Quantum Inequalities Research Articles

Some new bounds of Chebyshev and Grüss-type functionals on time scales

In this work, Korkine and Sonin's identities are defined on arbitrary time scales. These identities are utilized to establish the Chebyshev and Grüss-type inequalities on time scales. By applying these inequalities, we determine the bounds of the remainders in Montgomery identities that incorporate Taylor's formula on time scales. Moreover, we derive discrete and quantum inequalities based on these results.

Quantum Lp-spaces and inequalities of Hardy's type

Quantum Lp-spaces and inequalities of Hardy's type

Warping Spacetime: When Theoretical Physics Meets Science Fiction

Warp drives, though first envisioned in science fiction, have become a topic of scientific interest within theoretical physics. Miguel Alcubierre was the first one to propose a warp drive spacetime in 1994. His geometry is, however, considered unphysical: it violates all energy conditions and requires large amounts of exotic matter. Several modifications to his metric have been proposed which reduce these energy requirements, but it is still unlikely that warp drives will ever be technologically useful. Even so, they make for very interesting thought experiments, forcing physicists to face the limits of their current theories. Here we will discuss the Alcubierre drive, its energy condition violations, and its energy requirements as obtained from the quantum inequalities. The general definition of superluminal travel is briefly discussed. We also study the horizon problem and the possibility of closed timelike curves within this spacetime. Some recent progress in warp drive research is presented, followed by a conclusion which expands on the physical plausibility and research interest in warp drives.

On New Estimates of q-Hermite–Hadamard Inequalities with Applications in Quantum Calculus

In this paper, we first establish two quantum integral (q-integral) identities with the help of derivatives and integrals of the quantum types. Then, we prove some new q-midpoint and q-trapezoidal estimates for the newly established q-Hermite-Hadamard inequality (involving left and right integrals proved by Bermudo et al.) under q-differentiable convex functions. Finally, we provide some examples to illustrate the validity of newly obtained quantum inequalities.

On Some Generalizations of Reverse Dynamic Hardy Type Inequalities on Time Scales

In the present paper, we prove some new reverse type dynamic inequalities on T. Our main inequalities are proved by using the chain rule and Fubini’s theorem on time scales T. Our results extend some existing results in the literature. As special cases, we obtain some new discrete inequalities, quantum inequalities and integral inequalities.

Q1q2-Ostrowski-Type Integral Inequalities Involving Property of Generalized Higher-Order Strongly n-Polynomial Preinvexity

Quantum calculus has numerous applications in mathematics. This novel class of functions may be used to produce a variety of conclusions in convex analysis, special functions, quantum mechanics, related optimization theory, and mathematical inequalities. It can drive additional research in a variety of pure and applied fields. This article’s main objective is to introduce and study a new class of preinvex functions, which is called higher-order generalized strongly n-polynomial preinvex function. We derive a new q1q2-integral identity for mixed partial q1q2-differentiable functions. Because of the nature of generalized convexity theory, there is a strong link between preinvexity and symmetry. Utilizing this as an auxiliary result, we derive some estimates of upper bound for functions whose mixed partial q1q2-differentiable functions are higher-order generalized strongly n-polynomial preinvex functions on co-ordinates. Our results are the generalizations of the results in earlier papers. Quantum inequalities of this type and the techniques used to solve them have applications in a wide range of fields where symmetry is important.

Some Parameterized Quantum Simpson’s and Quantum Newton’s Integral Inequalities via Quantum Differentiable Convex Mappings

In this work, two generalized quantum integral identities are proved by using some parameters. By utilizing these equalities, we present several parameterized quantum inequalities for convex mappings. These quantum inequalities generalize many of the important inequalities that exist in the literature, such as quantum trapezoid inequalities, quantum Simpson’s inequalities, and quantum Newton’s inequalities. We also give some new midpoint-type inequalities as special cases. The results in this work naturally generalize the results for the Riemann integral.

Some New Simpson’s and Newton’s Formulas Type Inequalities for Convex Functions in Quantum Calculus

In this paper, using the notions of qκ2-quantum integral and qκ2-quantum derivative, we present some new identities that enable us to obtain new quantum Simpson’s and quantum Newton’s type inequalities for quantum differentiable convex functions. This paper, in particular, generalizes and expands previous findings in the field of quantum and classical integral inequalities obtained by various authors.

