The Minimal Length Uncertainty and the Nonextensive Thermodynamics

In this paper, we study the thermodynamics of quantum harmonic oscillator in the Tsallis framework and in the presence of a minimal length uncertainty. The existence of the minimal length is motivated by various theories such as string theory, loop quantum gravity, and black-hole physics. We analytically obtain the partition function, probability function, internal energy, and the specific heat capacity of the vibrational quantum system for \(1<q<\frac {3}{2}\) and compare the results with those of Tsallis and Boltzmann-Gibbs statistics without the minimal length scale.     

On the thermodynamics of relativistic ideal gases in the presence of a maximal length

We investigate the thermostatistics of relativistic ideal gases within the recently proposed deformed Heisenberg algebra (Perivolaropoulos, 2017), which includes a maximal length. By using the semiclassical method, the generalized canonical partition function is established and the modified thermodynamic functions are then calculated. It is explicitly shown that the maximal length, unlike the minimal length, modifies the equation of state of relativistic ideal gases. Furthermore, the ultrarelativistic and nonrelativistic regimes are discussed. In contrast to the minimal length corrections, the maximal length corrections do not depend on the considered regime. Such a result confirms that the effects of the maximal length and those of the minimal length are fundamentally different.     

Quantum theory of the generalised uncertainty principle

We extend significantly previous works on the Hilbert space representations of the generalized uncertainty principle (GUP) in 3 + 1 dimensions of the form \([X_i,P_j] = i F_{ij}\) where \(F_{ij} = f({\mathbf {P}}^2) \delta _{ij} + g({\mathbf {P}}^2) P_i P_j\) for any functions f. However, we restrict our study to the case of commuting X’s. We focus in particular on the symmetries of the theory, and the minimal length that emerge in some cases. We first show that, at the algebraic level, there exists an unambiguous mapping between the GUP with a deformed quantum algebra and a quadratic Hamiltonian into a standard, Heisenberg algebra of operators and an aquadratic Hamiltonian, provided the boost sector of the symmetries is modified accordingly. The theory can also be mapped to a completely standard Quantum Mechanics with standard symmetries, but with momentum dependent position operators. Next, we investigate the Hilbert space representations of these algebraically equivalent models, and focus specifically on whether they exhibit a minimal length. We carry the functional analysis of the various operators involved, and show that the appearance of a minimal length critically depends on the relationship between the generators of translations and the physical momenta. In particular, because this relationship is preserved by the algebraic mapping presented in this paper, when a minimal length is present in the standard GUP, it is also present in the corresponding Aquadratic Hamiltonian formulation, despite the perfectly standard algebra of this model. In general, a minimal length requires bounded generators of translations, i.e. a specific kind of quantization of space, and this depends on the precise shape of the function f defined previously. This result provides an elegant and unambiguous classification of which universal quantum gravity corrections lead to the emergence of a minimal length.     

Corrections to the Hawking Tunneling Radiation from MDR

We investigate some aspects of black hole (BH) thermodynamics in the framework of a modified dispersion relation. We calculate a minimal length and a maximal momentum to find a relation between spacetime dimensions and the presence of logarithmic prefactor in the black hole entropy relation. We show that the logarithmic prefactor appears not only in an even number of dimensions but also in an odd number of dimensions. In addition, the sign of the logarithmic factor is different for positive values of α in all dimensions. Using the corrected entropy, the black hole radiation probability is calculated in the tunneling formalism, which is corrected up to the same order of the Planck length and shows a more probable quantum tunneling.     

Planck-scale corrections to Friedmann equation

Recently, Verlinde proposed that gravity is an emergent phenomenon which originates from an entropic force. In this work, we extend Verlinde’s proposal to accommodate generalized uncertainty principles (GUP), which are suggested by some approaches to quantum gravity such as string theory, black hole physics and doubly special relativity (DSR). Using Verlinde’s proposal and two known models of GUPs, we obtain modifications to Newton’s law of gravitation as well as the Friedmann equation. Our modification to the Friedmann equation includes higher powers of the Hubble parameter which is used to obtain a corresponding Raychaudhuri equation. Solving this equation, we obtain a leading Planck-scale correction to Friedmann-Robertson-Walker (FRW) solutions for the p = ωp equation of state.     

