{
  "id": "1608.07333",
  "version": "v1",
  "published": "2016-08-25T23:11:06.000Z",
  "updated": "2016-08-25T23:11:06.000Z",
  "title": "Angular decomposition of tensor products of a vector",
  "authors": [
    "Gregory S. Adkins"
  ],
  "comment": "9 pages",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "The tensor product of $L$ copies of a single vector, such as $p_{i_1} ... p_{i_L}$, can be analyzed in terms of angular momentum. When $p_{i_1} ... p_{i_L}$ is decomposed into a sum of components $( p_{i_1} ... p_{i_L} )^L_\\ell$, each characterized by angular momentum $\\ell$, the components are in general complicated functions of the $p_i$ vectors, especially so for large $\\ell$. We obtain a compact expression for $( p_{i_1} ... p_{i_L} )^L_\\ell$ explicitly in terms of the $p_i$ valid for all $L$ and $\\ell$. We use this decomposition to perform three-dimensional Fourier transforms of functions like $p^n \\hat p_{i_1} ... \\hat p_{i_L}$ that are useful in describing particle interactions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-08-25T23:11:06.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "tensor product",
      "angular decomposition",
      "perform three-dimensional fourier transforms",
      "angular momentum",
      "particle interactions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}