{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:54:13Z","timestamp":1760241253895,"version":"build-2065373602"},"reference-count":76,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2019,12,27]],"date-time":"2019-12-27T00:00:00Z","timestamp":1577404800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We derive time evolution equations, namely the Klein\u2013Gordon equations for coherent fields and the Kadanoff\u2013Baym equations in quantum electrodynamics (QED) for open systems (with a central region and two reservoirs) as a practical model of quantum field theory of the brain. Next, we introduce a kinetic entropy current and show the H-theorem in the Hartree\u2013Fock approximation with the leading-order (LO) tunneling variable expansion in the 1st order approximation for the gradient expansion. Finally, we find the total conserved energy and the potential energy for time evolution equations in a spatially homogeneous system. We derive the Josephson current due to quantum tunneling between neighbouring regions by starting with the two-particle irreducible effective action technique. As an example of potential applications, we can analyze microtubules coupled to a water battery surrounded by a biochemical energy supply. Our approach can be also applied to the information transfer between two coherent regions via microtubules or that in networks (the central region and the     N res     reservoirs) with the presence of quantum tunneling.<\/jats:p>","DOI":"10.3390\/e22010043","type":"journal-article","created":{"date-parts":[[2019,12,27]],"date-time":"2019-12-27T11:42:47Z","timestamp":1577446967000},"page":"43","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Non-Equilibrium Quantum Electrodynamics in Open Systems as a Realizable Representation of Quantum Field Theory of the Brain"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0290-2994","authenticated-orcid":false,"given":"Akihiro","family":"Nishiyama","sequence":"first","affiliation":[{"name":"Graduate School of System Informatics, Kobe University, 1-1 Rokkodai, Nada-ku, Kobe 657-8501, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6659-2788","authenticated-orcid":false,"given":"Shigenori","family":"Tanaka","sequence":"additional","affiliation":[{"name":"Graduate School of System Informatics, Kobe University, 1-1 Rokkodai, Nada-ku, Kobe 657-8501, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9976-0429","authenticated-orcid":false,"given":"Jack A.","family":"Tuszynski","sequence":"additional","affiliation":[{"name":"Department of Oncology, University of Alberta, Cross Cancer Institute, Edmonton, AB T6G 1Z2, Canada"},{"name":"Department of Physics, University of Alberta, Edmonton, AB T6G 2J1, Canada"},{"name":"DIMEAS, Corso Duca degli Abruzzi, 24, Politecnico di Torino, 10129 Turin, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,12,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1319","DOI":"10.1038\/nn.2666","article-title":"Dna methylation and memory formation","volume":"13","author":"Day","year":"2010","journal-title":"Nat. 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