Nonperturbative effects of the minimal length uncertainty on the relativistic quantum mechanics

We study the nonperturbative effects of the minimal length on the energy spectrum of a relativistic particle in the context of the generalized uncertainty principle (GUP). This form of GUP is consistent with various candidates of quantum gravity such as string theory, loop quantum gravity, and black-hole physics and predicts a minimum measurable length proportional to the Planck length. Using a recently proposed formally self-adjoint representation, we solve the generalized Dirac and Klein–Gordon equations in various situations and find the corresponding exact energy eigenvalues and eigenfunctions. We show that for the Dirac particle in a box, the number of the solutions renders to be finite as a manifestation of both the minimal length and the theory of relativity. For the case of the Dirac oscillator and the wave equations with scalar and vector linear potentials, we indicate that the solutions can be obtained in a more simpler manner through the self-adjoint representation. It is also shown that, in the ultrahigh frequency regime, the partition function and the thermodynamical variables of the Dirac oscillator can be expressed in a closed analytical form. The Lorentz violating nature of the GUP-corrected relativistic wave equations is discussed finally.     

Gravity in non-commutative geometry

We study general relativity in the framework of non-commutative differential geometry. As a prerequisite we develop the basic notions of non-commutative Riemannian geometry, including analogues of Riemannian metric, curvature and scalar curvature. This enables us to introduce a generalized Einstein-Hilbert action for non-commutative Riemannian spaces. As an example we study a space-time which is the product of a four dimensional manifold by a two-point space, using the tools of non-commutative Riemannian geometry, and derive its generalized Einstein-Hilbert action. In the simplest situation, where the Riemannian metric is taken to be the same on the two copies of the manifold, one obtains a model of a scalar field coupled to Einstein gravity. This field is geometrically interpreted as describing the distance between the two points in the internal space.Dedicated to H. ArakiSupported in part by the Swiss National Foundation (SNF)     

Rotating ground states of trapped atoms in a Bose-Einstein condensate with arbitrary two-body interactions

In a k-dimensional system of weakly interacting Bose atoms trapped by a spherically symmetric and harmonic external potential, an exact expression is obtained for the rotating ground states at a fixed angular momentum. The result is valid for arbitrary interactions obeying minimal physical requirements. Depending on the sign of a modified scattering length, it reduces to either a collective rotation or a condensed vortex state, with no alternative. The ground state can undergo a kind of quantum phase transition when the shape of the interaction potential is smoothly varied.     

On Clifford Space Relativity,Black Hole Entropy,Rainbow Metrics,Generalized Dispersion and Uncertainty Relations

An analysis of some of the applications of Clifford space relativity to the physics behind the modified black hole entropy-area relations, rainbow metrics, generalized dispersion and minimal length stringy uncertainty relations is presented.     

Symmetry and symmetry breaking in generalized parastatistics

An analysis is made of the characteristics of internal symmetry and symmetry breaking in a quantum field theory with generalized parastatistics, defined by either double commutation relations or single commutation relations. The connection between the two statistics is clarified. We develop a formalism in which statistics is viewed as a dynamical or phase variable of quantum systems. It is shown that the types of Higgs phases possible depend upon statistics. Relationships between physical amplitudes implied by internal symmetry with normal statistics are violated in the case of generalized parastatistics.     

Generalized model of polypeptide chain with limitations on the length of helical fractions

A model is proposed described by a Hamiltonian consisting of two terms corresponding to Hamiltonians of the generalized model of polypeptide chain (GMPC) with different scales of interaction. We consider a case where the long-scale interaction corresponds to attraction, while the short-scale interaction corresponds to repulsion; this case models the melting of the system in compact packing conditions (globularized polypeptide or DNA in capsid). In the framework of the transfer-matrix approach we calculate the correlation length and the helicity degree. It is shown that the temperature dependence of the correlation length is determined by the complex second eigenvalue of the transfer-matrix. In this regime the system displays a new type of correlations when the melting interval cannot be a measure of the system cooperativity.     

Generalized synchronization of two unidirectionally coupled discrete stochastic dynamical systems

The existence of two kinds of generalized synchronization manifold in two unidirectionally coupled discrete stochastic dynamical systems is studied in this paper. When the drive system is chaotic and the modified response system collapses to an asymptotically stable equilibrium or asymptotically stable periodic orbit, under certain conditions, the existence of the generalized synchronization can be converted to the problem of a Lipschitz contractive fixed point or Schauder fixed point. Moreover, the exponential attractive property of generalized synchronization manifold is strictly proved. In addition, numerical simulations demonstrate the correctness of the present theory. The physical background and meaning of the results obtained in this paper are also discussed.%vspace1mm     

Some aspects of gravitational quantum mechanics

String theory, quantum geometry, loop quantum gravity and black hole physics all indicate the existence of a minimal observable length on the order of Planck length. This feature leads to a modification of Heisenberg uncertainty principle. Such a modified Heisenberg uncertainty principle is referred as gravitational uncertainty principle(GUP) in literatures. This proposal has some novel implications on various domains of theoretical physics. Here, we study some consequences of GUP in the spirit of Quantum mechanics. We consider two problem: a particle in an one-dimensional box and free particle wave function. In each case we will solve corresponding perturbational equations and compare the results with ordinary solutions.     

