Search Results (18)

We construct Wick products by creation and annihilation operators on Guichardet–Fock spaces $\mathcal{F}$ and obtain commutation relations and s—adapted characters about the Wick products. Meanwhile, we introduce a concept of singular Bogoliubov transformation on Guichardet–Fock space, and prove that the Wick product under the singular Bogoliubov transformation still satisfies the commutative relationships. Full article

The para-positronium system $\left({}^{1}S_{0}Ps\right)$ is described by means of specially constructed coherent states (CSs) in the Klauder–Perelomov sense. It is analyzed from the physical point of view and from the geometry underlying the relevant symmetry group establishing the dynamics of the processes. In this new theoretical context, the possibility of a gamma-ray laser emission is investigated within a QFT context, showing explicitly that, in addition to the oscillator solution based only on a Bogoliubov approximation for the condensate, there is a second phase or “squeezed” stage by which physical features beyond the classical ones appear. Explicitly, while the generated photons are in the active medium (e.g., Ps-BEC), the evolution is described by a Heisenberg–Weyl coherent state with displacement operators dependent on the interaction time, which is related to the condensate shape. After the interaction time has elapsed, we explicitly demonstrate that the displacement operator of the ${}^{1}S_{0}Ps$ is transformed into a squeezed operator of the photonic fields modulated by the matrix element of the Positronium decay ${\mathcal{M}}_{{}^{1}S_{0}Ps}\to 2\gamma$. We also show that this squeezed operator (belonging to the Metaplectic group) generates a non-classical radiation state spanning only even (s = 1/4) levels in the number of photons. The implications in astrophysical systems of interest, considering gamma-ray coherent emission and the possibility of an ${}^{1}S_{0}Ps-BEC$ in the context of pulsars, blazars, and quasars, are briefly discussed. Full article

We discuss local magnetic field quenches using perturbative methods of finite time path field theory (FTPFT) in the following spin chains: Ising and XY in a transverse magnetic field. Their common characteristics are: (i) they are integrable via mapping to a second quantized noninteracting fermion problem; and (ii) when the ground state is nondegenerate (true for finite chains except in special cases), it can be represented as a vacuum of Bogoliubov fermions. By switching on a local magnetic field perturbation at finite time, the problem becomes nonintegrable and must be approached via numeric or perturbative methods. Using the formalism of FTPFT based on Wigner transforms (WTs) of projected functions, we show how to: (i) calculate the basic “bubble” diagram in the Loschmidt echo (LE) of a quenched chain to any order in the perturbation; and (ii) resum the generalized Schwinger–Dyson equation for the fermion two-point retarded functions in the “bubble” diagram, hence achieving the resummation of perturbative expansion of LE for a wide range of perturbation strengths under certain analyticity assumptions. Limitations of the assumptions and possible generalizations beyond it and also for other spin chains are further discussed. Full article

Quantum field theory (QFT) in Rindler spacetime is a gateway to understanding unitarity and information loss paradoxes in curved spacetime. Rindler coordinates map Minkowski spacetime onto regions with horizons, effectively dividing accelerated observers into causally disconnected sectors. Employing standard quantum field theory techniques and Bogoliubov transformations between Minkowski and Rindler coordinates yields entanglement between states across these causally separated regions of spacetime. This results in a breakdown of unitarity, implying that information regarding the entangled partner may be irretrievably lost beyond the Rindler horizon. As a consequence, one has a situation of pure states evolving into mixed states. In this paper, we introduce a novel framework for comprehending this phenomenon using a recently proposed formulation of direct-sum quantum field theory (DQFT), which is grounded in superselection rules formulated by the parity and time reversal ($\mathcal{P}\mathcal{T}$) symmetry of Minkowski spacetime. In the context of DQFT applied to Rindler spacetime, we demonstrate that each Rindler observer can, in principle, access pure states within the horizon, thereby restoring unitarity. However, our analysis also reveals the emergence of a thermal spectrum of Unruh radiation. This prompts a reevaluation of entanglement in Rindler spacetime, where we propose a novel perspective on how Rindler observers may reconstruct complementary information beyond the horizon. Furthermore, we revisit the implications of the Reeh-Schlieder theorem within the framework of DQFT. Lastly, we underscore how our findings contribute to ongoing efforts aimed at elucidating the role of unitarity in quantum field theory within the context of de Sitter and black hole spacetimes. Full article

