{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:28:03Z","timestamp":1760239683000,"version":"build-2065373602"},"reference-count":85,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2020,12,13]],"date-time":"2020-12-13T00:00:00Z","timestamp":1607817600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100010198","name":"MINECO","doi-asserted-by":"publisher","award":["FIS2017-86497-C2-2-P"],"award-info":[{"award-number":["FIS2017-86497-C2-2-P"]}],"id":[{"id":"10.13039\/501100010198","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012634","name":"UNAM","doi-asserted-by":"publisher","award":["DGAPA-UNAM IN113618"],"award-info":[{"award-number":["DGAPA-UNAM IN113618"]}],"id":[{"id":"10.13039\/501100012634","id-type":"DOI","asserted-by":"publisher"}]},{"name":"COST Action","award":["CA16104 GWverse"],"award-info":[{"award-number":["CA16104 GWverse"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Universe"],"abstract":"<jats:p>In generic curved spacetimes, the unavailability of a natural choice of vacuum state introduces a serious ambiguity in the Fock quantization of fields. In this review, we study the case of fermions described by a Dirac field in non-stationary spacetimes, and present recent results obtained by us and our collaborators about well-motivated criteria capable to ensure the uniqueness in the selection of a vacuum up to unitary transformations, at least in certain situations of interest in cosmology. These criteria are based on two reasonable requirements. First, the invariance of the vacuum under the symmetries of the Dirac equations in the considered spacetime. These symmetries include the spatial isometries. Second, the unitary implementability of the Heisenberg dynamics of the annihilation and creation operators when the curved spacetime is treated as a fixed background. This last requirement not only permits the uniqueness of the Fock quantization but, remarkably, it also allows us to determine an essentially unique splitting between the phase space variables assigned to the background and the fermionic annihilation and creation variables. We first consider Dirac fields in 2 + 1 dimensions and then discuss the more relevant case of 3 + 1 dimensions, particularizing the analysis to cosmological spacetimes with spatial sections of spherical or toroidal topology. We use this analysis to investigate the combined, hybrid quantization of the Dirac field and a flat homogeneous and isotropic background cosmology when the latter is treated as a quantum entity, and the former as a perturbation. Specifically, we focus our study on a background quantization along the lines of loop quantum cosmology. Among the Fock quantizations for the fermionic perturbations admissible according to our criteria, we discuss the possibility of further restricting the choice of a vacuum by the requisite of a finite fermionic backreaction and, moreover, by the diagonalization of the fermionic contribution to the total Hamiltonian in the asymptotic limit of large wave numbers of the Dirac modes. Finally, we argue in support of the uniqueness of the vacuum state selected by the extension of this diagonalization condition beyond the commented asymptotic region, in particular proving that it picks out the standard Poincar\u00e9 and Bunch\u2013Davies vacua for fixed flat and de Sitter background spacetimes, respectively.<\/jats:p>","DOI":"10.3390\/universe6120241","type":"journal-article","created":{"date-parts":[[2020,12,13]],"date-time":"2020-12-13T23:39:36Z","timestamp":1607902776000},"page":"241","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Uniqueness Criteria for the Fock Quantization of Dirac Fields and Applications in Hybrid Loop Quantum Cosmology"],"prefix":"10.3390","volume":"6","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9705-6769","authenticated-orcid":false,"given":"Jer\u00f3nimo","family":"Cortez","sequence":"first","affiliation":[{"name":"Departamento de F\u00edsica, Facultad de Ciencias, Universidad Nacional Aut\u00f3noma de M\u00e9xico, Ciudad de M\u00e9xico 04510, Mexico"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6242-7814","authenticated-orcid":false,"given":"Beatriz Elizaga","family":"Navascu\u00e9s","sequence":"additional","affiliation":[{"name":"Institute