{
  "id": "quant-ph/9507004",
  "version": "v1",
  "published": "1995-07-07T16:27:35.000Z",
  "updated": "1995-07-07T16:27:35.000Z",
  "title": "Generalized uncertainty relations: Theory, examples, and Lorentz invariance",
  "authors": [
    "Samuel L. Braunstein",
    "Carlton M. Caves",
    "G. J. Milburn"
  ],
  "comment": "39 pages of text plus one figure; text formatted in LaTeX",
  "doi": "10.1006/aphy.1996.0040",
  "categories": [
    "quant-ph"
  ],
  "abstract": "The quantum-mechanical framework in which observables are associated with Hermitian operators is too narrow to discuss measurements of such important physical quantities as elapsed time or harmonic-oscillator phase. We introduce a broader framework that allows us to derive quantum-mechanical limits on the precision to which a parameter---e.g., elapsed time---may be determined via arbitrary data analysis of arbitrary measurements on $N$ identically prepared quantum systems. The limits are expressed as generalized Mandelstam-Tamm uncertainty relations, which involve the operator that generates displacements of the parameter---e.g., the Hamiltonian operator in the case of elapsed time. This approach avoids entirely the problem of associating a Hermitian operator with the parameter. We illustrate the general formalism, first, with nonrelativistic uncertainty relations for spatial displacement and momentum, harmonic-oscillator phase and number of quanta, and time and energy and, second, with Lorentz-invariant uncertainty relations involving the displacement and Lorentz-rotation parameters of the Poincar\\'e group.",
  "revisions": [
    {
      "version": "v1",
      "updated": "1995-07-07T16:27:35.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "generalized uncertainty relations",
      "lorentz invariance",
      "harmonic-oscillator phase",
      "hermitian operator",
      "elapsed time"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 39,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 397113
    }
  }
}