Konstantin G. Zloshchastiev

Known interpretations of quantum mechanics are somewhat disconnected from a theory of relativity, which is predominantly used to describe the dynamics of elementary particles and forms the foundation of the contemporary Standard Model of particle physics. Conventional quantum mechanics relies on Galilean-covariant evolution equations of the Schrödinger type, whereas relativistic wave equations are Lorentz-covariant, of the Klein-Gordon type, therefore their solutions do not have a straightforward probabilistic interpretation. The superfluid vacuum conjecture reconciles these two fundamental symmetries, integrating both within a unified formalism. It uses two formal mappings, which endow wavefunctions with a fluid-mechanical interpretation, relate the superfluid wave equation to Korteweg-type ideal fluid flow, describe its small fluctuations, and ultimately give rise to a four-dimensional Lorentzian spacetime. The latter naturally incorporates gravity and the equivalence principle, serving as a background for the propagation of small fluctuations of the superfluid vacuum and the emergence of relativistic fields. In this formalism, microscopic particles are described by small-amplitude waves of the background superfluid that obey relativistic field equations in curved spacetime with an induced metric, until they interact with a measurement apparatus and manifest particle-like behaviour. Furthermore, the superfluid vacuum approach suggests an alternative mechanism for the emergence of the classical world from a quantum substrate. While decoherence may account for classicality of systems of many weakly interacting particles, strongly interacting matter scales up primarily by overlapping wavefunctions and forming collective states. This naturally builds up the collective state's mass toward mesoscopic and macroscopic scales (maintaining the equivalence of gravitational and inertial mass due the aforementioned spacetime structure), whereas quantum superposition eventually breaks down. Apart from its fundamental value, Korteweg/logarithmic fluidics and its duality to quantum mechanics might be instrumental in building a kind of quantum computer based on acoustic oscillations in laboratory logarithmic fluids.

A unified quantum-statistical framework is proposed for open three-level atomic systems when two lower-energy states are coupled to a single upper state. We suggest a non-Hermitian generalization of the closed model’s Hamiltonian by the analytical continuation of real-valued parameters related to the energies into a complex domain, as well as by considering some terms previously neglected by the procedure of Hermitian conjugation. We show that such systems can have two types of density operator’s evolution, depending on whether they are expected to be sustainable, or in a process of irreversible decay. This approach not only explains previously heuristically introduced terms in equations of motion for the density matrix but also suggests new terms which predict new dissipative effects. We then consider a special case, namely the evolution of pure states described by a generalized Schrödinger equation, which turns out to be nonlinear with respect to the atomic wavefunctions. As a final step toward unification, in addition to the non-Hermitian Hamiltonian generalization, we also include Markovian quantum jumps by adding the Lindblad dissipator. The resulting equations not only expand the range of applicability of open three-level models, but also suggest that dissipative phenomena in quantum-optical systems in vivo are more diverse and significant than previously believed.

A reduced density operator approach is proposed for electron hopping between the nearest neighboring carbon atoms in the monolayer graphene lattice in the presence of the environment. To analytically describe at least the simplest of dissipative effects which can occur, we consider the tight‐binding approximation and propose the minimal non‐Hermitian extension of the band Hamiltonian, which models spontaneous emission or injection in bands as collective states of electrons in the presence of lattice atoms. We find that two types of dissipative evolution can exist in the open graphene's electronic structure, depending on whether it is sustainable (e.g. supported by particle injections of some kind) or undergoing an irreversible decay. In the first case, the master equation is nonlinear with respect to the density operator; whereas in the second case it is linear but does not conserve the trace norm. We also consider the evolution of pure states, which allows us to separate quantum‐mechanical properties from statistical ones, and derive the effective Hamiltonian and the Schrödinger equation which turns out to be nonlinear. The evolution equations which result indicate that electronic transport in graphene becomes significantly complex in the presence of even weak dissipative effects.

The height to which water ascends in trees can reach values, which are above those expected from classical hydraulic formulae by at least one order of magnitude. This problem requires a fluid-mechanical approach to long-distance water transport and capillarity in tree xylems beyond the Navier–Stokes approximation. In the leading-order approximation, such transport is described as a logarithmically nonlinear wave-mechanical many-body model isomorphic to the multiphase fluid with Korteweg-type capillarity and the ideal equation of state. This model indicates a significant reduction in the effective dynamic viscosity and predicts the existence of nonlinear, dispersive, non-dissipative waves of the Korteweg-de Vries type. The wave-mechanical duality suggests that the water molecules in the capillary do form a collective microfluidic state previously conjectured in Dixon–Joly's cohesion theory. The approach reveals the versatile role of transpiration in trees: aside from the vacuum pumping of water from roots to canopy, it creates temperature conditions, which suppress cavitation and thus sustain higher tension values. It is also responsible for creating and maintaining variations of temperature and density, which facilitates control of the direction and magnitude of the effective wave-mechanical force acting upon the flow. By regulating the temperature and density of the sap circulating via their xylem and phloem tissues, plants can adjust this force with respect to the force of gravity, making the long-distance transport of water and dissolved nutrients more efficient.

