{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T12:31:35Z","timestamp":1787315495591,"version":"build-2736575974"},"reference-count":78,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2018,3,19]],"date-time":"2018-03-19T00:00:00Z","timestamp":1521417600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Albert Einstein Center for Gravitation and Astrophysics, Czech Science Foundation","award":["14-37086G"],"award-info":[{"award-number":["14-37086G"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>A trajectory-based representation for the quantum theory of the gravitational field is formulated. This is achieved in terms of a covariant Generalized Lagrangian-Path (GLP) approach which relies on a suitable statistical representation of Bohmian Lagrangian trajectories, referred to here as GLP-representation. The result is established in the framework of the manifestly-covariant quantum gravity theory (CQG-theory) proposed recently and the related CQG-wave equation advancing in proper-time the quantum state associated with massive gravitons. Generally non-stationary analytical solutions for the CQG-wave equation with non-vanishing cosmological constant are determined in such a framework, which exhibit Gaussian-like probability densities that are non-dispersive in proper-time. As a remarkable outcome of the theory achieved by implementing these analytical solutions, the existence of an emergent gravity phenomenon is proven to hold. Accordingly, it is shown that a mean-field background space-time metric tensor can be expressed in terms of a suitable statistical average of stochastic fluctuations of the quantum gravitational field whose quantum-wave dynamics is described by GLP trajectories.<\/jats:p>","DOI":"10.3390\/e20030205","type":"journal-article","created":{"date-parts":[[2018,3,20]],"date-time":"2018-03-20T06:57:11Z","timestamp":1521529031000},"page":"205","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":19,"title":["Generalized Lagrangian Path Approach to Manifestly-Covariant Quantum Gravity Theory"],"prefix":"10.3390","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5276-8511","authenticated-orcid":false,"given":"Massimo","family":"Tessarotto","sequence":"first","affiliation":[{"name":"Department of Mathematics and Geosciences, University of Trieste, Via Valerio 12, 34127 Trieste, Italy"},{"name":"Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava, Bezru\u010dovo n\u00e1m.13, CZ-74601 Opava, Czech Republic"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2879-6085","authenticated-orcid":false,"given":"Claudio","family":"Cremaschini","sequence":"additional","affiliation":[{"name":"Institute of Physics and Research Center for Theoretical Physics and Astrophysics, Faculty of Philosophy and Science, Silesian University in Opava, Bezru\u010dovo n\u00e1m.13, CZ-74601 Opava, Czech Republic"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2018,3,19]]},"reference":[{"key":"ref_1","unstructured":"Messiah, A. 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