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   "source": [
    "# SU(2) → SO(3) homomorphism\n",
    "\n",
    "```{autolink-concat}\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The Cornwell theorem from the group theory (see for example Section 3, Chapter 5 of {cite}`Cornwell:1997ke`) gives the relation between the rotation of the transition amplitude and the physical vector of polarization sensitivity:\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "    R_{ij}(\\phi,\\theta,\\chi) &= \\frac{1}{2}\\mathrm{tr}\\left(\n",
    "    D^{1/2*}(\\phi,\\theta,\\chi) \\sigma^P_i D^{1/2*\\dagger}(\\phi,\\theta,\\chi) \\sigma^P_j\n",
    "    \\right)\\,,\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "where $\\mathrm{tr}$ represents the trace operation applied to the product of the two-dimensional matrices, $D$ and $\\sigma^P$, and $R_{ij}(\\phi,\\theta,\\chi)$ is a three-dimensional rotation matrix implementing the Euler transformation to a physical vector."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "jupyter": {
     "source_hidden": true
    },
    "mystnb": {
     "code_prompt_show": "SO(3) rotation with SymPy"
    },
    "tags": [
     "full-width",
     "hide-input"
    ]
   },
   "outputs": [],
   "source": [
    "from sympy import Matrix, cos, sin, symbols\n",
    "\n",
    "\n",
    "def Rz(α):\n",
    "    return Matrix(\n",
    "        [\n",
    "            [cos(α), -sin(α), 0],\n",
    "            [sin(α), cos(α), 0],\n",
    "            [0, 0, 1],\n",
    "        ]\n",
    "    )\n",
    "\n",
    "\n",
    "def Ry(α):\n",
    "    return Matrix(\n",
    "        [\n",
    "            [cos(α), 0, sin(α)],\n",
    "            [0, 1, 0],\n",
    "            [-sin(α), 0, cos(α)],\n",
    "        ]\n",
    "    )\n",
    "\n",
    "\n",
    "θ, ϕ, χ = symbols(\"theta phi chi\", positive=True)\n",
    "R_SO3 = Rz(ϕ) @ Ry(θ) @ Rz(χ)\n",
    "R_SO3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "jupyter": {
     "source_hidden": true
    },
    "mystnb": {
     "code_prompt_show": "SU(2) rotation with SymPy"
    },
    "tags": [
     "full-width",
     "hide-input"
    ]
   },
   "outputs": [],
   "source": [
    "from sympy import S, Trace, conjugate, expand, expand_trig, simplify\n",
    "from sympy.physics.matrices import msigma\n",
    "from sympy.physics.quantum.dagger import Dagger\n",
    "from sympy.physics.quantum.spin import WignerD\n",
    "\n",
    "\n",
    "def R(Dx, i, j):\n",
    "    return Trace(Dx @ msigma(i) @ Dagger(Dx) @ msigma(j)) / 2\n",
    "\n",
    "\n",
    "h = S(1) / 2\n",
    "Dx = conjugate(\n",
    "    Matrix([[WignerD(h, i, j, ϕ, θ, χ) for i in [+h, -h]] for j in [+h, -h]])\n",
    ")\n",
    "Rij = Matrix([[R(Dx, j, i).doit() for i in range(1, 4)] for j in range(1, 4)])\n",
    "hangle = {\n",
    "    cos(θ / 2) ** 2: (1 + cos(θ)) / 2,\n",
    "    sin(θ / 2) ** 2: (1 - cos(θ)) / 2,\n",
    "}\n",
    "R_SU2 = expand_trig(simplify(expand(Rij, complex=True).subs(hangle)))\n",
    "R_SU2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "assert R_SO3 == R_SU2"
   ]
  }
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