Uncertainty Principle Research Articles

Theoretical Quantum Mass

In the present theoretical work, attempts have been made to find the least mass, which quantifies the discrete mass, in terms of the reduced Planck constant and cosmological constant in the Heisenberg position-momentum uncertainty relation as well as by employing the time period of oscillating universes in mass-time inequality. It has been estimated and reported that the least mass exists in the range 5.98 × 10-74 < m < 2 × 10-68 kg, which quantifies discrete masses in the universe of size < 8.64 G ly.

Statistical uncertainty principle in Markov kinetics

A reciprocality between the statistical variance of observables of a thermodynamic state and that of their conjugate variables, as entropic forces, originates from the thermodynamic conjugacy with respect to an entropy function. This thermodynamic uncertainty principle in equilibrium can be derived from the Maximum Entropy principle and is independent upon underlying mechanistic details. We present, based on the Maximum Caliber principle as the dynamic generalization of Maximum Entropy, the formalism of the uncertainty principle in kinetics in time homogeneous Markov processes between transitional observables and their conjugate path entropic forces. A stochastic biophysical model for molecular motors is used as an illustrating example. The present work generalizes the phenomenological thermodynamics of uncertainties/fluctuations and is applicable to data ad infinitum.

Generalized uncertainty principle for entangled states of two identical particles

Generalized uncertainty principle for entangled states of two identical particles

Relativistic generalized uncertainty principle for a test particle in four-dimensional spacetime

The generalized uncertainty principle provides a promising phenomenological approach to reconciling the fundamentals of general relativity with quantum mechanics. As a result, noncommutativity, measurement uncertainty, and the fundamental theory of quantum mechanics are subject to finite gravitational fields. The generalized uncertainty principle (RGUP) on curved spacetime allows for the imposition of quantum-induced elements on general relativity (GR) in four dimensions. In the relativistic regimes, the determination of a test particle’s spacetime coordinates, [Formula: see text], becomes uncertain. There exists a specific range of coordinates where the accessibility to [Formula: see text] is notably limited. Consequently, the spacetime coordinates lack both smoothness and continuity. The quantum-mechanical calculations of the spacetime coordinates are directly linked to their measurement through the expectation value [Formula: see text]. This expectation value is dependent on [Formula: see text], which is itself limited by a minimum measurable length [Formula: see text]. A crucial finding presented in this script is the existence of a lower bound for the Hamiltonian, which implies the stability of the quantum nature of spacetime. The direct correlation between [Formula: see text] and [Formula: see text] is a significant discovery, suggesting a proportional relationship. The conjecture is made that the primary metric g carries all essential information regarding spacetime curvature and serves a role akin to the Jacobian determinant in general relativity. Moreover, the linear relationship between [Formula: see text] and the Planck length [Formula: see text] is established with a proportionality factor of [Formula: see text], where [Formula: see text] denotes the RGUP parameter. The discretization of spacetime coordinates results in the discontinuity of the test particle’s wavefunction [Formula: see text], leading to an unrealistic [Formula: see text]. It is also noted that the lower limit of [Formula: see text] is directly proportional to the fundamental tensor.

Synchronization-induced violation of thermodynamic uncertainty relations

Synchronization-induced violation of thermodynamic uncertainty relations

A note on the 1-D minimization problem related to solenoidal improvement of the uncertainty principle inequality