广义不确定关系与黑洞附近的热力学量

利用广义不确定关系修正的态密度计算了一般球对称静态黑洞附近无质量共形不变标量场、中微子场、电磁场、无质量Rarita-Schwinger场和引力场的热力学量.结果表明，黑洞附近的热力学量不仅依赖于黑洞的特征，还依赖于粒子的自旋和最小距离的尺度. 关键词： 广义不确定关系 一般球对称静态黑洞 热力学量     

Generalized Observables,Bell’s Inequalities and Mixtures in the ESR Model for QM

The extended semantic realism (ESR) model proposes a new theoretical perspective which embodies the mathematical formalism of standard (Hilbert space) quantum mechanics (QM) into a noncontextual framework, reinterpreting quantum probabilities as conditional instead of absolute. We provide in this review an overall view on the present status of our research on this topic. We attain in a new, shortened way a mathematical representation of the generalized observables introduced by the ESR model and a generalization of the projection postulate of elementary QM. Basing on these results we prove that the Bell-Clauser-Horne-Shimony-Holt (BCHSH) inequality, a modified BCHSH inequality and quantum predictions hold together in the ESR model because they refer to different parts of the picture of the physical world supplied by the model. Then we show that a new mathematical representation of mixtures must be introduced in the ESR model which does not coincide with the standard representation in QM and avoids some deep problems that arise from the representation of mixtures provided by QM. Finally we get a nontrivial generalization of the Lüders postulate, which is justified in a special case by introducing a reasonable physical assumption on the evolution of the compound system made up of the measured system and the measuring apparatus.     

Mechanisms behind the generalized synchronization conditions

A universal mechanism underlying generalized synchronization conditions in unidirectionally coupled stochastic oscillators is considered. The consideration is carried out in the framework of a modified system with additional dissipation. The approach developed is illustrated with model examples. The conclusion is reached that two types of the behavior of nonlinear dynamic systems known as generalized synchronization and noise-induced synchronization, which are viewed as different phenomena, actually represent a unique type of the synchronous behavior of stochastic oscillators and are caused by the same mechanism.     

Semi-Cats: Nonclassicality and Generation

We define generalized cat states as linear superpositions of the semi-coherent states. They can be considered as superpositions of two distinguishable components of the Schrödinger cat states. We study the statistical properties of the introduced states in detail. The physical properties of these states, like the sub-Poissonian statistics and normal-order as well as amplitude-squared squeezing effect, are discussed analytically. Moreover, we find some interesting properties of their optical tomogram derived in terms of the exponential function. Finally, we suggest a new theoretical framework for preparing generalized cat states.     

On External and Internal Properties of Extended Elementary Objects

The physical interpretation of induced representation intertwining as a process of materialization or localization is extrapolated to mappings (which are not intertwinings) between configuration and momentum representations. Propagation of extended particles composed of an external and an internal mode is a combination of two generalized materializations and two generalized localizations. Our aim is to submit, in the spinless case, the idea that mappings from external representations to internal ones are possible alternatives, probability amplitudes of which must be summed up in the whole propagator. However, it turns out that they lead to different final states and then correspond to 64 different propagators and not to alternatives of the same propagation. In addition to being transitions from one location to another, these propagations contain processes of interconversion between internal and external modes which are responsible for creation and annihilation of the extended particle. The physical interpretations are striking, while the mathematical stricture is poor. Inconsistencies of the model are outlined.     

Three-dimensional radiative transfer in an isotropically scattering, semi-infinite medium: generalized reflection function

The focus of this study is the generalized reflection function of multidimensional radiative transfer. The physical situation considered is spatially varying, collimated radiation incident on the upper boundary of an isotropically scattering, semi-infinite medium. An integral transform is used to reduce the three-dimensional transport equation to a one-dimensional form, and a modified Ambarzumian's method is applied to formulate a nonlinear integral equation for the generalized reflection function. The resulting equation is said to be in double-integral form because the integration is over both angular variables. Computational issues associated with this generalized reflection function formulation are investigated. The source function and reflection function formulations are compared, and the relative merits of the two approaches are discussed.