Generalized Uncertainty Relation Research Articles

Spin-charged point particle in a non-abelian external field with the generalized uncertainty relation

Spin-charged point particle in a non-abelian external field with the generalized uncertainty relation

Resonant detectors of gravitational wave in the linear and quadratic generalized uncertainty principle framework

In this work, we consider a resonant bar detector of gravitational waves in the generalized uncertainty principle (GUP) framework with linear and quadratic momentum uncertainties. The phonon modes in these detectors vibrate due to the interaction with the incoming gravitational wave. In this uncertainty principle framework, we calculate the resonant frequencies and transition rates induced by the incoming gravitational waves on these detectors. We observe that the energy eigenstates and the eigenvalues get modified by the GUP parameters. We also observe non-vanishing transition probabilities between two adjacent energy levels due to the existence of the linear order momentum correction in the generalized uncertainty relation which was not present in the quadratic GUP analysis (Class Quantum Gravity 37:195006, 2020). We finally obtain bounds on the dimensionless GUP parameters using the form of the transition rates obtained during this analysis.

Hawking radiation under generalized uncertainty principle

The generalized uncertainty relation is expected to be an essential element in a theory of quantum gravity. In this work, we examine its effect on the Hawking radiation of a Schwarzschild black hole formed from collapse by incorporating a minimal uncertainty length scale into the radial coordinate of the background. This is implemented in both the ingoing Vaidya coordinates and a family of freely falling coordinates. We find that, regardless of the choice of the coordinate system, Hawking radiation is turned off at around the scrambling time. Interestingly, this phenomenon occurs while the Hawking temperature remains largely unmodified.

Minimal length scale correction in the noise of gravitons

In this paper we have considered a quantized and linearly polarized gravitational wave interacting with a gravitational wave detector (interferometer detector) in the generalized uncertainty principle (GUP) framework. Following the analysis in Phys. Rev. Lett. 127:081602 (2021), we consider a quantized gravitational wave interacting with a gravitational wave detector (LIGO/VIRGO etc.) using a path integral approach. Although the incoming gravitational wave was quantized, no Planck-scale quantization effects were considered for the detector in earlier literatures. In our work, we consider a modified Heisenberg uncertainty relation with a quadratic order correction in the momentum variable between the two phase space coordinates of the detector. Using a path integral approach, we have obtained a stochastic equation involving the separation between two point-like objects. It is observed that random fluctuations (noises) and the correction terms due to the generalized uncertainty relation plays a crucial role in dictating such trajectories. Finally, we observe that the solution to the stochastic equation leads to time dependent standard deviation due to the GUP insertion, and for a primordial gravitational wave (where the initial state is a squeezed state) both the noise effect and the GUP effects exponentially enhance which may be possible to detect in future generation of gravitational wave detectors. We have also given a plot of the dimensionless standard deviation with time depicting that the GUP effect will carry a distinct signature which may be detectable in the future space based gravitational wave observatories.

Perspectives about Scales and Hierarchies in a Three-Dimensional Quantum Vacuum Ruled by Generalized Uncertainty Relations

Perspectives about Scales and Hierarchies in a Three-Dimensional Quantum Vacuum Ruled by Generalized Uncertainty Relations

Coherent states for generalized uncertainty relations as Tsallis probability amplitudes: New route to nonextensive thermostatistics

Coherent states for generalized uncertainty relations as Tsallis probability amplitudes: New route to nonextensive thermostatistics

Editorial: Generalized uncertainty relations: existing paradigms and new approaches

Editorial: Generalized uncertainty relations: existing paradigms and new approaches

Coherent states for generalized uncertainty relations and their cosmological implications

We study coherent states associated with a generalized uncertainty principle (GUP). Our particular focus is on the negative deformation parameter β. We show that the ensuing coherent state can be identified with Tsallis’ probability amplitude with the non-extensivity parameter q being a monotonically increasing function of β. Furthermore, for β < 0, we reformulate the GUP in terms of a one-parameter class of Tsallis entropy based uncertainty relations, which are again saturated by the GUP coherent states. We argue that this combination of coherent states with Tsallis entropy offers a natural conceptual framework allowing to study the quasi-classical regime of GUP in terms of non-extensive thermostatistics. We bolster this claim by discussing a generalization of Verlinde’s entropic force and the ensuing implications in the late-inflation epoch. The corresponding dependence of the β parameter on the cosmological time is derived for the reheating epoch. The obtained β is consistent with both values predicted by string-theory models and the naturalness principle.