for Quantum Gravity, Friedrich-Alexander University Erlangen-N\u00fcrnberg, Staudstra\u00dfe 7, 91058 Erlangen, Germany"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3378-9610","authenticated-orcid":false,"given":"Guillermo A.","family":"Mena Marug\u00e1n","sequence":"additional","affiliation":[{"name":"Instituto de Estructura de la Materia, IEM-CSIC, Serrano 121, 28006 Madrid, Spain"}]},{"given":"Santiago","family":"Prado","sequence":"additional","affiliation":[{"name":"Instituto de Estructura de la Materia, IEM-CSIC, Serrano 121, 28006 Madrid, Spain"}]},{"given":"Jos\u00e9 M.","family":"Velhinho","sequence":"additional","affiliation":[{"name":"Faculdade de Ci\u00eancias and FibEnTech, Universidade da Beira Interior, R. Marqu\u00eas D\u2019\u00c1vila e Bolama, 6201-001 Covilh\u00e3, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2020,12,13]]},"reference":[{"key":"ref_1","unstructured":"Peskin, M., and Schroeder, D. (1995). An Introduction to Quantum Field Theory, Westview Press."},{"key":"ref_2","unstructured":"Weinberg, S. (2005). The Quantum Theory of Fields: Volume 1, Foundations, Cambridge University Press."},{"key":"ref_3","unstructured":"Wald, R.M. (1994). Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics, Chicago University Press."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Galindo, A., and Pascual, P. (1990). Quantum Mechanics I, Springer.","DOI":"10.1007\/978-3-642-83854-5"},{"key":"ref_5","unstructured":"Streater, R.F. (1972). Topics in Functional Analysis, Academic Press."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Baez, J.C., Segal, I.V., and Zhou, Z. (1992). Introduction to Algebraic and Constructive Quantum Field Theory, Princeton University Press.","DOI":"10.1515\/9781400862504"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1090\/S0002-9947-1962-0137504-6","article-title":"Linear symmetries of free boson fields","volume":"103","author":"Shale","year":"1962","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_8","first-page":"65","article-title":"Introduction to representations of the canonical commutation and anticommutation relations","volume":"Volume 695","author":"Siedentop","year":"2006","journal-title":"Large Coulomb Systems. Lecture Notes in Physics"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1007\/BF01940330","article-title":"Linear spin-zero quantum fields in external gravitational and scalar fields I. A one particle structure for the stationary case","volume":"62","author":"Kay","year":"1978","journal-title":"Commun. Math. Phys."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"36","DOI":"10.1016\/j.aop.2015.09.016","article-title":"Quantum unitary dynamics in cosmological spacetimes","volume":"363","author":"Cortez","year":"2015","journal-title":"Ann. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"064020","DOI":"10.1103\/PhysRevD.72.064020","article-title":"Feasibility of a unitary quantum dynamics in the Gowdy T3 cosmological model","volume":"72","author":"Cortez","year":"2005","journal-title":"Phys. Rev. D"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"084020","DOI":"10.1103\/PhysRevD.73.084020","article-title":"Quantum Gowdy T3 model: A unitary description","volume":"73","author":"Corichi","year":"2006","journal-title":"Phys. Rev. D"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"6301","DOI":"10.1088\/0264-9381\/23\/22\/014","article-title":"Quantum Gowdy T3 model: A uniqueness result","volume":"23","author":"Corichi","year":"2006","journal-title":"Class. Quantum Grav."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"124031","DOI":"10.1103\/PhysRevD.76.124031","article-title":"Quantum Gowdy T3 model: Schr\u00f6dinger representation with unitary dynamics","volume":"76","author":"Corichi","year":"2007","journal-title":"Phys. Rev. D"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"084027","DOI":"10.1103\/PhysRevD.75.084027","article-title":"Uniqueness of the Fock quantization of the Gowdy T3 model","volume":"75","author":"Cortez","year":"2007","journal-title":"Phys. Rev. D"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"105005","DOI":"10.1088\/0264-9381\/25\/10\/105005","article-title":"Uniqueness of the Fock representation of the Gowdy S1 \u00d7 S2 and S3 models","volume":"25","author":"Cortez","year":"2008","journal-title":"Class. Quantum Grav."