Introducing physical warp drives

The Alcubierre warp drive is an exotic solution in general relativity. It allows for superluminal travel at the cost of enormous amounts of matter with negative mass density. For this reason, the Alcubierre warp drive has been widely considered unphysical. In this study, we develop a model of a general warp drive spacetime in classical relativity that encloses all existing warp drive definitions and allows for new metrics without the most serious issues present in the Alcubierre solution. We present the first general model for subliminal positive-energy, spherically symmetric warp drives; construct superluminal warp-drive solutions which satisfy quantum inequalities; provide optimizations for the Alcubierre metric that decrease the negative energy requirements by two orders of magnitude; and introduce a warp drive spacetime in which space capacity and the rate of time can be chosen in a controlled manner. Conceptually, we demonstrate that any warp drive, including the Alcubierre drive, is a shell of regular or exotic material moving inertially with a certain velocity. Therefore, any warp drive requires propulsion. We show that a class of subluminal, spherically symmetric warp drive spacetimes, at least in principle, can be constructed based on the physical principles known to humanity today

Quantum Inequalities of Hermite–Hadamard Type for r -Convex Functions

In this present study, we first establish Hermite–Hadamard type inequalities for r -convex functions via q κ 2 -definite integrals. Then, we prove some quantum inequalities of Hermite–Hadamard type for product of two r -convex functions. Finally, by using these established inequalities and the results given by (Brahim et al. 2015), we prove several quantum Hermite–Hadamard type inequalities for coordinated r -convex functions and for the product of two coordinated r -convex functions.

Quantum Inequalities and Particle Creation in the Presence of an External, Time-Dependent Mamaev-Trunov Potential

In 2011, Mr. Dan Solomon proposed a model of a quantized scalar field interacting with a time-dependent Mamaev-Trunov potential in two-dimensional Minkowski spacetime. This model is governed by the Klein-Gordon wave equation with a time-dependent potential. Mr. Solomon claims that this model violates both the classical energy conditions of special relativity and the quantum energy conditions of quantum field theory in curved spacetime. Every classical energy condition can be violated, and their natural replacements are known as quantum inequalities. Mr. Solomon attempted to prove violations of the spatial and temporal quantum inequalities, and he correctly assumed that the negative energy splits into two fluxes at the Cauchy surface, where the potential is turned off. Unfortunately, Solomon neglects the contribution to the energy density due to particle creation when the potential is turned off at time t = 0. In this project, we calculate the contribution to the stress energy tensor due to particle creation. We show that while the classical energy conditions are violated, the quantum energy inequalities hold, contrary to Mr. Solomon’s statements.

Erratum: Quantum inequalities in two dimensional curved spacetimes [Phys. Rev. D 66 , 104007 (2002)

Erratum: Quantum inequalities in two dimensional curved spacetimes [Phys. Rev. D <b>66</b> , 104007 (2002)

Uniformity of dynamic inequalities constituted on time Scales

In this article, we present extensions of some well-known inequalities such as Young's inequality and Qi's inequality on fractional calculus of time scales. To find generalizations of such types of dynamic inequalities, we apply the time scale Riemann-Liouville type fractional integrals. We investigate dynamic inequalities on delta calculus and their symmetric nabla results. The theory of time scales is utilized to combine versions in one comprehensive form. The calculus of time scales unifies and extends some continuous forms and their discrete and quantum inequalities. By applying the calculus of time scales, results can be generated in more general form. This hybrid theory is also extensively practiced on dynamic inequalities.

Quasiprobability Distribution for Heat Fluctuations in the Quantum Regime

The standard approach to deriving fluctuation theorems fails to capture the effect of quantum correlation and coherence in the initial state of the system. Here we overcome this difficulty and derive heat exchange fluctuation theorem in the full quantum regime by showing that the energy exchange between two locally thermal states in the presence of initial quantum correlations is faithfully captured by a quasiprobability distribution. Its negativities, being associated with proofs of contextuality, are proxys of non-classicality. We discuss the thermodynamic interpretation of negative probabilities, and provide heat flow inequalities that can only be violated in their presence. Remarkably, testing these fully quantum inequalities, at arbitrary dimension, is not more difficult than testing traditional fluctuation theorems. We test these results on data collected in a recent experiment studying the heat transfer between two qubits, and give examples for the capability of witnessing negative probabilities at higher dimensions.

