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

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.

Formulation Of Quantum Mechanics On Poincaré Disks

“…This approach leads naturally to covariant thermodynamic fluctuation theory correlated with Riemannian thermodynamic curvature in information geometry (Ruppeiner 1995). 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 under evaluation and the surrounding environment (Cartas-Fuentevilla et al, 2020). By an experimental standpoint, a step towards on-chip quantum simulation of materials science and interacting particles has been recently achieved in hyperbolic curved space.…”

Deformations of thermo-algebras and of Fock spaces on a ring, and open quantum systems

“…The fundamental observation of our approach is to realize that such a Lagrangian symmetry induces naturally its own complex structure, namely, the hyperbolic complex plane with its own complex unit. It has been demonstrated in [4] that this hyperbolic structure restricts strongly the quantization process, enforcing in fact towards a non-canonical scheme for the quantum description of open systems. Hence, the description of the hyperbolic complex plane will be develop in the next section 2, motivating its introduction by the sequence, mirror image → hyperbolic rotations → hyperbolic complex plane → non-canonical quantization.…”

“…Since the conventional schemes for quantum field theories have been constructed for describing closed systems, they cannot encompass in satisfactory way the description of open systems that will show dissipation; in this sense the quantum physics of the damping harmonic oscillator has played an profound influential role in the well-established formalism for the dissipation in quantum field theory. However, such a formulation has suffered of long-lived problems that nowadays even persist, but some of which have been addressed and overcome recently [4], by invoking a new algebraic setting that implies the use of two complex units, namely, the conventional one required by the quantization, and an additional (hyperbolic) unit that provides an enveloping algebra for encompassing the dynamics of the system of interest and of the environment. The incorporation of the new complex unit is not artificial, rather it is invoked by symmetry principles coming from the Lagrangian formulation of dissipative field theories; moreover at quantum level such a complex unit provides unitarity in the evolution of the whole system.…”

“…The dissipative dynamics can be obtained variationally from a Lagrangian that encodes the doubling of the degrees of freedom in phase space, with the idea of incorporating the dynamics of the environment; specifically a time reflected image of the system of interest is incorporated; hence the structure of such a Lagrangian that consists of the system plus its mirror image generates naturally an internal symmetry of the whole system, the hyperbolic rotations, which have been revealed recently [4]. The fundamental observation of our approach is to realize that such a Lagrangian symmetry induces naturally its own complex structure, namely, the hyperbolic complex plane with its own complex unit.…”

Informational Approaches Lead to Formulations of Quantum Mechanics on Poincaré Disks

“…To provide an analogy, think to the inspection of a single "frozen" frame extracted from a long video sequence. Paraphrasing [11], pathologies associated with the possible existence of many unitarity inequivalent representations on the hyperbolic disk disappear, since the whole operation of QM measurement is realized by choosing only one representation, i.e., the ψa located in the central tile.…”