Generalized uncertainty relation between thermodynamic variables in quantum thermodynamics

Macroscopic thermodynamics, via the weak coupling approximation, assumes that the equilibrium properties of a system are not affected by interactions with its environment. However, this assumption may not hold for quantum systems, where the strength of interaction between the system and the environment may become non-negligible in a strong coupling regime. In such a regime, the equilibrium properties of the system depend on the interaction energy and the system state is no longer of the Gibbs form. Regarding such interactions, using tools from the quantum estimation theory, we derive the thermodynamic uncertainty relation between intensive and extensive variables valid at all coupling regimes through the generalized Gibbs ensemble. Where we demonstrate the lower bound on the uncertainty of intensive variables increases in presence of quantum fluctuations. Also, we calculate the general uncertainty relations for several ensembles to corroborate the literature results, thus showing the versatility of our method.

New deformed Heisenberg algebra from the μ-deformed model of dark matter

Recently, the μ-deformation-based approach to modeling dark matter, which exploits μ-deformed thermodynamics, was extended to the study of galaxy halo density profile and of the rotation curves of a number of (dwarf or low brightness) galaxies. For that goal, μ-deformed analogs of the Lane–Emden equation (LEE) have been proposed, and their solutions describing density profiles obtained. There are two seemingly different versions of μ-deformed LEE which possess the same solution, and so we deal with their equivalence. From the latter property we derive new, rather unusual, μ-deformed Heisenberg algebra (HA) for the position and momentum operators, and present the μ-HA in few possible forms (each one at μ → 0 recovers usual HA). The generalized uncertainty relation linked with the new μ-HA is studied, along with its interesting implications including the appearance of the quadruple of both maximal and minimal lengths and momenta.

Generalized uncertainty relations, particles, black holes, and Casimir effect in the three-dimensional quantum vacuum

Generalized uncertainty relations, particles, black holes, and Casimir effect in the three-dimensional quantum vacuum

Uncertainty relation based on metric-adjusted skew information with quantum memory

The uncertainty principle is one of the most essential features of quantum mechanics. Recently, uncertainty relations of skew information have been widely studied. In this paper, general and conditional uncertainty relations based on metric-adjusted skew information are put forward to study the case of the uncertainty relation with the existence of a quantum memory for the bipartite quantum system. These uncertainty relations include the product form and the sum form. The results show that the lower bounds contain two parts: one is characterizing the degree of compatibility of two measurements, and the other one is the quantum correlation between the measured system and the quantum memory. These results may play an important role in witnessing entanglement and quantum key distribution.

Generalized uncertainty relations for multiple measurements

The uncertainty relation is regarded as a remarkable feature of quantum mechanics differing from the classical counterpart, and it plays a backbone role in the region of quantum information theory. In principle, the uncertainty relation offers a nontrivial limit to predict the outcome of arbitrarily incompatible observed variables. Therefore, to pursue a more general uncertainty relations ought to be considerably important for obtaining accurate predictions of multi-observable measurement results in genuine multipartite systems. In this article, we derive a generalized entropic uncertainty relation (EUR) for multi-measurement in a multipartite framework. It is proved that the bound we proposed is stronger than the one derived from Renes et al. in [Phys. Rev. Lett. 103,020402(2009) ] for the arbitrary multipartite case. As an illustration, we take several typical scenarios that confirm that our proposed bound outperforms that presented by Renes et al. Hence, we believe our findings provide generalized uncertainty relations with regard to multi-measurement setting, and facilitate the EUR’s applications on quantum precision measurement regarding genuine multipartite systems.

Supersymmetric quantum mechanics and coherent states for a deformed oscillator with position-dependent effective mass

We study the classical and quantum oscillator in the context of a non-additive (deformed) displacement operator associated with a position-dependent effective mass by means of the supersymmetric formalism. From the supersymmetric partner Hamiltonians and the shape invariance technique, we obtain the eigenstates and the eigenvalues along with the ladders operators, thus showing a preservation of the supersymmetric structure in terms of the deformed counterpartners. The deformed space in supersymmetry allows to characterize position-dependent effective mass and uniform field interactions and to obtain a generalized uncertainty relation (GUP) that behaves as a distinguishability measure for the coherent states, these latter satisfying a periodic evolution for the corrections of the GUP.

Signature of universal fast scrambling in the transient response of a driven mott insulator system

The scrambling rate λ s, a measure of the early growth of decoherence in an interacting quantum system, has been conjectured to have a universal saturation bound, λ s ⩽ 2πk B T/ℏ, where T is the temperature. This decoherence arises from the spread of quantum information over a large number of untracked degrees of freedom. The commonly studied indicator of scrambling is the out of time-ordered correlator (OTOC) of noncommuting quantum operators, in-turn related to generalized uncertainty relations, and reminiscent of the Lyapunov exponent of classically chaotic systems. From a practical measurement point of view, other quantities besides OTOCs, that are also sensitive to these generalized uncertainty relations, may capture the scrambling behavior. Here, using a large- Keldysh field theory approach, we show that the nonequilibrium current response of a Mott insulator system consisting of a mesoscopic quantum dot array, when subjected to an electric field quench, reveals this phenomenon on account of number-phase uncertainty. Both ac and dc field quenches are considered. The passage from the initial Mott insulator phase with well-defined charge excitations, to the final nonequilibrium steady current state, is revealed in the transient current response that has Bloch-like oscillations. We find that the amplitude of these oscillations decreases at the universal rate, 2πk B T/ℏ, associated with fast scramblers. Our Mott insulator model provides a new example of a fast scrambler in addition to the known ones such as extremal black holes and the Sachdev–Ye–Kitaev (SYK) model.

