{
  "id": "1807.07007",
  "version": "v1",
  "published": "2018-07-18T15:48:09.000Z",
  "updated": "2018-07-18T15:48:09.000Z",
  "title": "Path integrals with discarded degrees of freedom",
  "authors": [
    "Luke M. Butcher"
  ],
  "comment": "7 pages; sequel to arXiv:1707.05789",
  "categories": [
    "quant-ph"
  ],
  "abstract": "Whenever variables $\\phi=(\\phi^1,\\phi^2,\\ldots)$ are discarded from a system, and the discarded information capacity $\\mathcal{S}(x)$ depends on the value of an observable $x$, a quantum correction $\\Delta V_\\mathrm{eff}(x)$ appears in the effective potential [arXiv:1707.05789]. Here I examine the origins and implications of $\\Delta V_\\mathrm{eff}$ within the path integral, which I construct using Synge's world function. I show that the $\\phi$ variables can be `integrated out' of the path integral, reducing the propagator to a sum of integrals over observable paths $x(t)$ alone. The phase of each path is equal to the semiclassical action (divided by $\\hbar$) including the same correction $\\Delta V_\\mathrm{eff}$ as previously derived. This generalises the prior results beyond the limits of the Schr\\\"odinger equation; in particular, it allows us to consider discarded variables with a history-dependent information capacity $\\mathcal{S}=\\mathcal{S}(x(t),\\int^t f(x(t'))\\mathrm{d} t')$. History dependence does not alter the formula for $\\Delta V_\\mathrm{eff}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-07-18T15:48:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "path integral",
      "discarded degrees",
      "history-dependent information capacity",
      "synges world function",
      "quantum correction"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}