を計算します。\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "5b0ba917",
   "metadata": {},
   "outputs": [
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       "$\\begin{pmatrix} 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & -1 \\\\ 0 & 0 & 0 & 0 \\\\ \\end{pmatrix}$"
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   "source": [
    "hodge (wedge dt dx)\n"
   ]
  },
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   "cell_type": "markdown",
   "id": "0a1e435e",
   "metadata": {},
   "source": [
    "空間方向だけからなる面積形式の双対は、時間・空間方向の面積形式になります。\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "dfcf5583",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$\\begin{pmatrix} 0 & 1 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\\\ \\end{pmatrix}$"
      ]
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     "output_type": "display_data"
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   "source": [
    "hodge (wedge dy dz)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cc98eb43",
   "metadata": {},
   "source": [
    "このローレンツ計量の規約では、これらの符号から2形式上で $\\star^2=-1$ が従います。\n",
    "また、電磁場テンソルの双対を取るときに電場成分と磁場成分を交換する符号でもあります。\n"
   ]
  }
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