The Generalized Uncertainty Principle

AbstractThe uncertainty principle lies at the heart of quantum physics, and is widely thought of as a fundamental limit of the measurement precision of incompatible observables. Here it is shown that the traditional uncertainty relation in fact belongs to the leading order approximation of a generalized uncertainty relation. That is, the leading order linear dependence of observables gives the Heisenberg type of uncertainty relations, while higher order nonlinear dependence may reveal more different and interesting correlation properties. Applications of the generalized uncertainty relation and the high order nonlinear dependence between observables in quantum information science are also discussed.

Uncertainty Relations in Hydrodynamics

The qualitative behaviors of uncertainty relations in hydrodynamics are numerically studied for fluids with low Reynolds numbers in 1+1 dimensional system. We first give a review for the formulation of the generalized uncertainty relations in the stochastic variational method (SVM), following the work by two of the present authors [Phys. Lett. A 382, 1472 (2018)]. In this approach, the origin of the finite minimum value of uncertainty is attributed to the non-differentiable (virtual) trajectory of a quantum particle and then both of the Kennard and Robertson-Schrödinger inequalities in quantum mechanics are reproduced. The same non-differentiable trajectory is applied to the motion of fluid elements in the Navier-Stokes-Fourier equation or the Navier-Stokes-Korteweg equation. By introducing the standard deviations of position and momentum for fluid elements, the uncertainty relations in hydrodynamics are derived. These are applicable even to the Gross-Pitaevskii equation and then the field-theoretical uncertainty relation is reproduced. We further investigate numerically the derived relations and find that the behaviors of the uncertainty relations for liquid and gas are qualitatively different. This suggests that the uncertainty relations in hydrodynamics are used as a criterion to classify liquid and gas in fluid.

Covariance matrix entanglement criterion for an arbitrary set of operators

A criterion for entanglement detection based on covariance matrices for an arbitrary set of observables is formulated. The method generalizes the covariance matrix entanglement criterion by Simon (2000 Phys. Rev. Lett. 84 2726) to a more general set of operators using the positive partial transpose test for the covariance matrix. The relation is found by starting from the generalized uncertainty relation for multiple operators, and taking the partial transpose on the bipartition. The method is highly efficient and versatile in the sense that the set of measurement operators can be freely chosen, and there is no constraint on the commutation relations. The main restriction on the chosen set of measurement operators is that the correlators and expectation values of the partially transposed observable operators can be evaluated. The method is particularly suited for systems with higher dimensionality since the computations do not scale with the dimension of the Hilbert space—rather they scale with the number of chosen observables. We illustrate the approach by examining the entanglement between two spin ensembles, and show that it detects entanglement in a basis independent way.

Generalized uncertainty relations for semi-Markov processes

The thermodynamic and kinetic uncertainty relations provide finite-time bounds on the observable fluctuation in Markov processes. Herein, we generalize these bounds for semi-Markov processes. Specifically, we prove that, unlike in the Markovian case, the fluctuation of time-antisymmetric observables is bounded not only by entropy production but also by a memory term. For generic observables, we analogously show that the fluctuation is bounded by both dynamical activity and a memory term. Our results indicate that memory plays an important role in the bounds. Interestingly, with a proper form of the waiting-time distribution, the memory can decrease the observable fluctuation. When the waiting-time distribution is Poissonian (i.e., the process becomes Markov), the memory terms vanish, and the derived bounds reduce to the conventional bounds.

Remarks on the uncertainty relations

We analyze general uncertainty relations and we show that there can exist such pairs of non-commuting observables [Formula: see text] and [Formula: see text] and such vectors that the lower bound for the product of standard deviations [Formula: see text] and [Formula: see text] calculated for these vectors is zero: [Formula: see text]. We also show that for some pairs of non-commuting observables the sets of vectors for which [Formula: see text] can be complete (total). The Heisenberg, [Formula: see text], and Mandelstam–Tamm (MT), [Formula: see text], time–energy uncertainty relations ([Formula: see text] is the characteristic time for the observable [Formula: see text]) are analyzed too. We show that the interpretation [Formula: see text] for eigenvectors of a Hamiltonian [Formula: see text] does not follow from the rigorous analysis of MT relation. We show also that contrary to the position–momentum uncertainty relation, the validity of the MT relation is limited: It does not hold on complete sets of eigenvectors of [Formula: see text] and [Formula: see text].