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Haba Z. Functional Formulation of Quantum Theory of a Scalar Field in a Metric with Lorentzian and Euclidean Signatures. ENTROPY (BASEL, SWITZERLAND) 2024; 26:329. [PMID: 38667883 PMCID: PMC11049123 DOI: 10.3390/e26040329] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/28/2024] [Revised: 03/31/2024] [Accepted: 04/07/2024] [Indexed: 04/28/2024]
Abstract
We study the Schrödinger equation in quantum field theory (QFT) in its functional formulation. In this approach, quantum correlation functions can be expressed as classical expectation values over (complex) stochastic processes. We obtain a stochastic representation of the Schrödinger time evolution on Wentzel-Kramers-Brillouin (WKB) states by means of the Wiener integral. We discuss QFT in a flat expanding metric and in de Sitter space-time. We calculate the evolution kernel in an expanding flat metric in the real-time formulation. We discuss a field interaction in pseudoRiemannian and Riemannian metrics showing that an inversion of the signature leads to some substantial simplifications of the singularity problems in QFT.
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Affiliation(s)
- Zbigniew Haba
- Institute of Theoretical Physics, University of Wroclaw, Plac Maxa Borna 9, 50-204 Wroclaw, Poland
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2
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Shi Z, Abe S. Quantum Weak Invariants: Dynamical Evolution of Fluctuations and Correlations. ENTROPY 2020; 22:e22111219. [PMID: 33286987 PMCID: PMC7711532 DOI: 10.3390/e22111219] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Subscribe] [Scholar Register] [Received: 09/08/2020] [Revised: 10/15/2020] [Accepted: 10/21/2020] [Indexed: 11/16/2022]
Abstract
Weak invariants are time-dependent observables with conserved expectation values. Their fluctuations, however, do not remain constant in time. On the assumption that time evolution of the state of an open quantum system is given in terms of a completely positive map, the fluctuations monotonically grow even if the map is not unital, in contrast to the fact that monotonic increases of both the von Neumann entropy and Rényi entropy require the map to be unital. In this way, the weak invariants describe temporal asymmetry in a manner different from the entropies. A formula is presented for time evolution of the covariance matrix associated with the weak invariants in cases where the system density matrix obeys the Gorini-Kossakowski-Lindblad-Sudarshan equation.
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Affiliation(s)
- Zeyi Shi
- Department of Physics, College of Information Science and Engineering, Huaqiao University, Xiamen 361021, China;
| | - Sumiyoshi Abe
- Department of Physics, College of Information Science and Engineering, Huaqiao University, Xiamen 361021, China;
- Institute of Physics, Kazan Federal University, Kazan 420008, Russia
- Department of Natural and Mathematical Sciences, Turin Polytechnic University in Tashkent, Tashkent 100095, Uzbekistan
- ESIEA, 9 Rue Vesale, 75005 Paris, France
- Correspondence:
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Abe S. Weak invariants in dissipative systems: action principle and Noether charge for kinetic theory. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2020; 378:20190196. [PMID: 32223402 PMCID: PMC7134955 DOI: 10.1098/rsta.2019.0196] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 08/03/2019] [Indexed: 05/03/2023]
Abstract
In non-equilibrium classical thermostatistics, the state of a system may be described by not only dynamical/thermodynamical variables but also a kinetic distribution function. This 'double structure' bears some analogy with that in quantum thermodynamics, where both dynamical variables and the Hilbert space are involved. Recently, the concept of weak invariants has repeatedly been discussed in the context of quantum thermodynamics. A weak invariant is defined in such a way that its value changes in time but its expectation value is conserved under time evolution prescribed by a kinetic equation. Here, a new aspect of a weak invariant is revealed for the classical Fokker-Planck equation as an example of classical kinetic equations. The auxiliary field formalism is applied to the construction of the action for the kinetic equation. Then, it is shown that the auxiliary field is a weak invariant and is the Noether charge. The action is invariant under the transformation generated by the weak invariant. The result may shed light on possible roles of the symmetry principle in the kinetic descriptions of non-equilibrium systems. This article is part of the theme issue 'Fundamental aspects of nonequilibrium thermodynamics'.