AbstractThis paper gives a second way to solve the one-dimensional minimization problem of the form : $$\begin{aligned} \min _{f\not \equiv 0}\frac{\displaystyle \int \limits _0^\infty \left( f''\right) ^2x^{\mu +1}dx\int \limits _0^\infty \left( {x}^2\left( f'\right) ^2 -\varepsilon f^2\right) {{x}}^{\mu -1}d{x}}{\displaystyle \left( \int \limits _0^\infty \left( f'\right) ^2 {{x}}^{\mu }d{x}\right) ^2} \end{aligned}$$ min f ≢ 0 ∫ 0 ∞ f ′ ′ 2 x μ + 1 d x ∫ 0 ∞ x 2 f ′ 2 - ε f 2 x μ - 1 d x ∫ 0 ∞ f ′ 2 x μ d x 2 for scalar-valued functions f on the half line, where $$f'$$ f ′ and $$f''$$ f ′ ′ are its derivatives and $$\varepsilon $$ ε and $$\mu $$ μ are positive parameters with $$\varepsilon < \frac{\mu ^2}{4}.$$ ε < μ 2 4 . This problem plays an essential part of the calculation of the best constant in Heisenberg’s uncertainty principle inequality for solenoidal vector fields. The above problem was originally solved by using (generalized) Laguerre polynomial expansion; however, the calculation was complicated and long. In the present paper, we give a simpler method to obtain the same solution, the essential part of which was communicated in Theorem 5.1 of the preprint (Hamamoto, arXiv:2104.02351v4, 2021).

Uncertainty and information in physiological signals: Explicit physical trade-off with log-normal wavelets

Physiological recordings contain a great deal of information about the underlying dynamics of Life. The practical statistical treatment of these single-trial measurements is often hampered by the inadequacy of overly strong assumptions. Heisenberg’s uncertainty principle allows for more parsimony, trading off statistical significance for localization. By decomposing signals into time–frequency atoms and recomposing them into local quadratic estimates, we propose a concise and expressive implementation of these fundamental concepts based on the choice of a geometric paradigm and two physical parameters. Starting from the spectrogram based on two fixed timescales and Gabor’s normal window, we then build its scale-invariant analogue, the scalogram based on two quality factors and Grossmann’s log-normal wavelet. These canonical estimators provide a minimal and flexible framework for single trial time–frequency statistics, which we apply to polysomnographic signals: EEG representations, HRV extraction from ECG, coherence and mutual information between heart rate and respiration.

Hypergeometric Gabor transform and uncertainty principles

Hypergeometric Gabor transform and uncertainty principles

Discrete linear canonical transform on graphs: Uncertainty principle and sampling

With an increasing influx of classical signal processing methodologies into the field of graph signal processing, approaches grounded in discrete linear canonical transform have found application in graph signals. In this paper, we initially propose the uncertainty principle of the graph linear canonical transform (GLCT), which is based on a class of graph signals maximally concentrated in both vertex and graph spectral domains. Subsequently, leveraging the uncertainty principle, we establish conditions for recovering bandlimited signals of the GLCT from a subset of samples, thereby formulating the sampling theory for the GLCT. We elucidate interesting connections between the uncertainty principle and sampling. Further, by employing sampling set selection and experimental design sampling strategies, we introduce optimal sampling operators in the GLCT domain. Finally, we evaluate the performance of our methods through simulations and numerical experiments across applications.

The corrections to the metric of Reissner-Nordström black hole from the generalized uncertainty principle

The emergence of the minimal observable length is commonly accepted in most of the quantum gravitational candidates. The existence of such a minimal length in high energy physics necessitates some revisions to the standard uncertainty principle. The generalized uncertainty principle is particularly suitable for incorporating such a finite resolution of the space-time and may provide a useful phenomenological approach to study the physics of quantum gravity. It is possible to use the generalized uncertainty principle to modify the black hole thermodynamics straightforwardly. However, it is also possible to use the generalized uncertainty principle to modify the black hole metric itself. In this paper, we are going to modify the Reissner-Nordström metric in the presence of the quantum gravitational effects via the generalized uncertainty principle. Then, we use the modified charged black hole metric to study the black hole thermodynamics. The modified metric is also applied to study the light deflection angle.

Generalized Trace Inequalities for Q Uncertainty Relations

Iin 2015 we obtained non-hermitian extensions of Heisenberg type and Schrödinger type uncertainty relations for generalized metric adjusted skew information or generalized metric adjusted correlation measure and gave the results of Dou-Du in 2013 and 2014 as corollaries. In this paper, we define generalized quasi-metric adjusted Q skew information for different two generalized states and obtain corresponding uncertainty relation. The result is applied to the inequalities related to fidelity and trace distance for different two generalized states which were given by Audenaert, et al. in 2009 and 2008; and Powers-Strmer in 1970.

