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   "source": [
    "# $U(1)$ Yang--Mills Theory as Electromagnetism\n",
    "\n",
    "For an Abelian gauge field, the connection is a spacetime one-form $A$\n",
    "and its curvature is the electromagnetic two-form\n",
    "\n",
    "$$F=dA.$$\n",
    "\n",
    "Maxwell's equations become the Bianchi identity $dF=0$ and the source\n",
    "equation $\\delta F=J$.  In vacuum, $J=0$.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b6d09020",
   "metadata": {},
   "source": [
    "## Minkowski spacetime and form operators\n",
    "\n",
    "We use signature $(-,+,+,+)$.  The codifferential is built from the\n",
    "Hodge star and the exterior derivative, so all metric signs enter in a\n",
    "single place.\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": [
    "A quick basis check fixes the Lorentzian orientation and sign convention:\n",
    "$\\star(dt\\wedge dx)=-dy\\wedge dz$.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "e3bdf5d4",
   "metadata": {},
   "outputs": [
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      "text/html": [
       "$\\begin{pmatrix} 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & -1 \\\\ 0 & 0 & 0 & 0 \\\\ \\end{pmatrix}$"
      ]
     },
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   ],
   "source": [
    "hodge (wedge [| 1, 0, 0, 0 |] [| 0, 1, 0, 0 |])\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "68aa9474",
   "metadata": {},
   "source": [
    "## Gauge potential and curvature\n",
    "\n",
    "Let $A=\\varphi\\,dt+A_x\\,dx+A_y\\,dy+A_z\\,dz$, with arbitrary symbolic\n",
    "component functions.  Antisymmetrizing $dA$ exposes the electric and\n",
    "magnetic field-strength combinations.\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": {},
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       "$\\begin{pmatrix} 0 & \\frac{-1}{2} \\frac{\\partial ϕ}{\\partial 2} + \\frac{1}{2} \\frac{\\partial Ax}{\\partial 1} & \\frac{-1}{2} \\frac{\\partial ϕ}{\\partial 3} + \\frac{1}{2} \\frac{\\partial Ay}{\\partial 1} & \\frac{-1}{2} \\frac{\\partial ϕ}{\\partial 4} + \\frac{1}{2} \\frac{\\partial Az}{\\partial 1} \\\\ \\frac{1}{2} \\frac{\\partial ϕ}{\\partial 2} + \\frac{-1}{2} \\frac{\\partial Ax}{\\partial 1} & 0 & \\frac{1}{2} \\frac{\\partial Ay}{\\partial 2} + \\frac{-1}{2} \\frac{\\partial Ax}{\\partial 3} & \\frac{1}{2} \\frac{\\partial Az}{\\partial 2} + \\frac{-1}{2} \\frac{\\partial Ax}{\\partial 4} \\\\ \\frac{1}{2} \\frac{\\partial ϕ}{\\partial 3} + \\frac{-1}{2} \\frac{\\partial Ay}{\\partial 1} & \\frac{-1}{2} \\frac{\\partial Ay}{\\partial 2} + \\frac{1}{2} \\frac{\\partial Ax}{\\partial 3} & 0 & \\frac{1}{2} \\frac{\\partial Az}{\\partial 3} + \\frac{-1}{2} \\frac{\\partial Ay}{\\partial 4} \\\\ \\frac{1}{2} \\frac{\\partial ϕ}{\\partial 4} + \\frac{-1}{2} \\frac{\\partial Az}{\\partial 1} & \\frac{-1}{2} \\frac{\\partial Az}{\\partial 2} + \\frac{1}{2} \\frac{\\partial Ax}{\\partial 4} & \\frac{-1}{2} \\frac{\\partial Az}{\\partial 3} + \\frac{1}{2} \\frac{\\partial Ay}{\\partial 4} & 0 \\\\ \\end{pmatrix}$"
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   "source": [
    "dfNormalize (d potential)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "067854b4",
   "metadata": {},
   "source": [
    "## Maxwell tensor\n",
    "\n",
    "To display the field equations in their familiar variables, define the\n",
    "antisymmetric tensor $F_{\\mu\\nu}$ from symbolic electric and magnetic\n",
    "components.  Egison stores a differential two-form as\n",
    "$\\tfrac12F_{\\mu\\nu}dx^\\mu\\wedge dx^\\nu$, so the full antisymmetric\n",
    "component matrix carries an explicit factor of one half.\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": [
    "The dual Bianchi expression packages $\\nabla\\cdot B=0$ and\n",
    "$\\nabla\\times E=-\\partial_tB$.\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"
   ]
  },
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   "cell_type": "markdown",
   "id": "bc32663e",
   "metadata": {},
   "source": [
    "The codifferential packages Gauss's law and the Ampere--Maxwell law.\n",
    "Setting this vector equal to a current one-form gives $\\delta F=J$.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "fe97bce5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$\\begin{pmatrix} -\\frac{\\partial Ez}{\\partial 4} - \\frac{\\partial Ey}{\\partial 3} - \\frac{\\partial Ex}{\\partial 2} \\\\ -\\frac{\\partial Ex}{\\partial 1} + \\frac{\\partial Bz}{\\partial 3} - \\frac{\\partial By}{\\partial 4} \\\\ -\\frac{\\partial Ey}{\\partial 1} - \\frac{\\partial Bz}{\\partial 2} + \\frac{\\partial Bx}{\\partial 4} \\\\ -\\frac{\\partial Ez}{\\partial 1} + \\frac{\\partial By}{\\partial 2} - \\frac{\\partial Bx}{\\partial 3}\\\\ \\end{pmatrix}$"
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   ],
   "source": [
    "δ F\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "583f31ad",
   "metadata": {},
   "source": [
    "The one-half component convention now agrees with\n",
    "`dfNormalize (d potential)`.  The derivative combinations are the\n",
    "homogeneous and sourced Maxwell equations.  This is the Abelian\n",
    "$U(1)$ Yang--Mills system, where the nonlinear $A\\wedge A$ term vanishes.\n"
   ]
  }
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