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Affiliation(s)
- Sumiyoshi Abe
- Department of Physics, College of Information Science and Engineering, Huaqiao University, Xiamen 361021, People's Republic of China
- Institute of Physics, Kazan Federal University, Kazan 420008, Russia
- Department of Natural and Mathematical Sciences, Turin Polytechnic University in Tashkent, Tashkent 100095, Uzbekistan
- ESIEA, 9 Rue Vesale, Paris 75005, France
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Pinheiro MJ. A reformulation of mechanics and electrodynamics. Heliyon 2017; 3:e00365. [PMID: 28795163 PMCID: PMC5541156 DOI: 10.1016/j.heliyon.2017.e00365] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/05/2017] [Revised: 06/19/2017] [Accepted: 07/19/2017] [Indexed: 11/25/2022] Open
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5
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A variational method in out-of-equilibrium physical systems. Sci Rep 2013; 3:3454. [PMID: 24316718 PMCID: PMC3856405 DOI: 10.1038/srep03454] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/17/2013] [Accepted: 11/21/2013] [Indexed: 12/01/2022] Open
Abstract
We propose a new variational principle for out-of-equilibrium dynamic systems that are fundamentally based on the method of Lagrange multipliers applied to the total entropy of an ensemble of particles. However, we use the fundamental equation of thermodynamics on differential forms, considering U and S as 0-forms. We obtain a set of two first order differential equations that reveal the same formal symplectic structure shared by classical mechanics, fluid mechanics and thermodynamics. From this approach, a topological torsion current emerges of the form , where Aj and ωk denote the components of the vector potential (gravitational and/or electromagnetic) and where ω denotes the angular velocity of the accelerated frame. We derive a special form of the Umov-Poynting theorem for rotating gravito-electromagnetic systems. The variational method is then applied to clarify the working mechanism of particular devices.
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Abe S. Variational principle for fractional kinetics and the Lévy Ansatz. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:022142. [PMID: 24032810 DOI: 10.1103/physreve.88.022142] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/09/2013] [Indexed: 05/20/2023]
Abstract
A variational principle is developed for fractional kinetics based on the auxiliary-field formalism. It is applied to the Fokker-Planck equation with spatiotemporal fractionality, and a variational solution is obtained with the help of the Lévy Ansatz. It is shown how the whole range from subdiffusion to superdiffusion is realized by the variational solution as a competing effect between the long waiting time and the long jump. The motion of the center of the probability distribution is also analyzed in the case of a periodic drift.
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Affiliation(s)
- Sumiyoshi Abe
- Department of Physical Engineering, Mie University, Mie 514-8507, Japan
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Parreira JE, Fonseca-Romero KM, Nemes MC. Information content of nonautonomous free fields in curved space-time. Int J Clin Exp Med 2011. [DOI: 10.1103/physrevd.83.064033] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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8
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Cao FJ, Feito M. Out of equilibrium quantum field dynamics of an initial thermal state after a change in the external field. Int J Clin Exp Med 2006. [DOI: 10.1103/physrevd.73.045017] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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9
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Lee W, Charng YY, Lee DS, Fang LZ. Off-equilibrium dynamics of the primordial perturbations in the inflationary universe: TheO(N)model. Int J Clin Exp Med 2004. [DOI: 10.1103/physrevd.69.123522] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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10
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Abe S. Dilatation symmetry of the Fokker-Planck equation and anomalous diffusion. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 69:016102. [PMID: 14995662 DOI: 10.1103/physreve.69.016102] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/30/2003] [Indexed: 05/24/2023]
Abstract
Based on the canonical formalism, the dilatation symmetry is implemented to the Fokker-Planck equation for the Wigner distribution that describes atomic motion in an optical lattice. This reveals the symmetry principle underlying the recent result obtained by Lutz [Phys. Rev. A 67, 051402(R) (2003)] on the connection between anomalous transport in the optical lattice and Tsallis statistics in the high-energy regime. A condition is derived for the general linear Fokker-Planck equation to admit a nonstationary solution distribution that decays as a power law.