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"044037","DOI":"10.1103\/PhysRevD.81.044037","article-title":"Fock quantization of a scalar field with time dependent mass on the three-sphere: Unitarity and uniqueness","volume":"81","author":"Cortez","year":"2010","journal-title":"Phys. Rev. D"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"084040","DOI":"10.1103\/PhysRevD.79.084040","article-title":"Uniqueness of the Fock quantization of a free scalar field on S1 with time dependent mass","volume":"79","author":"Cortez","year":"2009","journal-title":"Phys. Rev. D"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"030","DOI":"10.1088\/1475-7516\/2010\/10\/030","article-title":"A unique Fock quantization for fields in non-stationary spacetimes","volume":"10","author":"Cortez","year":"2010","journal-title":"J. Cosmol. Astropart. Phys."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"025002","DOI":"10.1103\/PhysRevD.83.025002","article-title":"Uniqueness of the Fock quantization of fields with unitary dynamics in nonstationary spacetimes","volume":"83","author":"Cortez","year":"2011","journal-title":"Phys. Rev. D"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"172001","DOI":"10.1088\/0264-9381\/28\/17\/172001","article-title":"A uniqueness criterion for the Fock quantization of scalar fields with time dependent mass","volume":"28","author":"Cortez","year":"2011","journal-title":"Class. Quantum Grav."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"104003","DOI":"10.1103\/PhysRevD.86.104003","article-title":"Criteria for the determination of time dependent scalings in the Fock quantization of scalar fields with a time dependent mass in ultrastatic spacetimes","volume":"86","author":"Cortez","year":"2012","journal-title":"Phys. Rev. D"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"103525","DOI":"10.1103\/PhysRevD.85.103525","article-title":"Unique Fock quantization of scalar cosmological perturbations","volume":"85","author":"Olmedo","year":"2012","journal-title":"Phys. Rev. D"},{"key":"ref_24","first-page":"001","article-title":"Uniqueness of the Fock quantization of scalar fields in spatially flat cosmological spacetimes","volume":"1211","author":"Cortez","year":"2012","journal-title":"J. Cosmol. Astropart. Phys."},{"key":"ref_25","first-page":"43","article-title":"Unitary evolution and uniqueness of the Fock quantization in flat cosmologies with compact spatial sections","volume":"11","author":"Cortez","year":"2014","journal-title":"Electron. J. Theor. Phys."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"044013","DOI":"10.1103\/PhysRevD.87.044013","article-title":"Uniqueness of the Fock quantization of scalar fields under mode preserving canonical transformations varying in time","volume":"87","author":"Cortez","year":"2013","journal-title":"Phys. Rev. D"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"075015","DOI":"10.1088\/0264-9381\/30\/7\/075015","article-title":"Massless scalar field in de Sitter spacetime: Unitary quantum time evolution","volume":"30","author":"Cortez","year":"2013","journal-title":"Class. Quantum Grav."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"084052","DOI":"10.1103\/PhysRevD.89.084052","article-title":"Uniqueness of the Fock quantization of scalar fields and processes with signature change in cosmology","volume":"89","year":"2014","journal-title":"Phys. Rev. D"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"105019","DOI":"10.1103\/PhysRevD.94.105019","article-title":"Uniqueness of the Fock quantization of scalar fields in a Bianchi I cosmology with unitary dynamics","volume":"94","author":"Cortez","year":"2016","journal-title":"Phys. Rev. D"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"124010","DOI":"10.1103\/PhysRevD.91.124010","article-title":"Unitarity and ultraviolet regularity in cosmology","volume":"91","author":"Ashtekar","year":"2015","journal-title":"Phys. Rev. D"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1057","DOI":"10.1103\/PhysRev.183.1057","article-title":"Quantized fields and particle creation in expanding universes. I","volume":"183","author":"Parker","year":"1969","journal-title":"Phys. Rev."