{
  "id": "1506.08432",
  "version": "v1",
  "published": "2015-06-28T18:14:45.000Z",
  "updated": "2015-06-28T18:14:45.000Z",
  "title": "Advancing the case for $PT$ Symmetry -- the Hamiltonian is always $PT$ Symmetric",
  "authors": [
    "Philip D. Mannheim"
  ],
  "comment": "11 pages, revtex4",
  "categories": [
    "quant-ph",
    "hep-ph"
  ],
  "abstract": "While a Hamiltonian can be both Hermitian and $PT$ symmetric, it is $PT$ symmetry that is the more general, as it can lead to real energy eigenvalues even if the Hamiltonian is not Hermitian. We discuss some specific ways in which $PT$ symmetry goes beyond Hermiticity and is more far reaching than it. We show that simply by virtue of being the generator of time translations, the Hamiltonian must always be $PT$ symmetric, regardless of whether or not it might be Hermitian. We show that the reality of the Euclidean time path integral is a necessary and sufficient condition for $PT$ symmetry of a quantum field theory, with Hermiticity only being a sufficient condition. We show that in order to construct the correct classical action needed for a path integral quantization one must impose $PT$ symmetry on each classical path, a requirement that has no counterpart in any Hermiticity condition since Hermiticity of a Hamiltonian is only definable after the quantization has been performed and the quantum Hilbert space has been constructed. With the spacetime metric being $PT$ even we show that a covariant action must always be $PT$ symmetric. Unlike Hermiticity, $PT$ symmetry does not need to be postulated as it is derivable from Poincare invariance. Hermiticity is just a particular realization of $PT$ symmetry, one in which the eigenspectrum is real and complete.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-06-28T18:14:45.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hamiltonian",
      "hermiticity",
      "euclidean time path integral",
      "sufficient condition",
      "real energy eigenvalues"
    ],
    "note": {
      "typesetting": "RevTeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150608432M",
      "inspire": 1380207
    }
  }
}