Minimal Length Scale Research Articles

Global Length and Overhang Control for Level Set and Density Approaches via Perimeter Minimization

ABSTRACTTopology optimization is probably one of the most efficient techniques for structural design. However, running topology optimization without geometry control provides complex designs, which often are manufactured with additive manufacturing methods. Consequently, a fundamental aspect in topology optimization is to consider the following additive manufacturing constraints: minimal length scale and overhang. The aim of this paper is to propose a new numerical method to control globally such additive manufacturing constraints. The idea relies on penalizing a regularized version of the perimeter: an isotropic version for global length and an anisotropic version for overhang. Besides, we show that the method may be used with density and level set approaches. Several numerical examples, including compliant mechanisms and material design, show that some bars have been removed, decreasing the complexity of the shape, and vertical tendency orientation of boundaries is generally obtained.

Minimal length scale in quantum mechanics from a deformation quantization perspective

Abstract We develop a deformation quantization approach to quantum mechanics that exhibits a nonzero minimal uncertainty in position through the modification of commutation relations for the operators of position and momentum. Deformation quantization is a method of quantizing a classical Hamiltonian system by suitably deforming the Poisson algebra of the system. The developed theory of deformation quantization is non-formal. An appropriate integral formula for the star-product is introduced, along with a suitable space of functions on which the star-product is well defined. A C*-algebra of observables and a space of states are constructed. Moreover, an operator representation in momentum space is presented. Finally, an example of states of maximal localization is given in terms of generalized Wigner functions.

A covariant tapestry of linear GUP, metric-affine gravity, their Poincaré algebra and entropy bound

Motivated by the potential connection between metric-affine gravity and linear generalized uncertainty principle (GUP) in the phase space, we develop a covariant form of linear GUP and an associated modified Poincaré algebra, which exhibits distinctive behavior, nearing nullity at the minimal length scale proposed by linear GUP. We use three-torus geometry to visually represent linear GUP within a covariant framework. The three-torus area provides an exact geometric representation of Bekenstein’s universal bound. We depart from Bousso’s approach, which adapts Bekenstein’s bound by substituting the Schwarzschild radius ( rs ) with the radius (R) of the smallest sphere enclosing the physical system, thereby basing the covariant entropy bound on the sphere’s area. Instead, our revised covariant entropy bound is described by the area of a three-torus, determined by both the inner radius rs and outer radius R where rs⩽R due to gravitational stability. This approach results in a more precise geometric representation of Bekenstein’s bound, notably for larger systems where Bousso’s bound is typically much larger than Bekensetin’s universal bound. Furthermore, we derive an equation that turns the standard uncertainty inequality into an equation when considering the contribution of the three-torus covariant entropy bound, suggesting a new avenue of quantum gravity.

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.

Constraints via the Event Horizon Telescope for Black Hole Solutions with Dark Matter under the Generalized Uncertainty Principle Minimal Length Scale Effect

AbstractFour spherically symmetric but non‐asymptotically flat black hole solutions surrounded with spherical dark matter distribution perceived under the minimal length scale effect is derived via the generalized uncertainty principle. Here, the effect of this quantum correction, described by the parameter , is considered on a toy model galaxy with dark matter and the three well‐known dark matter distributions: the cold dark matter, scalar field dark matter, and the universal rotation curve. The aim is to find constraints to by applying these solutions to the known supermassive black holes: Sagittarius A (Sgr. A*) and Messier 87* (M87*), in conjunction with the available Event Horizon telescope. The effect of is then examined on the event horizon, photonsphere, and shadow radii, where unique deviations from the Schwarzschild case are observed. As for the shadow radii, bounds are obtained for the values of on each black hole solution at confidence level. The results revealed that under minimal length scale effect, black holes can give positive (larger shadow) and negative values (smaller shadow) of , which are supported indirectly by laboratory experiments and astrophysical or cosmological observations, respectively.

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.