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Birrell, N.D., and Davies, P.C.W. (1982). Quantum Fields in Curved Space, Cambridge University Press.","DOI":"10.1017\/CBO9780511622632"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"3136","DOI":"10.1103\/PhysRevD.32.3136","article-title":"Vacuum states in de Sitter space","volume":"32","author":"Allen","year":"1985","journal-title":"Phys. Rev. D"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Fulling, S.A. (1989). Aspects of Quantum Field Theory in Curved Spacetime, Cambridge University Press.","DOI":"10.1017\/CBO9781139172073"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1642007","DOI":"10.1142\/S0218271816420074","article-title":"Hybrid models in loop quantum cosmology","volume":"25","year":"2016","journal-title":"Int. J. Mod. Phys. D"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Elizaga Navascu\u00e9s, B., and Mena Marug\u00e1n, G.A. (2020). Hybrid loop quantum cosmology: An overview. arXiv.","DOI":"10.3389\/fspas.2021.624824"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"044023","DOI":"10.1103\/PhysRevD.96.044023","article-title":"Fermions in hybrid loop quantum cosmology","volume":"96","year":"2017","journal-title":"Phys. Rev. D"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"063535","DOI":"10.1103\/PhysRevD.98.063535","article-title":"Backreaction of fermionic perturbations in the Hamiltonian of hybrid loop quantum cosmology","volume":"98","year":"2018","journal-title":"Phys. Rev. D"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"063535","DOI":"10.1103\/PhysRevD.99.063535","article-title":"Asymptotic diagonalization of the fermionic Hamiltonian in hybrid loop quantum cosmology","volume":"99","author":"Prado","year":"2019","journal-title":"Phys. Rev. D"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"185010","DOI":"10.1088\/1361-6382\/ab32af","article-title":"Hamiltonian diagonalization in hybrid quantum cosmology","volume":"36","author":"Thiemann","year":"2019","journal-title":"Class. Quantum Grav."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1098\/rspa.1978.0060","article-title":"Quantum field theory in de Sitter space: Renormalization by point-splitting","volume":"360","author":"Bunch","year":"1978","journal-title":"Proc. R. Soc. Lond. A"},{"key":"ref_42","unstructured":"Lawson, H.B., and Michelson, M.L. (1989). Spin Geometry, Princeton University Press."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1739","DOI":"10.1063\/1.1664507","article-title":"Spinor structure of space-times in General Relativity. I","volume":"9","author":"Geroch","year":"1968","journal-title":"J. Math. Phys."},{"key":"ref_44","unstructured":"Misner, C.W., Thorne, K.S., and Wheeler, J.A. (1973). Gravitation, W.H. Freeman."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1090\/S0002-9947-1982-0637032-8","article-title":"Dirac quantum fields on a manifold","volume":"269","author":"Dimock","year":"1982","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"490","DOI":"10.1016\/0003-4916(79)90135-0","article-title":"Existence of the S-matrix in quantum field theory in curved space-time","volume":"118","author":"Wald","year":"1979","journal-title":"Ann. Phys."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"955","DOI":"10.1103\/PhysRevD.4.955","article-title":"Canonical quantization of cylindrical gravitational waves","volume":"4","year":"1971","journal-title":"Phys. Rev. D"},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Israel, W. (1973). Canonical quantization of gravity. Relativity, Astrophysics and Cosmology, D. Reidel Publishing Company.","DOI":"10.1007\/978-94-010-2639-0"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"041502","DOI":"10.1103\/PhysRevD.73.041502","article-title":"Unitary evolution in Gowdy cosmology","volume":"73","author":"Corichi","year":"2006","journal-title":"Phys. Rev. D"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"012151","DOI":"10.1088\/1742-6596\/410\/1\/012151","article-title":"Unitary evolution and uniqueness of the Fock quantization in flat cosmologies","volume":"410","year":"2013","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Cortez, J., Mena Marug\u00e1n, G.A., and Velhinho, J. (2020). Quantum