[Jupyter] Generalized Gell-Mann matrices

GGellMann.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "from IPython.display import display, Math"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "def range1(n):\n",
    "    return range(1, n + 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "https://en.wikipedia.org/wiki/Generalizations_of_Pauli_matrices\n",
    "\n",
    "## Generalized Gell-Mann matrices (Hermitian)\n",
    "\n",
    "### Construction\n",
    "\n",
    "Let $E_{jk}$ be the matrix with $1$ in the $jk$-th entry and $0$ elsewhere. Consider the space of $d×d$ complex matrices, $ℂ^{d×d}$, for a fixed $d$.\n",
    "\n",
    "Define the following matrices, \n",
    "\n",
    "$f_{k,j}^d =$<br>\n",
    "&nbsp;&nbsp;&nbsp;&nbsp;$E_{kj} + E_{jk}$, for $k < j$.<br>\n",
    "&nbsp;&nbsp;&nbsp;&nbsp;$−i (E_{jk} − E_{kj})$, for $k > j$.\n",
    "\n",
    "$h_k^d =$<br>\n",
    "&nbsp;&nbsp;&nbsp;&nbsp;$I_d$, the identity matrix, for $k = 1$.<br>\n",
    "&nbsp;&nbsp;&nbsp;&nbsp;$h_k^{d−1} ⊕ 0$, for $1 < k < d$.<br>\n",
    "&nbsp;&nbsp;&nbsp;&nbsp;$\\displaystyle {\\sqrt {\\tfrac {2}{d(d-1)}}}\\left(h_{1}^{d-1}\\oplus (1-d)\\right)={\\sqrt {\\tfrac {2}{d(d-1)}}}\\left(I_{d-1}\\oplus (1-d)\\right)$, for $k = d$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "def E(j, k, d):\n",
    "    return Matrix([[1 if (j, k) == (y, x) else 0 for x in range1(d)] for y in range1(d)])\n",
    "def f(k, j, d):\n",
    "    if k < j:\n",
    "        return E(k, j, d) + E(j, k, d)\n",
    "    if k > j:\n",
    "        return -I * (E(j, k, d) - E(k, j, d))\n",
    "def h(k, d):\n",
    "    if k == 1:\n",
    "        return 1, eye(d)\n",
    "    if 1 < k < d:\n",
    "        n, m = h(k, d - 1)\n",
    "        return n, diag(m, 0)\n",
    "    if k == d:\n",
    "        return int(d * (d - 1) / 2), diag(eye(d - 1), 1 - d)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "def generate(d):\n",
    "    def g():\n",
    "        for k in range1(d):\n",
    "            for j in range1(d):\n",
    "                if j == k:\n",
    "                    yield \"h_{%d}^{%d}\" % (k, d), h(k, d)\n",
    "                else:\n",
    "                    yield \"f_{%d,%d}^{%d}\" % (k, j, d), (1, f(k, j, d))\n",
    "    s = r\"\\begin{alignedat}{%d}\" % (d * 2)\n",
    "    for i, (fn, (n, m)) in enumerate(g()):\n",
    "        if i % d > 0:\n",
    "            s += r\",&\\ \"\n",
    "        elif i > 0:\n",
    "            s += r\"\\\\\"\n",
    "        s += f\"{fn}&=\"\n",
    "        if n != 1:\n",
    "            s += r\"\\frac1{\\sqrt{%s}}\" % latex(n)\n",
    "        s += \"&&\" + latex(m)\n",
    "    display(Math(s + r\"\\end{alignedat}\"))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\begin{alignedat}{4}h_{1}^{2}&=&&\\left[\\begin{matrix}1 & 0\\\\0 & 1\\end{matrix}\\right],&\\ f_{1,2}^{2}&=&&\\left[\\begin{matrix}0 & 1\\\\1 & 0\\end{matrix}\\right]\\\\f_{2,1}^{2}&=&&\\left[\\begin{matrix}0 & - i\\\\i & 0\\end{matrix}\\right],&\\ h_{2}^{2}&=&&\\left[\\begin{matrix}1 & 0\\\\0 & -1\\end{matrix}\\right]\\end{alignedat}$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "generate(2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\begin{alignedat}{6}h_{1}^{3}&=&&\\left[\\begin{matrix}1 & 0 & 0\\\\0 & 1 & 0\\\\0 & 0 & 1\\end{matrix}\\right],&\\ f_{1,2}^{3}&=&&\\left[\\begin{matrix}0 & 1 & 0\\\\1 & 0 & 0\\\\0 & 0 & 0\\end{matrix}\\right],&\\ f_{1,3}^{3}&=&&\\left[\\begin{matrix}0 & 0 & 1\\\\0 & 0 & 0\\\\1 & 0 & 0\\end{matrix}\\right]\\\\f_{2,1}^{3}&=&&\\left[\\begin{matrix}0 & - i & 0\\\\i & 0 & 0\\\\0 & 0 & 0\\end{matrix}\\right],&\\ h_{2}^{3}&=&&\\left[\\begin{matrix}1 & 0 & 0\\\\0 & -1 & 0\\\\0 & 0 & 0\\end{matrix}\\right],&\\ f_{2,3}^{3}&=&&\\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & 1\\\\0 & 1 & 0\\end{matrix}\\right]\\\\f_{3,1}^{3}&=&&\\left[\\begin{matrix}0 & 0 & - i\\\\0 & 0 & 0\\\\i & 0 & 0\\end{matrix}\\right],&\\ f_{3,2}^{3}&=&&\\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & - i\\\\0 & i & 0\\end{matrix}\\right],&\\ h_{3}^{3}&=\\frac1{\\sqrt{3}}&&\\left[\\begin{matrix}1 & 0 & 0\\\\0 & 1 & 0\\\\0 & 0 & -2\\end{matrix}\\right]\\end{alignedat}$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "generate(3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\begin{alignedat}{8}h_{1}^{4}&=&&\\left[\\begin{matrix}1 & 0 & 0 & 0\\\\0 & 1 & 0 & 0\\\\0 & 0 & 1 & 0\\\\0 & 0 & 0 & 1\\end{matrix}\\right],&\\ f_{1,2}^{4}&=&&\\left[\\begin{matrix}0 & 1 & 0 & 0\\\\1 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\end{matrix}\\right],&\\ f_{1,3}^{4}&=&&\\left[\\begin{matrix}0 & 0 & 1 & 0\\\\0 & 0 & 0 & 0\\\\1 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\end{matrix}\\right],&\\ f_{1,4}^{4}&=&&\\left[\\begin{matrix}0 & 0 & 0 & 1\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\1 & 0 & 0 & 0\\end{matrix}\\right]\\\\f_{2,1}^{4}&=&&\\left[\\begin{matrix}0 & - i & 0 & 0\\\\i & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\end{matrix}\\right],&\\ h_{2}^{4}&=&&\\left[\\begin{matrix}1 & 0 & 0 & 0\\\\0 & -1 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\end{matrix}\\right],&\\ f_{2,3}^{4}&=&&\\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & 0 & 1 & 0\\\\0 & 1 & 0 & 0\\\\0 & 0 & 0 & 0\\end{matrix}\\right],&\\ f_{2,4}^{4}&=&&\\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & 0 & 0 & 1\\\\0 & 0 & 0 & 0\\\\0 & 1 & 0 & 0\\end{matrix}\\right]\\\\f_{3,1}^{4}&=&&\\left[\\begin{matrix}0 & 0 & - i & 0\\\\0 & 0 & 0 & 0\\\\i & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\end{matrix}\\right],&\\ f_{3,2}^{4}&=&&\\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & 0 & - i & 0\\\\0 & i & 0 & 0\\\\0 & 0 & 0 & 0\\end{matrix}\\right],&\\ h_{3}^{4}&=\\frac1{\\sqrt{3}}&&\\left[\\begin{matrix}1 & 0 & 0 & 0\\\\0 & 1 & 0 & 0\\\\0 & 0 & -2 & 0\\\\0 & 0 & 0 & 0\\end{matrix}\\right],&\\ f_{3,4}^{4}&=&&\\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 1\\\\0 & 0 & 1 & 0\\end{matrix}\\right]\\\\f_{4,1}^{4}&=&&\\left[\\begin{matrix}0 & 0 & 0 & - i\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\i & 0 & 0 & 0\\end{matrix}\\right],&\\ f_{4,2}^{4}&=&&\\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & 0 & 0 & - i\\\\0 & 0 & 0 & 0\\\\0 & i & 0 & 0\\end{matrix}\\right],&\\ f_{4,3}^{4}&=&&\\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & - i\\\\0 & 0 & i & 0\\end{matrix}\\right],&\\ h_{4}^{4}&=\\frac1{\\sqrt{6}}&&\\left[\\begin{matrix}1 & 0 & 0 & 0\\\\0 & 1 & 0 & 0\\\\0 & 0 & 1 & 0\\\\0 & 0 & 0 & -3\\end{matrix}\\right]\\end{alignedat}$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "generate(4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\begin{alignedat}{10}h_{1}^{5}&=&&\\left[\\begin{matrix}1 & 0 & 0 & 0 & 0\\\\0 & 1 & 0 & 0 & 0\\\\0 & 0 & 1 & 0 & 0\\\\0 & 0 & 0 & 1 & 0\\\\0 & 0 & 0 & 0 & 1\\end{matrix}\\right],&\\ f_{1,2}^{5}&=&&\\left[\\begin{matrix}0 & 1 & 0 & 0 & 0\\\\1 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\end{matrix}\\right],&\\ f_{1,3}^{5}&=&&\\left[\\begin{matrix}0 & 0 & 1 & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\\\1 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\end{matrix}\\right],&\\ f_{1,4}^{5}&=&&\\left[\\begin{matrix}0 & 0 & 0 & 1 & 0\\\\0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\\\1 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\end{matrix}\\right],&\\ f_{1,5}^{5}&=&&\\left[\\begin{matrix}0 & 0 & 0 & 0 & 1\\\\0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\\\1 & 0 & 0 & 0 & 0\\end{matrix}\\right]\\\\f_{2,1}^{5}&=&&\\left[\\begin{matrix}0 & - i & 0 & 0 & 0\\\\i & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\end{matrix}\\right],&\\ h_{2}^{5}&=&&\\left[\\begin{matrix}1 & 0 & 0 & 0 & 0\\\\0 & -1 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\end{matrix}\\right],&\\ f_{2,3}^{5}&=&&\\left[\\begin{matrix}0 & 0 & 0 & 0 & 0\\\\0 & 0 & 1 & 0 & 0\\\\0 & 1 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\end{matrix}\\right],&\\ f_{2,4}^{5}&=&&\\left[\\begin{matrix}0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 1 & 0\\\\0 & 0 & 0 & 0 & 0\\\\0 & 1 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\end{matrix}\\right],&\\ f_{2,5}^{5}&=&&\\left[\\begin{matrix}0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 1\\\\0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\\\0 & 1 & 0 & 0 & 0\\end{matrix}\\right]\\\\f_{3,1}^{5}&=&&\\left[\\begin{matrix}0 & 0 & - i & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\\\i & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\end{matrix}\\right],&\\ f_{3,2}^{5}&=&&\\left[\\begin{matrix}0 & 0 & 0 & 0 & 0\\\\0 & 0 & - i & 0 & 0\\\\0 & i & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\end{matrix}\\right],&\\ h_{3}^{5}&=\\frac1{\\sqrt{3}}&&\\left[\\begin{matrix}1 & 0 & 0 & 0 & 0\\\\0 & 1 & 0 & 0 & 0\\\\0 & 0 & -2 & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\end{matrix}\\right],&\\ f_{3,4}^{5}&=&&\\left[\\begin{matrix}0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 1 & 0\\\\0 & 0 & 1 & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\end{matrix}\\right],&\\ f_{3,5}^{5}&=&&\\left[\\begin{matrix}0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 1\\\\0 & 0 & 0 & 0 & 0\\\\0 & 0 & 1 & 0 & 0\\end{matrix}\\right]\\\\f_{4,1}^{5}&=&&\\left[\\begin{matrix}0 & 0 & 0 & - i & 0\\\\0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\\\i & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\end{matrix}\\right],&\\ f_{4,2}^{5}&=&&\\left[\\begin{matrix}0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & - i & 0\\\\0 & 0 & 0 & 0 & 0\\\\0 & i & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\end{matrix}\\right],&\\ f_{4,3}^{5}&=&&\\left[\\begin{matrix}0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & - i & 0\\\\0 & 0 & i & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\end{matrix}\\right],&\\ h_{4}^{5}&=\\frac1{\\sqrt{6}}&&\\left[\\begin{matrix}1 & 0 & 0 & 0 & 0\\\\0 & 1 & 0 & 0 & 0\\\\0 & 0 & 1 & 0 & 0\\\\0 & 0 & 0 & -3 & 0\\\\0 & 0 & 0 & 0 & 0\\end{matrix}\\right],&\\ f_{4,5}^{5}&=&&\\left[\\begin{matrix}0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 1\\\\0 & 0 & 0 & 1 & 0\\end{matrix}\\right]\\\\f_{5,1}^{5}&=&&\\left[\\begin{matrix}0 & 0 & 0 & 0 & - i\\\\0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\\\i & 0 & 0 & 0 & 0\\end{matrix}\\right],&\\ f_{5,2}^{5}&=&&\\left[\\begin{matrix}0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & - i\\\\0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\\\0 & i & 0 & 0 & 0\\end{matrix}\\right],&\\ f_{5,3}^{5}&=&&\\left[\\begin{matrix}0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & - i\\\\0 & 0 & 0 & 0 & 0\\\\0 & 0 & i & 0 & 0\\end{matrix}\\right],&\\ f_{5,4}^{5}&=&&\\left[\\begin{matrix}0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & - i\\\\0 & 0 & 0 & i & 0\\end{matrix}\\right],&\\ h_{5}^{5}&=\\frac1{\\sqrt{10}}&&\\left[\\begin{matrix}1 & 0 & 0 & 0 & 0\\\\0 & 1 & 0 & 0 & 0\\\\0 & 0 & 1 & 0 & 0\\\\0 & 0 & 0 & 1 & 0\\\\0 & 0 & 0 & 0 & -4\\end{matrix}\\right]\\end{alignedat}$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "generate(5)"
   ]
  }
 ],
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