We present a theoretical study on the energy dispersion of an ultrathin film of periodically-aligned single-walled carbon nanotubes (SWCNTs) with the help of the Bogoliubov–Valatin transformation. The Hamiltonian of the film was derived using the many-particle green function technique in the Matsubara frequency formalism. The periodic array of SWCNTs was embedded in a dielectric with comparatively higher permittivity than the substrate and the superstrate such that the SWCNT film became independent with the axis of quantization but keeps the thickness as the variable parameter, making the film neither two-dimensional nor three-dimensional, but transdimensional. It was revealed that the energy dispersion of the SWCNT film is thickness dependent. Full article

We prove the consistency of the different approaches for deriving the black hole radiation for the spherically symmetric case inside the theory of Massive Gravity. By comparing the results obtained by using the Bogoliubov transformations with those obtained by using the Path Integral formulation, we find that in both cases, the presence of the extra-degrees of freedom creates the effect of extra-particles creation due to the distortions on the definitions of time defined by the different observers at large scales. This, however, does not mean extra-particle creation at the horizon level. Instead, the apparent additional particles perceived at large scales emerge from how distant observers define their time coordinate, which is distorted due to the existence of extra-degrees of freedom. Full article

In this investigation, two different models for two coupled asymmetrical oscillators, known as, coupled forced damped Duffing oscillators (FDDOs) are reported. The first model of coupled FDDOs consists of a nonlinear forced damped Duffing oscillator (FDDO) with a linear oscillator, while the second model is composed of two nonlinear FDDOs. The Krylov–Bogoliubov–Mitropolsky (KBM) method, is carried out for analyzing the coupled FDDOs for any model. To do that, the coupled FDDOs are reduced to a decoupled system of two individual FDDOs using a suitable linear transformation. After that, the KBM method is implemented to find some approximations for both unforced and forced damped Duffing oscillators (DDOs). Furthermore, the KBM analytical approximations are compared with the fourth-order Runge–Kutta (RK4) numerical approximations to check the accuracy of all obtained approximations. Moreover, the RK4 numerical approximations to both coupling and decoupling systems of FDDOs are compared with each other. Full article

In the actual panorama of machine learning (ML) algorithms, the issue of the real-time information extraction/classification/manipulation/analysis of data streams (DS) is acquiring an ever-growing relevance. They arrive generally at high speed and always require an unsupervised real-time analysis for individuating long-range and higher order correlations among data that are continuously changing over time (phase transitions). This emphasizes the infinitary character of the issue, i.e., the continuous change of the signifying number of degrees of freedom characterizing the statistical representation function, challenging the classical ML algorithms, both in their classical and quantum versions, as far as all are based on the (stochastic) search for the global minimum of some cost/energy function. The physical analogue must be studied in the realm of quantum field theory (QFT) for dissipative systems as biological and neural systems, which are able to map between different phases of quantum fields, using the formalism of the Bogoliubov transform (BT). By applying the BT in a reversed way, on the system-thermal bath energetically balanced states, it is possible to define the powerful computational tool of the “doubling of the degrees of freedom” (DDF), making the choice of the signifying finite number of the degrees of freedom dynamic and then automatic, so to suggest a different class of unsupervised ML algorithms for solving the DS issue. Full article

We study the quantum radiation of particle production by vacuum from an ultra-relativistic moving mirror (dynamical Casimir effect) solution that allows (possibly for the first time) analytically calculable time evolution of particle creation and an Airy particle spectral distribution. The reality of the beta Bogoliubov coefficients is responsible for the simplicity, and the mirror is asymptotically inertial at the speed of light, with finite energy production. We also discuss general relations regarding negative energy flux, the transformation to the 1-D Schrödinger equation, and the incompleteness of entanglement entropy. Full article

We review the canonical transformation in quantum physics known as the Bogoliubov transformation and present its application to the general theory of quantum field mixing and oscillations with an arbitrary number of mixed particles with either boson or fermion statistics. The mixing relations for quantum states are derived directly from the definition of mixing for quantum fields and the unitary inequivalence of the Fock space of energy and flavor eigenstates is shown by a straightforward algebraic method. The time dynamics of the interacting fields is then explicitly solved and the flavor oscillation formulas are derived in a unified general formulation with emphasis on antiparticle content and effect introduced by nontrivial flavor vacuum. Full article