linear scalar fields with time dependent potentials: Overview and applications to cosmology. Mathematics, 8.","DOI":"10.3390\/math8010115"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"025024","DOI":"10.1103\/PhysRevD.96.025024","article-title":"Uniqueness of the Fock quantization of Dirac fields in 2+1 dimensions","volume":"96","author":"Cortez","year":"2017","journal-title":"Phys. Rev. D"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"105013","DOI":"10.1103\/PhysRevD.92.105013","article-title":"Unitary evolution and uniqueness of the Fock representation of Dirac fields in cosmological spacetimes","volume":"92","author":"Cortez","year":"2015","journal-title":"Phys. Rev. D"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"084053","DOI":"10.1103\/PhysRevD.93.084053","article-title":"Unique Fock quantization of a massive fermion field in a cosmological scenario","volume":"93","author":"Cortez","year":"2016","journal-title":"Phys. Rev. D"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"76","DOI":"10.1016\/j.aop.2016.11.005","article-title":"Dirac fields in flat FLRW cosmology: Uniqueness of the Fock quantization","volume":"376","author":"Cortez","year":"2017","journal-title":"Ann. Phys."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"2450816","DOI":"10.1155\/2018\/2450816","article-title":"On the uniqueness of the Fock quantization of the Dirac field in the closed FRW cosmology","volume":"2018","author":"Cortez","year":"2018","journal-title":"Adv. Math. Phys."},{"key":"ref_57","first-page":"3","article-title":"Richtungsfelder und fernparallelismus in mannigfaltigkeiten","volume":"8","author":"Stiefel","year":"1936","journal-title":"Comm. Math. Helv."},{"key":"ref_58","doi-asserted-by":"crossref","unstructured":"Isham, C.J. (1999). Modern Differential Geometry for Physicists, World Scientific. [2nd ed.].","DOI":"10.1142\/3867"},{"key":"ref_59","unstructured":"Roe, J. (1999). Elliptic Operators, Topology and Asymptotic Methods, Chapman & Hall\/CRC. [2nd ed.]."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1007\/BF02748689","article-title":"On the quantization of systems with anticommuting variables","volume":"33","author":"Casalbuoni","year":"1976","journal-title":"Nuovo Cimento A"},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"86","DOI":"10.1016\/0003-4916(78)90005-2","article-title":"Hamiltonian formulation of the theory of interacting gravitational and electron fields","volume":"116","author":"Nelson","year":"1978","journal-title":"Ann. Phys."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"1100","DOI":"10.1103\/PhysRevD.35.1100","article-title":"Fermions in quantum cosmology","volume":"35","author":"Halliwell","year":"1987","journal-title":"Phys. Rev. D"},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1007\/BF01199016","article-title":"The Dirac operator on homogeneous spaces and its spectrum on 3-dimensional lens spaces","volume":"59","year":"1992","journal-title":"Arch. Math."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"453","DOI":"10.3836\/tjm\/1255958682","article-title":"A representation of Spin(4) on the eigenspinors of the Dirac operator on S3","volume":"23","author":"Homma","year":"2000","journal-title":"Tokyo J. Math."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"57","DOI":"10.4064\/cm-48-1-57-62","article-title":"Zur abh\u00e4ngigkeit des Dirac-operators von der spin-struktur","volume":"48","author":"Friedrich","year":"1984","journal-title":"Colloq. Math."},{"key":"ref_66","doi-asserted-by":"crossref","unstructured":"Ginoux, N. (2009). The Dirac Spectrum, Springer.","DOI":"10.1007\/978-3-642-01570-0"},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1007\/BF01601813","article-title":"Ueber den begriff der vollst\u00e4ndigen differentialgeometrischen fl\u00e4che","volume":"3","author":"Hopf","year":"1931","journal-title":"Comment. Math. Helv."},{"key":"ref_68","doi-asserted-by":"crossref","unstructured":"Eisberg, R., and Resnick, R. (1985). Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, Wiley. [2nd ed.].","DOI":"10.1063\/1.2814935"},{"key":"ref_69","unstructured":"Dirac, P.A.M. (1964). Lectures on Quantum Mechanics, Belfer Graduate School of Science, Yeshiva University."