{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,8]],"date-time":"2026-10-08T22:41:19Z","timestamp":1791499279314,"version":"4.3.4"},"reference-count":39,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2019,7,8]],"date-time":"2019-07-08T00:00:00Z","timestamp":1562544000000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Quantum"],"abstract":"<jats:p>\n                    This article generalizes the conditional probability interpretation of time in which time evolution is realized through entanglement between a clock and a system of interest. This formalism is based upon conditioning a solution to the Wheeler-DeWitt equation on a subsystem of the Universe, serving as a clock, being in a state corresponding to a time\n                    <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <mml:mi>t<\/mml:mi>\n                    <\/mml:math>\n                    . Doing so assigns a conditional state to the rest of the Universe\n                    <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                        <mml:mo stretchy=\"false\">|<\/mml:mo>\n                      <\/mml:mrow>\n                      <mml:msub>\n                        <mml:mi>\u03c8<\/mml:mi>\n                        <mml:mi>S<\/mml:mi>\n                      <\/mml:msub>\n                      <mml:mo stretchy=\"false\">(<\/mml:mo>\n                      <mml:mi>t<\/mml:mi>\n                      <mml:mo stretchy=\"false\">)<\/mml:mo>\n                      <mml:mo fence=\"false\" stretchy=\"false\">\u27e9<\/mml:mo>\n                    <\/mml:math>\n                    , referred to as the system. We demonstrate that when the total Hamiltonian appearing in the Wheeler-DeWitt equation contains an interaction term coupling the clock and system, the conditional state\n                    <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                        <mml:mo stretchy=\"false\">|<\/mml:mo>\n                      <\/mml:mrow>\n                      <mml:msub>\n                        <mml:mi>\u03c8<\/mml:mi>\n                        <mml:mi>S<\/mml:mi>\n                      <\/mml:msub>\n                      <mml:mo stretchy=\"false\">(<\/mml:mo>\n                      <mml:mi>t<\/mml:mi>\n                      <mml:mo stretchy=\"false\">)<\/mml:mo>\n                      <mml:mo fence=\"false\" stretchy=\"false\">\u27e9<\/mml:mo>\n                    <\/mml:math>\n                    satisfies a time-nonlocal Schr\u00f6dinger equation in which the system Hamiltonian is replaced with a self-adjoint integral operator. This time-nonlocal Schr\u00f6dinger equation is solved perturbatively and three examples of clock-system interactions are examined. One example considered supposes that the clock and system interact via Newtonian gravity, which leads to the system's Hamiltonian developing corrections on the order of\n                    <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <mml:mi>G<\/mml:mi>\n                      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                        <mml:mo>\/<\/mml:mo>\n                      <\/mml:mrow>\n                      <mml:msup>\n                        <mml:mi>c<\/mml:mi>\n                        <mml:mn>4<\/mml:mn>\n                      <\/mml:msup>\n                    <\/mml:math>\n                    and inversely proportional to the distance between the clock and system.\n                  <\/jats:p>","DOI":"10.22331\/q-2019-07-08-160","type":"journal-article","created":{"date-parts":[[2019,7,8]],"date-time":"2019-07-08T12:55:21Z","timestamp":1562590521000},"page":"160","source":"Crossref","is-referenced-by-count":89,"title":["Quantizing time: Interacting clocks and systems"],"prefix":"10.22331","volume":"3","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4618-4832","authenticated-orcid":false,"given":"Alexander R. 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