On Possible Minimal Length Deformation of Metric Tensor, Levi-Civita Connection, and the Riemann Curvature Tensor

The minimal length conjecture is merged with a generalized quantum uncertainty formula, where we identify the minimal uncertainty in a particle’s position as the minimal measurable length scale. Thus, we obtain a quantum-induced deformation parameter that directly depends on the chosen minimal length scale. This quantum-induced deformation is conjectured to require the generalization of Riemannian spacetime geometry underlying the classical theory of general relativity to an eight-dimensional spacetime fiber bundle, which dictates the deformation of the line element, metric tensor, Levi-Civita connection, Riemann curvature tensor, etc. We calculate the deformation thus produced in the Levi-Civita connection and find it to explicitly and exclusively depend on the product of the minimum measurable length and the particle’s spacelike four-acceleration vector, L2x¨2. We find that the deformed Levi-Civita connection preserves all properties of its undeformed counterpart, such as torsion freedom and metric compatibility. Accordingly, we have constructed a deformed version of the Riemann curvature tensor whose expression can be factorized in all its terms with different functions of L2x¨2. We also show that the classical four-manifold status of being Riemannian is preserved when the quantum-induced deformation is negligible. We study the dependence of a parallel-transported tangent vector on the spacelike four-acceleration. We illustrate the impact of the minimal-length-induced quantum deformation on the classical geometrical objects of the general theory of relativity using the unit radius two-sphere example. We conclude that the minimal length deformation implies a correction to the spacetime curvature and its contractions, which is manifest in the additional curvature terms of the corrected Riemann tensor. Accordingly, quantum-induced effects endow an additional spacetime curvature and geometrical structure.

Estimation of Correction of Ground State Energy of Hydrogen Atom in Presence of Quadratic GUP

Abstract: String Theory, Quantum Geometry, Loop Quantum Gravity and Black Hole physic all predict the existence of a observable minimal length at Planck scale. For example, in case of string theory it is conjectured that a particle described as a string does not interact at distances smaller than its size. As a consequence, the HUP has to be generalized to take into account this aspect. The models which are designed to implement the minimal length scale and/or the maximum momentum in different physical systems entered into the literature as the Generalized Uncertainty Principle (GUP). Here, quadratic GUP model has been used to estimate the quantum-gravitational correction of ground state energy of hydrogen atom (H-atom).

Topology optimization of a rectangular phase change material module

Latent heat storages have numerous times been demonstrated to benefit from including highly conductive fin structures for elevated charging and discharging properties. However, to this point it is still unclear how fins should be designed for optimal charging performance while meeting constraints on minimum storage density and manufacturability. Therefore, in the present work, we use topology optimization to obtain optimal designs for rectangular Phase Change Material (PCM) modules charged by a water flow, based on a conductive heat transfer model for the PCM. The optimal topologies are designed for maximum mean charging power of the PCM over a desired charge time. The optimal designs feature tree-like metal structures with an increased amount of branching for higher power requirements. Moreover, the optimal topologies are compared with straight fins with an equal minimal length scale. This comparison reveals the sub-optimality of the obtained tree-like designs for conduction driven phase-change problems.

Minimal length: A cut-off in disguise?

The minimal-length paradigm, a possible implication of quantum gravity at low energies, is commonly understood as a phenomenological modification of Heisenberg's uncertainty relation. We show that this modification is equivalent to a cut-off in the space conjugate to the position representation, i.e. the space of wave numbers, which does not necessarily correspond to momentum space. This result is generalized to several dim ensions and noncommutative geometries once a suitable definition of the wave number is provided. Furthermore, we find a direct relation between the ensuing bound in wave-number space and the minimal-length scale. For scenarios in which the existence of the minimal length cannot be explicitly verified, the proposed framework can be used to clarify the situation. Indeed, applying it to common models, we find that one of them does, against all expectations, allow for arbitrary precision in position measurements. In closing, we comment on general implications of our findings for the field. In particular, we point out that the minimal length is purely kinematical such that, effectively, it is not influenced by the overlying dynamics and the choice of Hamiltonian.

Effect of minimal length uncertainty on neutrino oscillation

Abstract In this paper, we study the effect of the minimal length on neutrino oscillation in a static magnetic field. In the framework of the generalized uncertainty principle, we reformulate the Hamiltonian for a relativistic neutrino moving in a magnetic field oriented along the z-direction of Cartesian coordinates. Using the modified energy spectrum, we obtain the oscillation probability for different neutrino flavors. In addition, we obtain the energy differences for the neutrino-mass eigenstates. We find that the energy and energy difference depend on the minimal length parameter α, and the energy difference becomes independent of α when the magnetic field is not present. In addition, we find that the modified probability of oscillation differs from the usual probability of oscillation if a magnetic field is present. Using the current experimental result, we estimate the upper bound on the deformation parameter and the minimal length, and find that the upper bound on the minimal length scale is less than the electroweak scale. If the minimal length is at Planck scale, the minimal length formalism leads to the same result as a quantum theory of gravity with an S U 2 L ⊗ U 1 $SU{\left(2\right)}_{L}\otimes U\left(1\right)$ effective invariant dimension-5 Lagrangian including neutrino and Higgs fields.