},{"key":"ref_70","unstructured":"Wallach, N.R. (1973). Harmonic Analysis on Homogeneous Spaces, Marcel Dekker."},{"key":"ref_71","doi-asserted-by":"crossref","unstructured":"Kirillov, A.A. (1976). Elements of the Theory of Representations, Springer.","DOI":"10.1007\/978-3-642-66243-0"},{"key":"ref_72","doi-asserted-by":"crossref","unstructured":"Halzen, F., and Martin, A.D. (1984). Quarks and Leptons: An Introductory Course in Modern Particle Physics, John Wiley & Sons.","DOI":"10.1119\/1.14146"},{"key":"ref_73","first-page":"045","article-title":"Gauge-invariant perturbations in hybrid quantum cosmology","volume":"1506","year":"2015","journal-title":"J. Cosmol. Astropart. Phys."},{"key":"ref_74","doi-asserted-by":"crossref","unstructured":"Carrier, G.F., and Pearson, C.E. (1976). Partial Differential Equations: Theory and Technique, Academic Press.","DOI":"10.1016\/B978-0-12-160450-9.50008-9"},{"key":"ref_75","doi-asserted-by":"crossref","first-page":"123530","DOI":"10.1103\/PhysRevD.101.123530","article-title":"Unique fermionic vacuum in de Sitter spacetime from hybrid quantum cosmology","volume":"101","author":"Prado","year":"2020","journal-title":"Phys. Rev. D"},{"key":"ref_76","doi-asserted-by":"crossref","unstructured":"Andrews, G.E., Askey, R., and Roy, R. (1999). Special Functions, Cambridge University Press.","DOI":"10.1017\/CBO9781107325937"},{"key":"ref_77","unstructured":"Abramovitz, M., and Stegun, I.A. (1970). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables."},{"key":"ref_78","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1007\/BF01454972","article-title":"Spinor two-point functions in maximally symmetric spaces","volume":"106","author":"Allen","year":"1986","journal-title":"Commun. Math. Phys."},{"key":"ref_79","doi-asserted-by":"crossref","first-page":"024002","DOI":"10.1103\/PhysRevD.71.024002","article-title":"Fermionic \u03b1-vacua","volume":"71","author":"Collins","year":"2005","journal-title":"Phys. Rev. D"},{"key":"ref_80","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1103\/PhysRevD.9.341","article-title":"Adiabatic regularization of the energy-momentum tensor of a quantized field in homogeneous spaces","volume":"9","author":"Parker","year":"1974","journal-title":"Phys. Rev. D"},{"key":"ref_81","doi-asserted-by":"crossref","first-page":"2963","DOI":"10.1103\/PhysRevD.36.2963","article-title":"Adiabatic regularization in closed Robertson-Walker universes","volume":"36","author":"Anderson","year":"1987","journal-title":"Phys. Rev. D"},{"key":"ref_82","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1007\/BF02102088","article-title":"Local quasiequivalence and adiabatic vacuum states","volume":"134","author":"Roberts","year":"1990","journal-title":"Commun. Math. Phys."},{"key":"ref_83","doi-asserted-by":"crossref","first-page":"635","DOI":"10.1007\/s002200000350","article-title":"The Hadamard condition for Dirac fields and adiabatic states on Robertson-Walker spacetimes","volume":"216","author":"Hollands","year":"2001","journal-title":"Commun. Math. Phys."},{"key":"ref_84","doi-asserted-by":"crossref","first-page":"025016","DOI":"10.1103\/PhysRevD.98.025016","article-title":"Adiabatic expansions for Dirac fields, renormalization, and anomalies","volume":"98","author":"Barbero","year":"2018","journal-title":"Phys. Rev. D"},{"key":"ref_85","doi-asserted-by":"crossref","first-page":"125003","DOI":"10.1103\/PhysRevD.100.125003","article-title":"Fock quantization of the Dirac field in hybrid quantum cosmology: Relation with adiabatic states","volume":"100","author":"Prado","year":"2019","journal-title":"Phys. Rev. D"}],"container-title":["Universe"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2218-1997\/6\/12\/241\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:44:36Z","timestamp":1760179476000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2218-1997\/6\/12\/241"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,12,13]]},"references-count":85,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2020,12]]}},"alternative-id":["universe6120241"],"URL":"https:\/\/doi.org\/10.3390\/universe6120241","relation":{},"ISSN":["2218-1997"],"issn-type":[{"type":"electronic","value":"2218-1997"}],"subject":[],"published":{"date-parts":[[2020,12,13]]}}}