Here, classical and quantum field theory of dipolar, axisymmetric quadrupolar and octupolar Bose gases is considered within a general approach. Dipole, axisymmetric quadrupole and octupole interaction potentials in the momentum representation are calculated. These results clearly demonstrate attraction and repulsion areas in corresponding gases. Then the Gross–Pitaevskii (GP) equation, which plays a key role in the present paper, is derived from the corresponding functional. The zoology of the form factors appearing in the GP equation is studied in details. The proper classes for the description of spatially non-uniform condensates form factors are chosen. In the Thomas–Fermi approximation a general solution of the GP equation with a quasilocal form factor is obtained. This solution has an interesting form in terms of a double rapidly converging series that universally includes all the interactions considered. Plots of condensate density functions for the exponential-trigonometric form factor are given. For the sake of completeness, in this paper we consider the GP equation with an optical lattice potential in the limit of small condensate densities. This limit does not distinguish between dipolar, quadrupolar and octupolar gases. An important analysis of the condensate stability, in other words the study of condensate excitations, is also performed in this paper. In the Gaussian approximation (from the Gross–Pitaevskii functional), a functional describing the perturbations of the condensate is derived in detail. This problem is an analog of the Bogolubov transformation used in the study of quantum Bose gases in operator formalism. For a probe wave function in the form of a plane wave, a spectrum of (Bogoliubov) excitations was obtained, from which an equation describing the threshold momentum for the emergence of instability was derived. An important result of this paper is the dependence of the threshold on the momentum of a stationary condensate. For completeness of the presentation, the approximating expression in the form of a rapidly converging series is obtained for the corresponding dependence, and plots of the corresponding series for the exponential-trigonometric form factor are given. Finally, in the conclusion a quantum hydrodynamic theory for dipolar, axisymmetric quadrupolar and octupolar gases is briefly presented, giving a clue to the experimental determination of the form factors. Full article

We analyze the loss of information and the loss of learning capability inside an arrangement of neural networks. Our method is based on the formulation of the Bogoliubov transformations in order to connect the information between different points of the arrangement. Similar methods translated to the physics of black-holes, reproduce the Hawking radiation effect. From this perspective we can conclude that the black-holes are objects reproducing naturally the bottleneck effect, which is fundamental in neural networks in order to perceive the useful information, eliminating in this way the noise. Full article

This paper addresses the fate of extended space-time symmetries, in particular conformal symmetry and supersymmetry, in two-dimensional Rindler space-time appropriate to a uniformly accelerated non-inertial frame in flat 1+1-dimensional space-time. Generically, in addition to a conformal co-ordinate transformation, the transformation of fields from Minkowski to Rindler space is accompanied by local conformal and Lorentz transformations of the components, which also affect the Bogoliubov transformations between the associated Fock spaces. I construct these transformations for massless scalars and spinors, as well as for the ghost and super-ghost fields necessary in theories with local conformal and supersymmetries, as arising from coupling to two-dimensional (2-D) gravity or supergravity. Cancellation of the anomalies in Minkowski and in Rindler space requires theories with the well-known critical spectrum of particles that arise in string theory in the limit of infinite strings, and it is relevant for the equivalence of Minkowski and Rindler frame theories. Full article

For a scalar field in an exponentially expanding universe, constituent modes of elementary excitation become unstable consecutively at shorter wavelength. After canonical quantization, a Bogoliubov transformation reduces the minimally coupled scalar field to independent 1D modes of two inequivalent types, leading eventually to a cosmological partitioning of energy. Due to accelerated expansion of the coupled space-time, each underlying mode transits from an attractive oscillator with discrete energy spectrum to a repulsive unit with continuous unbounded energy spectrum. The underlying non-autonomous Schrödinger equation is solved here as the wave function evolves through the attraction-repulsion transition and ceases to oscillate. Full article

We theoretically study the dynamical Casimir effect (DCE), i.e., parametric amplification of a quantum vacuum, in an optomechanical cavity interacting with a photonic crystal, which is considered to be an ideal system to study the microscopic dissipation effect on the DCE. Starting from a total Hamiltonian including the photonic band system as well as the optomechanical cavity, we have derived an effective Floquet–Liouvillian by applying the Floquet method and Brillouin–Wigner–Feshbach projection method. The microscopic dissipation effect is rigorously taken into account in terms of the energy-dependent self-energy. The obtained effective Floquet–Liouvillian exhibits the two competing instabilities, i.e., parametric and resonance instabilities, which determine the stationary mode as a result of the balance between them in the dissipative DCE. Solving the complex eigenvalue problem of the Floquet–Liouvillian, we have determined the stationary mode with vanishing values of the imaginary parts of the eigenvalues. We find a new non-local multimode DCE represented by a multimode Bogoliubov transformation of the cavity mode and the photon band. We show the practical advantage for the observation of DCE in that we can largely reduce the pump frequency when the cavity system is embedded in a narrow band photonic crystal with a bandgap. Full article