Uncertainty of Helium Ion Momentum (𝑯𝒆+) by Using the Heisenberg Uncertainty Approach on Quantum Numbers 𝒏 ≤ 𝟑

Uncertainty of Helium Ion Momentum (𝑯𝒆+) by Using the Heisenberg Uncertainty Approach on Quantum Numbers 𝒏 ≤ 𝟑

Incompatible observables in classical physics: A closer look at measurement in Hamiltonian mechanics.

Quantum theory famously entails the existence of incompatible measurements, pairs of system observables which cannot be simultaneously measured to arbitrary precision. Incompatibility is widely regarded to be a uniquely quantum phenomenon, linked to failure to commute of quantum operators. Even in the face of deep parallels between quantum commutators and classical Poisson brackets, no connection has been established between the Poisson algebra and any intrinsic limitations to classical measurement. Here I examine measurement in classical Hamiltonian physics as a process involving the joint evolution of an object-system and a finite-temperature measuring apparatus. Instead of the ideal measurement capable of extracting information without disturbing the system, I find a Heisenberg-like precision-disturbance relation: Measuring an observable leaves all Poisson-commuting observables undisturbed but inevitably disturbs all non-Poisson-commuting observables. In this classical uncertainty relation the role of h-bar is played by an apparatus-specific quantity, q-bar. While this is not a universal constant, the analysis suggests that q-bar takes a finite positive value for any apparatus that can be built. (Specifically: q-bar vanishes in the model only in the unreachable limit of zero absolute temperature.) I show that a classical version of Ozawa's model of quantum measurement [Ozawa, Phys. Rev. Lett. 60, 385 (1988)0031-900710.1103/PhysRevLett.60.385], originally proposed as a means to violate Heisenberg's relation, does not violate the classical relation. If this result were to generalize to all models of measurement, then incompatibility would prove to be a feature not only of quantum, but of classical physics too. Put differently: The approach presented here points the way to studying the (Bayesian) epistemology of classical physics, which was until now assumed to be trivial. It now seems possible that it is nontrivial and bears a resemblance to the quantum formalism. The present findings may be of interest to researchers working on foundations of quantum mechanics, particularly for ψ-epistemic interpretations. More practically, there may be applications in the fields of precision measurement, nanoengineering, and molecular machines.

The quantum uncertainty relations of quantum channels

The quantum uncertainty relations of quantum channels

Heisenberg's uncertainty relations for a hydrogen atom confined by an impenetrable spherical cavity

Heisenberg's uncertainty relations for a hydrogen atom confined by an impenetrable spherical cavity

Variance-based uncertainty relations and entanglement amplification for particles constrained on a torus

We formulate the variance-based uncertainty relations (URs) via the Robertson’s inequality, for a 2-particle entangled system constrained on a torus and subject to a stationary magnetic field B→\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {\\vec {B}}$$\\end{document}. We explore the system’s parameter space and show that these new URs have field-tunable uncertainty bounds. Our analysis reveals that A→\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {\\vec {A}}$$\\end{document} (vector potential) induces a phase shift in the state, due to the Aharonov–Bohm effect, leading to a perturbed system dynamics which results in asymmetric product of variance (POV\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {POV}$$\\end{document}). Additionally, we give the critical range of A→\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {\\vec {A}}$$\\end{document} and B→\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {\\vec {B}}$$\\end{document} where the system acts as an entanglement amplifier; this amplification is also discussed under various geometric parameters. The possibility of reducing the POV\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {POV}$$\\end{document} of the conjugate pair [q, p] below the known benchmark value by the Generalized Uncertainty Relation (GUR) is also demonstrated. Finally, we check the susceptibility of state coherence to B→\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {\\vec {B}}$$\\end{document} by saturating the angular momentum uncertainty relation and identify the critical coherence value Bc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {B}}_c$$\\end{document} such that when B≠Bc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {B}}\ e {\\mathcal {B}}_c$$\\end{document}, the state decoheres.