We conjecture that during the time period preceding the inflationary epoch, the background matter was initially a condensate formed from a many-body system of indistinguishable particles whose states were in a quantum superposition. This resulted in the occurrence of a statistical ensemble of spacetimes, thus causing the probabilistic uncertainty in the spacetime geometry of the pre-inflationary multiverse. Then, at a certain moment in time, a measurement event occurred, which broke the linear superposition and reduced the primordial geometrical multiverse to a single state. This process can be described as a quantum Shannon information transfer, which induces logarithmic nonlinearity in the evolution equations of the background system. The latter, therefore, transformed into a logarithmic quantum liquid of a superfluid type and formed the physical vacuum. This measurement also generated the primary mass energy necessary for the Universe’s further evolution into the inflationary epoch, followed by the contemporary “dark energy” era.

In this paper, we review the various origins of the logarithmic Schrödinger equation: quantum informational and thermodynamic (based on quantum information entropy), statistical (pertinent to strongly correlated and interacting systems), hydrodynamic (ideal fluid), empirical (acoustics and scattering inside liquid helium and Bose–Einstein condensates of alkali atoms), gravitational (long range of induced gravity in superfluid vacuum theory and its compatibility with relativity), cosmological (accelerated expansion of the universe), and others. We argue that the statistical origin is the most fundamental one from an axiomatic point of view, because it is based on assumptions which are universally valid for a large class of materials. In fact, it can lay the foundations of a nonperturbative approach to strongly correlated and interacting systems. The empirical origin is instrumental in establishing the role of logarithmic nonlinearity in the standard theory of quantum Bose liquids and in placing experimental constraints upon its parameters. We also discuss the origins of the Lorentz-covariant version of the model, the logarithmic Klein–Gordon equation, which occurs in dilatation-covariant local relativistic scalar field models, non-Abelian gauge theories with radiative corrections in dimensional regularization and quantum astrophysics.

The laminar constant-velocity superflow of a physical vacuum modelled by logarithmic quantum Bose liquid is considered. We demonstrate that this three-dimensional non-relativistic quantum flow generates a four-dimensional relativistic quinton system, which comprises the dilaton and quintom (a combination of the quintessence and tachyonic phantom fields); all three fields are thus shown to be projections of the dynamical evolution of superfluid vacuum density and its fluctuations onto the measuring apparatus of a relativistic observer. The unified model describes the transition from the inflationary period in the early universe to the contemporary accelerating expansion of the universe, commonly referred to as the “dark energy” period. The quintessence and tachyonic scalar components of the derived model turn out to be non-minimally coupled, which is a hitherto unexplored generalization of cosmological phantom models.

We consider a general relativistic model of a self-interacting complex scalar field with logarithmic nonlinearity motivated by studies of laboratory superfluids and Bose-Einstein condensates. Spherically-symmetric gravitational equilibria are shown in this model, which do not have event horizons but which are regular, singularity-free and asymptotically flat. They can be thus interpreted as compact stars whose stability against gravitational collapse is enhanced not only by the Heisenberg uncertainty principle but also by the property of superfluidity itself, their ``darkness'' comes naturally as a result of suppressed dissipative excitations. Such objects do not obey any absolute upper mass limit of a Tolman-Oppenheimer-Volkoff type, while their relativisticity and effective compactness values are comparable to those of black holes. Their spatial density distribution drops abruptly (at the Gaussian-like rate), which can be mistaken in realistic astronomical observations for the presence of an exact material surface. We therefore present logarithmic superfluid stars as dark compact objects and black hole mimickers.

Within the framework of the quantum-statistical approach, utilizing both non-Hermitian Hamiltonian and Lindblad’s jump operators, one can derive various generalizations of the von Neumann equation for reduced density operators, also known as hybrid master equations. If one considers the evolution of pure states only, i.e., disregarding the coherence between states and spontaneous transitions from pure to mixed states, then one can resort to quantum-mechanical equations of the Schrödinger type. We derive them from the hybrid master equations and study their main properties, which indicate that our equations have a larger range of applicability compared to other generalized Schrödinger equations proposed hitherto. Among other features, they can describe not only systems which remain in the stationary eigenstates of the Hamiltonian as time passes, but also those which evolve from those eigenstates. As an example, we consider a simple but important model, a quantum harmonic oscillator driven by both Hamiltonian and non-Hamiltonian terms, and derive its classical limit, which turns out to be the damped harmonic oscillator. Using this model, we demonstrate that the effects of dissipative environments of different types can cancel each other, thus resulting in an effectively dissipation-free classical system. Another discussed phenomenon is whether a non-trivial quantum system can reduce to a classical system in free motion, i.e., without experiencing any classical Newtonian forces. This uncovers a large class of quantum-mechanical non-Hamiltonian systems whose dynamics are not determined by conventional mechanics’ potentials and forces, but rather come about through quantum statistical effects caused by the system’s environment.