Locally detecting UV cutoffs on a sphere with particle detectors

The potential breakdown of the notion of a metric at high energy scales could imply the existence of a fundamental minimal length scale below which distances cannot be resolved. One approach to realizing this minimum length scale is construct a quantum field theory with a bandlimit on the field. We report on an investigation of the effects of imposing a bandlimit on a field on a curved and compact spacetime and how best to detect such a bandlimit if it exists. To achieve this operationally, we couple two Gaussian-smeared UDW detectors to a scalar field on a $S^2 \times R$ spherical spacetime through delta-switching. The bandlimit is implemented through a cut-off of the allowable angular momentum modes of the field. We observe that a number of features of single detector response in the spherical case are similar to those in flat spacetime, including the dependence on the geometry of the detector, and that smaller detectors couple more strongly to the field, leading to an optimal size for bandlimit detection. We find that in flat spacetime squeezed detectors are more senstive to the bandlimit provided they are larger than the optimal size; however, in spherical spacetime the bandlimit itself determines if squeezing improves the sensitivity. We also explore setups with two detectors, noting that in the spherical case, due to its compact nature, there is a lack of dissipation of any perturbation to the field, which results in locally excited signals being refocused at the poles. Quite strikingly, this feature can be exploited to significantly improve bandlimit detection via field mediated signalling. Moreover, we find that squeezing on a sphere introduces extra anisotropies that could be exploited to amplify or weaken the response of the second detector.

“The more things change the more they stay the same” Minimum lengths with unmodified uncertainty principle and dispersion relation

Broad arguments indicate that quantum gravity should have a minimal length scale. In this paper, we construct a minimum length model by generalizing the time-position and energy–momentum operators while keeping much of the structure of quantum mechanics and relativity intact: the standard position-momentum commutator, the special relativistic time-position, and energy–momentum relationships all remain the same. Since the time-position and energy–momentum relationships for the modified operators remain the same, we retain a form of Lorentz symmetry. This avoids the constraints on these theories coming from lack of photon dispersion while holding the potential to address the Greisen–Zatsepin–Kuzmin (GZK) puzzle of ultra high-energy cosmic rays.

Effect of RGUP on the nonlinear Klein-Gordon model with spontaneous symmetry breaking

In the framework of the relativistic generalized uncertainty principle (RGUP), we study the nonlinear Klein-Gordon field with ϕ4 self-interaction. This generalization comes from the quantum gravity theories that predict the existence of a minimal measurable length scale. The quantum gravity effects slightly modify the momentum and change the Lagrangian density of the field. This model, specifically in the context of spontaneous symmetry breaking, is so fruitful and leads us to many interesting phenomena. Due to the complexity of the consequent equations, the model is normally studied via perturbation mechanisms. However, here, we apply a generalized tanh-method which supposed that the field solutions should be sech-functions of segments of space and time. By this extended method, we find some more general solutions with respect to our previous work [1]. Several solutions can be found but we select only the normalizable and bounded solitary fields. The energy spectrum of each field solution is obtained in an analytical form and discussed. The modification parameter of the corresponding RGUP is estimated by using the rest energy of the mass of the Higgs boson.

Decoherence limit of quantum systems obeying generalized uncertainty principle: New paradigm for Tsallis thermostatistics

The generalized uncertainty principle (GUP) is a phenomenological model whose purpose is to account for a minimal length scale (e.g., Planck scale or characteristic inverse-mass scale in effective quantum description) in quantum systems. In this Letter, we study possible observational effects of GUP systems in their decoherence domain. We first derive coherent states associated to GUP and unveil that in the momentum representation they coincide with Tsallis' probability amplitudes, whose non-extensivity parameter $q$ monotonically increases with the GUP deformation parameter $\beta$. Secondly, for $\beta < 0$ (i.e., $q < 1$), we show that, due to Bekner-Babenko inequality, the GUP is fully equivalent to information-theoretic uncertainty relations based on Tsallis-entropy-power. Finally, we invoke the Maximal Entropy principle known from estimation theory to reveal connection between the quasi-classical (decoherence) limit of GUP-related quantum theory and non-extensive thermostatistics of Tsallis. This might provide an exciting paradigm in a range of fields from quantum theory to analog gravity. For instance, in some quantum gravity theories, such as conformal gravity, aforementioned quasi-classical regime has relevant observational consequences. We discuss some of the implications.