Planck Length Emerging as the Invariant Quantum Minimum Effective Length Determined by the Heisenberg Uncertainty Principle in Manifestly Covariant Quantum Gravity Theory

The meaning of the quantum minimum effective length that should distinguish the quantum nature of a gravitational field is investigated in the context of manifestly covariant quantum gravity theory (CQG-theory). In such a framework, the possible occurrence of a non-vanishing minimum length requires one to identify it necessarily with a 4-scalar proper length s.It is shown that the latter must be treated in a statistical way and associated with a lower bound in the error measurement of distance, namely to be identified with a standard deviation. In this reference, the existence of a minimum length is proven based on a canonical form of Heisenberg inequality that is peculiar to CQG-theory in predicting massive quantum gravitons with finite path-length trajectories. As a notable outcome, it is found that, apart from a numerical factor of O1, the invariant minimum length is realized by the Planck length, which, therefore, arises as a constitutive element of quantum gravity phenomenology. This theoretical result permits one to establish the intrinsic minimum-length character of CQG-theory, which emerges consistently with manifest covariance as one of its foundational properties and is rooted both on the mathematical structure of canonical Hamiltonian quantization, as well as on the logic underlying the Heisenberg uncertainty principle.

Operator size growth in Lindbladian SYK

We investigate the growth of operator size in the Lindbladian Sachdev-Ye-Kitaev model with q-body interaction terms and linear jump terms at finite dissipation strength. We compute the operator size as well as its distribution numerically at finite q and analytically at large q. With dissipative (productive) jump terms, the size converges to a value smaller (larger) than half the number of Majorana fermions. At weak dissipation, the evolution of operator size displays a quadratic-exponential-plateau behavior. The plateau value is determined by the ratios between the coupling of the interaction and the linear jump term in the large q limit. The operator size distribution remains localized in the finite size region even at late times, contrasting with the unitary case. Moreover, we also derived the time-independent orthogonal basis for operator expansion which exhibits the operator size concentration at finite dissipation. Finally, we observe that the uncertainty relation for operator size growth is saturated at large q, leading to classical dynamics of the operator size growth with dissipation.

On the Speed of Light as a Key Element in the Structure of Quantum Mechanics

We follow the assumption that relativistic causality is a key element in the structure of quantum mechanics and integrate the speed of light, c, into quantum mechanics through the postulate that the (reduced) Planck constant is a function of c with a leading order of the form ℏc∼Λ/cp for a constant Λ>0, and p>1. We show how the limit c→∞ implies classicality in quantum mechanics and explain why p has to be larger than 1. As the limit c→∞ breaks down both relativity theory and quantum mechanics, as followed by the proposed model, it can then be understood through similar conceptual physical laws. We further show how the position-dependent speed of light gives rise to an effective curved space in quantum systems and show that a stronger gravitational field implies higher quantum uncertainties, followed by the varied c. We then discuss possible ways to find experimental evidence of the proposed model using set-ups to test the varying speed of light models and examine analogies of the model based on electrons in semiconductor heterostructures.

N-Dimensional non-commutative GUP quantization and application to the Bianchi I model

We analyse a n-dimensional Generalized Uncertainty Principle (GUP) quantization framework, characterized by a non-commutative nature of the configurational variables. First, we identify a set of states which are maximally localized only along a single direction, at the expense of being less localized in all the other ones. Subsequently, in order to recover information about localization on the whole configuration space, we use the only state of the theory which exhibits maximal localization simultaneously in every direction to construct a satisfactory quasi-position representation, by virtue of a suitable translational operator. The resultant quantum framework is then applied to model the dynamics of the Bianchi I cosmology. The corresponding Wheeler–DeWitt equation is reduced to Schrödinger dynamics for the two anisotropy degrees of freedom, using a WKB representation for the volume-like variable of the Universe, in accordance with the Vilenkin scenario. The main result of our cosmological implementation of the constructed quantum theory demonstrates how the dynamics of a wave packet peaked at some point in the configuration space represented in the quasi-position variables favours as the most probable configuration exactly the initial one for a relatively long time, if compared with the ordinary quantum theory. This preference arises from the different dynamical behavior exhibited by wave packets in the two quantum theories.