New Estimates of q1q2-Ostrowski-Type Inequalities within a Class of n-Polynomial Prevexity of Functions

In this article, we develop a novel framework to study for a new class of preinvex functions depending on arbitrary nonnegative function, which is called n-polynomial preinvex functions. We use the n-polynomial preinvex functions to develop q1q2-analogues of the Ostrowski-type integral inequalities on coordinates. Different features and properties of excitement for quantum calculus have been examined through a systematic way. We are discussing about the suggestions and different results of the quantum inequalities of the Ostrowski-type by inferring a new identity for q1q2-differentiable function. However, the problem has been proven to utilize the obtained identity, we give q1q2-analogues of the Ostrowski-type integrals inequalities which are connected with the n-polynomial preinvex functions on coordinates. Our results are the generalizations of the results in earlier papers.

A Class of Quantum Briot–Bouquet Differential Equations with Complex Coefficients

Quantum inequalities (QI) are local restraints on the magnitude and range of formulas. Quantum inequalities have been established to have a different range of applications. In this paper, we aim to introduce a study of QI in a complex domain. The idea basically, comes from employing the notion of subordination. We shall formulate a new q-differential operator (generalized of Dunkl operator of the first type) and employ it to define the classes of QI. Moreover, we employ the q-Dunkl operator to extend the class of Briot–Bouquet differential equations. We investigate the upper solution and exam the oscillation solution under some analytic functions.

New parameterized quantum integral inequalities via \u03b7-quasiconvexity

We establish new quantum Hermite–Hadamard and midpoint types inequalities via a parameter mu in [0,1] for a function F whose |{}_{alpha }D_{q}F|^{u} is η-quasiconvex on [alpha ,beta ] with ugeq 1. Results obtained in this paper generalize, sharpen, and extend some results in the literature. For example, see (Noor et al. in Appl. Math. Comput. 251:675–679, 2015; Alp et al. in J. King Saud Univ., Sci. 30:193–203, 2018) and (Kunt et al. in Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 112:969–992, 2018). By choosing different values of μ, loads of novel estimates can be deduced. We also present some illustrative examples to show how some consequences of our results may be applied to derive more quantum inequalities.

Testing a Quantum Inequality with a Meta-analysis of Data for Squeezed Light

In quantum field theory, coherent states can be created that have negative energy density, meaning it is below that of empty space, the free quantum vacuum. If no restrictions existed regarding the concentration and permanence of negative energy regions, it might, for example, be possible to produce exotic phenomena such as Lorentzian transversable wormholes, warp drives, time machines, violations of the second law of thermodynamics, and naked singularities. Quantum Inequalities (QIs) have been proposed that restrict the size and duration of the regions of negative quantum vacuum energy that can be accessed by observers. However, QIs generally are derived for situations in cosmology and are very difficult to test. Direct measurement of vacuum energy is difficult and to date no QI has been tested experimentally. We test a proposed QI for squeezed light by a meta-analysis of published data obtained from experiments with optical parametric amplifiers (OPA) and balanced homodyne detection. Over that last three decades, researchers in quantum optics have been trying to maximize the squeezing of light and the quantum vacuum and have succeeded in reducing the variance in the quantum vacuum fluctuations to -15 dB. To apply the QI, a time sampling function is required. In our meta-analysis different time sampling functions for the QI were examined, but in all physically reasonable cases the QI is violated by much or all of the measured data. This brings into question the basis for QI. Possible explanations are given for this surprising result.

CHSH-Type Inequality Involving Commuting Continuous Variables

The correlation of projections of the momentum operators of two particles is used to derive a quantum inequality for continuous variables, which must be satisfied by any bipartite system in a pure state. This inequality resembles a Clauser–Horne–Shimony–Holt (CHSH)-type inequality except for additional terms related to the imaginary component of the weak value of the momentum, which normally remains concealed in the usual quantum description but turns out to be of relevance for entangled states. Our results shed new light on the link between noncommutativity, entanglement and nonlocality of the quantum description.

Entanglement entropy: Holography and renormalization group

Entanglement entropy plays a variety of roles in quantum field theory, including the connections between quantum states and gravitation through the holographic principle. This article provides a review of entanglement entropy from a mixed viewpoint of field theory and holography. A set of basic methods for the computation is developed and illustrated with simple examples such as free theories and conformal field theories. The structures of the ultraviolet divergences and the universal parts are determined and compared with the holographic descriptions of entanglement entropy. The utility of quantum inequalities of entanglement are discussed and shown to derive the C-theorem that constrains renormalization group flows of quantum field theories in diverse dimensions.