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Affiliation(s)
- Sumiyoshi Abe
- Institute of Physics, University of Tsukuba, Ibaraki 305-8571, Japan
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Faussurier G, Blancard C, Berthier E. Nonlocal thermodynamic equilibrium self-consistent average-atom model for plasma physics. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 63:026401. [PMID: 11308579 DOI: 10.1103/physreve.63.026401] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/26/2000] [Indexed: 05/23/2023]
Abstract
A time-dependent collisional-radiative average-atom model is presented to study statistical properties of highly charged ion plasmas in off-equilibrium conditions. The time evolution of electron populations and the electron covariance matrix is obtained as approximate solutions of a master equation. Atomic structure is described either with a screened-hydrogenic model including l splitting, or by calculating one-electron states in a self-consistent average-atom potential. Collisional and radiative excitation/deexcitation and ionization/recombination rates, as well as autoionization and dielectronic recombination rates, are formulated within the average-configuration framework. Local thermodynamic equilibrium is obtained as a specific steady-state solution. The influence of atomic structure and the role of autoionization and dielectronic recombination processes are studied by calculating steady-state average ionization and ionization variance of hot plasmas with or without radiation field.
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Affiliation(s)
- G Faussurier
- Commissariat à l'Energie Atomique, 91680 Bruyères-le-Châtel, Boîte Postale 12, France
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Cruz-Pacheco G, Minzoni A, Padilla P, Corichi A, Rosenbaum M, Ryan MP, Smyth NF. Effect of low momentum quantum fluctuations on a coherent field structure. Int J Clin Exp Med 2000. [DOI: 10.1103/physrevd.61.105011] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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Lee GH, Lee TH, Yee JH. Perturbative expansion around the Gaussian effective potential of the fermion field theory. Int J Clin Exp Med 1998. [DOI: 10.1103/physrevd.58.125001] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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Cho KH, Ji JY, Kim SP, Lee CH, Ryu JY. Heisenberg-picture approach to the evolution of the scalar fields in an expanding universe. Int J Clin Exp Med 1997. [DOI: 10.1103/physrevd.56.4916] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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15
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Minzoni AA, Rosenbaum M, Ryan MP. Stability of Gaussian approximations in minisuperspace: A variational approach. Int J Clin Exp Med 1997. [DOI: 10.1103/physrevd.56.2144] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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Natti PL. Initial-condition problem for a chiral Gross-Neveu system. PHYSICAL REVIEW. D, PARTICLES AND FIELDS 1996; 54:7867-7878. [PMID: 10020795 DOI: 10.1103/physrevd.54.7867] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Rajagopal AK. Time-dependent variational principle and the effective action in density-functional theory and Berry's phase. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1996; 54:3916-3922. [PMID: 9913939 DOI: 10.1103/physreva.54.3916] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Walhout TS, Cenni R, Fabrocini A, Fantoni S. Path integral variational methods for strongly correlated systems. PHYSICAL REVIEW. C, NUCLEAR PHYSICS 1996; 54:1622-1635. [PMID: 9971509 DOI: 10.1103/physrevc.54.1622] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Burgess M. Quantum fields in disequilibrium: Neutral scalar bosons with long-range, inhomogeneous perturbations. Int J Clin Exp Med 1995; 52:7103-7120. [PMID: 10019251 DOI: 10.1103/physrevd.52.7103] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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Martin C. Variational approximation for two-time correlation functions in Phi 4 theory: Optimization of the dynamics. Int J Clin Exp Med 1995; 52:7121-7128. [PMID: 10019252 DOI: 10.1103/physrevd.52.7121] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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Lee H, Na K, Yee JH. Finite temperature quantum field theory in the functional Schrödinger picture. Int J Clin Exp Med 1995; 51:3125-3128. [PMID: 10018784 DOI: 10.1103/physrevd.51.3125] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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Gleiser M, Ramos RO. Microphysical approach to nonequilibrium dynamics of quantum fields. PHYSICAL REVIEW. D, PARTICLES AND FIELDS 1994; 50:2441-2455. [PMID: 10017878 DOI: 10.1103/physrevd.50.2441] [Citation(s) in RCA: 16] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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23