A Subtle Aspect of Minimal Lengths in the Generalized Uncertainty Principle

In this work, we point out an overlooked and subtle feature of the generalized uncertainty principle (GUP) approach to quantizing gravity: namely that different pairs of modified operators with the same modified commutator, [X^,P^]=iħ(1+βp2), may have different physical consequences such as having no minimal length at all. These differences depend on how the position and/or momentum operators are modified rather than only on the resulting modified commutator. This provides guidance when constructing GUP models since it distinguishes those GUPs that have a minimal length scale, as suggested by some broad arguments about quantum gravity, versus GUPs without a minimal length scale.

Reconciling a quantum gravity minimal length with lack of photon dispersion

Generic arguments lead to the idea that quantum gravity has a minimal length scale. A possible observational signal of such a minimal length scale is that photons should exhibit dispersion. In 2009, the observation of a short gamma ray burst seemed to push the minimal length scale to distances smaller than the Planck length. This poses a challenge for such minimal distance models. Here we propose a modification of the position and momentum operators, xˆ and pˆ, which lead to a minimal length scale, but preserve the photon energy-momentum relationship E=pc. In this way there is no dispersion of photons with different energies. Additionally, this can be accomplished without modifying the commutation relationship [xˆ,pˆ]=iħ.

Image of the Electron Suggested by Nonlinear Electrodynamics Coupled to Gravity

We present a systematic review of the basic features that were adopted for different electron models and show, in a brief overview, that, for electromagnetic spinning solitons in nonlinear electrodynamics minimally coupled to gravity (NED-GR), all of these features follow directly from NED-GR dynamical equations as model-independent generic features. Regular spherically symmetric solutions of NED-GR equations that describe electrically charged objects have obligatory de Sitter center due to the algebraic structure of stress–energy tensors for electromagnetic fields. By the Gürses-Gürsey formalism, which includes the Newman–Janis algorithm, they are transformed to axially symmetric solutions that describe regular spinning objects asymptotically Kerr–Newman for a distant observer, with the gyromagnetic ratio g=2. Their masses are determined by the electromagnetic density, related to the interior de Sitter vacuum and to the breaking of spacetime symmetry from the de Sitter group. De Sitter center transforms to the de Sitter vacuum disk, which has properties of a perfect conductor and ideal diamagnetic. The ring singularity of the Kerr–Newman geometry is replaced with the superconducting current, which serves as the non-dissipative source for exterior fields and source of the intrinsic magnetic momentum for any electrically charged spinning NED-GR object. Electromagnetic spinning soliton with the electron parameters can shed some light on appearance of a minimal length scale in the annihilation reaction e+e−→γγ(γ).

The Cell Adaptation Time Sets a Minimum Length Scale for Patterned Substrates

The structure and dynamics of tissue cultures depend strongly on the physical and chemical properties of the underlying substrate. Inspired by previous advances in the context of inorganic materials, the use of patterned culture surfaces has been proposed as an effective way to induce space-dependent properties in cell tissues. However, cells move and diffuse, and the transduction of external stimuli to biological signals is not instantaneous. Here, we show that the fidelity of patterns to demix tissue cells depends on the relation between the diffusion (τD) and adaptation (τ) times. Numerical results for the self-propelled Voronoi model reveal that the fidelity decreases with τ/τD, a result that is reproduced by a continuum reaction-diffusion model. Based on recent experimental results for single cells, we derive a minimal length scale for the patterns in the substrate that depends on τ/τD and can be much larger than the cell size.

Thermal properties of a two-dimensional Duffin–Kemmer–Petiau oscillator under an external magnetic field in the presence of a minimal length

Generalized uncertainty principle puts forward the existence of the shortest distances and/or maximum momentum at the Planck scale for consideration. In this article, we investigate the solutions of a two-dimensional Duffin–Kemmer–Petiau (DKP) oscillator within an external magnetic field in a minimal length (ML) scale. First, we obtain the eigensolutions in ordinary quantum mechanics. Then, we examine the DKP oscillator in the presence of an ML for the spin-zero and spin-one sectors. We determine an energy eigenvalue equation in both cases with the corresponding eigenfunctions in the non-relativistic limit. We show that in the ordinary quantum mechanic limit, where the ML correction vanishes, the energy eigenvalue equations become identical with the habitual quantum mechanical ones. Finally, we employ the Euler–Mclaurin summation formula and obtain the thermodynamic functions of the DKP oscillator in the high-temperature scale.