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arXiv:2006.04774 · The European Physical Journal C

Hyperbolic ring based formulation for thermo field dynamics, quantum dissipation, entanglement, and holography

R. Cartas-Fuentevilla, Jasel Berra–Montiel, O. Meza-Aldama

Published Jul 1, 2020• 8 citations
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Abstract

Abstract The classical and quantum formulations for open systems related to dissipative dynamics are constructed on a complex hyperbolic ring, following universal symmetry principles, and considering the double thermal fields approach for modeling the system of interest, and the environment. The hyperbolic rotations are revealed as an underlying internal symmetry for the dissipative dynamics, and a chemical potential is identified as conjugate variable to the charge operator, and thus a grand partition function is constructed. As opposed to the standard scheme, there are not patologies associated with the existence of many unitarity inequivalent representations on the hyperbolic ring, since the whole of the dissipative quantum dynamics is realized by choosing only one representation of the field commutation relations. Entanglement entropy operators for the subsystem of interest and the environment, are constructed as a tool for study the entanglement generated from the dissipation. The holographic perspectives of our results are discussed.

DOI
10.1140/epjc/s10052-020-8161-x

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Abstract The classical and quantum formulations for open systems related to dissipative dynamics are constructed on a complex hyperbolic ring, following universal symmetry principles, and considering the double thermal fields approach for modeling the system of interest, and the environment. The hyperbolic rotations are revealed as an underlying internal symmetry for the dissipative dynamics, and a chemical potential is identified as conjugate variable to the charge operator, and thus a grand partition function is constructed. As opposed to the standard scheme, there are not patologies associated with the existence of many unitarity inequivalent representations on the hyperbolic ring, since the whole of the dissipative quantum dynamics is realized by choosing only one representation of the field commutation relations. Entanglement entropy operators for the subsystem of interest and the environment, are constructed as a tool for study the entanglement generated from the dissipation. The holographic perspectives of our results are discussed.
Abstract Using a hyperbolic ring based formalism, it is shown that special unitary groups correspond to thermo-algebras for closed quantum systems, while for open systems, the full unitary algebra is the suitable underlying structure. Specifically the groups U (2) and U (1, 1) are realized as the dynamical symmetry groups for open quantum systems through continuous tensorial products; these unitary groups are obtained by mean of abelian extensions through U (1)-factors, from the standard thermo-algebras SU (2), and SU (1, 1).
A microscopic description of an open system is generally expressed by the Hamiltonian of the form: H tot = H sys + H environ + H sys − environ . We developed a microscopic theory of entropy and derived a general formula, so-called "entropy-Hamiltonian relation" (EHR), that connects the entropy of the system to the interaction Hamiltonian represented by H sys - environ for a nonequilibrium open quantum system. To derive the EHR formula, we mapped the open quantum system to the representation space of the Liouville-space formulation or thermo field dynamics (TFD), and thus worked on the representation space := ⊗ , where denotes the ordinary Hilbert space while the tilde Hilbert space conjugates to . We show that the natural transformation (mapping) of nonequilibrium open quantum systems is accomplished within the theoretical structure of TFD. By using the obtained EHR formula, we also derived the equation of motion for the distribution function of the system. We demonstrated that by knowing the microscopic description of the interaction, namely, the specific form of H sys − environ on the representation space , the EHR formulas enable us to evaluate the entropy of the system and to gain some information about entropy for nonequilibrium open quantum systems.
In this paper, we study the relation between entanglement and global internal symmetries in de Sitter space. We consider two symmetric causally disconnected regions in the hyperbolic chart on de Sitter space. Since entanglement measures characterizes correlations, the study of entanglement between the two causally disconnected regions gives information about the long range correlations in de Sitter space. When a theory possesses an additive global internal symmetry, the entanglement measures for a state with fixed global charge may be decomposed into local charge sectors in either subsystem and thus providing a finer resolution of entanglement. Here, we will consider two theories: free complex scalar field and free Dirac field on de Sitter space. Both theories possess global internal U(1)U(1)U(1) symmetry. We study the symmetry resolved entanglement entropy for both theories in the Bunch-Davies vacuum state. We find that the symmetry resolved entanglement entropy has equipartition into local charge sectors up to the terms that scale as VH30{V}_{{H}_{3}}^{0}VH3​0​ in the limit of large VH3{V}_{{H}_{3}}VH3​​, where VH3{V}_{{H}_{3}}VH3​​ is the volume of either region. This equipartition however is only broken by the terms of order O(1/VH3)O(1/{V}_{{H}_{3}})O(1/VH3​​). Consequently, we have equipartition of symmetry resolved entanglement entropy in the limit of infinite volume.
General relationship among the density matrix formalism, the double-space formulation and thermo field dynamics is discussed in non-equilibrium dissipative quantum systems. The concept of weakly equivalent operators in the double space formulation is introduced to review many well-known results and it is shown to be useful in mapping between the density matrix formalism and the double space formulation. A new non-equilibrium thermo field dynamics is formulated to discuss dissipative quantum systems. A simple example of damped harmonic oscillators is discussed in the present representation.
A fundamental framework of thermo field dynamics (TFD), including non-equilibrium situations as well as equilibrium stuations, is formulated through a compact axiomatic form, in which no reference to reservoirs is needed, by considering quantum fields. The semi-free quantum fields in TFD are constructed in a way parallel to that of the usual quantum field theory without thermal degrees of freedom, i.e. in terms of orthonormalization relations among wave functions, the sum rule for wave functions, the divisor operator and the particle-antiparticle conjugation (C-conjugation).
We study entanglement entropy (EE) in conformal field theories (CFTs) in Minkowski space with a planar boundary or with a planar defect of any codimension. In any such boundary CFT or defect CFT (DCFT), we consider the reduced density matrix and associated EE obtained by tracing over the degrees of freedom outside of a (hemi)sphere centered on the boundary or defect. Following Casini, Huerta, and Myers, we map the reduced density matrix to a thermal density matrix of the same theory on hyperbolic space. The EE maps to the thermal entropy of the theory on hyperbolic space. For boundary CFTs and DCFTs dual holographically to Einstein gravity theories, the thermal entropy is equivalent to the Bekenstein--Hawking entropy of a hyperbolic black brane. We show that the horizon of the hyperbolic black brane coincides with the minimal area surface used in Ryu and Takayanagi's conjecture for the holographic calculation of EE. We thus prove their conjecture in these cases. We use our results to compute the R\'enyi entropies and EE in DCFTs in which the defect corresponds to a probe brane in a holographic dual.