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Boyanovsky D, Lee DS, Singh A. Phase transitions out of equilibrium: Domain formation and growth. Int J Clin Exp Med 1993; 48:800-815. [PMID: 10016309 DOI: 10.1103/physrevd.48.800] [Citation(s) in RCA: 96] [Impact Index Per Article: 3.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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24
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Boyanovsky D. Quantum rolling down out of equilibrium. PHYSICAL REVIEW. D, PARTICLES AND FIELDS 1993; 47:2343-2355. [PMID: 10015824 DOI: 10.1103/physrevd.47.2343] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Yong LC. Kinetic approach to the initial-value problem in phi4 field theory. PHYSICAL REVIEW. D, PARTICLES AND FIELDS 1992; 46:742-756. [PMID: 10014985 DOI: 10.1103/physrevd.46.742] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Lee GH, Yee JH. Covariant-gauge quantization of gauge theories in the functional Schrödinger picture. PHYSICAL REVIEW. D, PARTICLES AND FIELDS 1992; 46:865-867. [PMID: 10014999 DOI: 10.1103/physrevd.46.865] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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27
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Samiullah M, Éboli O, Pi SY. Quantum-field-theoretic analysis of inflation dynamics in a (2+1)-dimensional universe. PHYSICAL REVIEW. D, PARTICLES AND FIELDS 1991; 44:2335-2355. [PMID: 10014115 DOI: 10.1103/physrevd.44.2335] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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28
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Kim SK, Namgung W, Soh KS, Yee JH. Equivalence between the covariant, Weyl, and Coulomb gauges in the functional Schrödinger picture. Int J Clin Exp Med 1991; 43:2046-2049. [PMID: 10013584 DOI: 10.1103/physrevd.43.2046] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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Kim SK, Namgung W, Soh KS, Yee JH. Equivalence between the Weyl, Coulomb, and unitary gauges in the functional Schrödinger picture. Int J Clin Exp Med 1990; 41:3792-3795. [PMID: 10012321 DOI: 10.1103/physrevd.41.3792] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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30
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Kim SK, Namgung W, Soh KS, Yee JH. Abelian Higgs mechanism in the Schrödinger picture. Int J Clin Exp Med 1990; 41:1209-1215. [PMID: 10012454 DOI: 10.1103/physrevd.41.1209] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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31
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Redmount IH. Excited-state spectra of de Sitter-space scalar fields. PHYSICAL REVIEW. D, PARTICLES AND FIELDS 1989; 40:3343-3356. [PMID: 10011702 DOI: 10.1103/physrevd.40.3343] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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32
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Lawrie ID. Perturbative description of dissipation in nonequilibrium field theory. PHYSICAL REVIEW. D, PARTICLES AND FIELDS 1989; 40:3330-3342. [PMID: 10011701 DOI: 10.1103/physrevd.40.3330] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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33
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Kandrup HE. Nonequilibrium statistical quantum field theory for open systems. Int J Clin Exp Med 1989; 39:2253-2257. [PMID: 9959905 DOI: 10.1103/physrevd.39.2253] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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Redmount IH, Ruiz FR. Thermal equilibrium in de Sitter space. PHYSICAL REVIEW. D, PARTICLES AND FIELDS 1989; 39:2289-2296. [PMID: 9959909 DOI: 10.1103/physrevd.39.2289] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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35
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Symmetry behavior in the Einstein universe: Gaussian approximation in the Schrödinger picture. Int J Clin Exp Med 1988; 38:1853-1858. [PMID: 9959335 DOI: 10.1103/physrevd.38.1853] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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