Uncertainty Principles for the Generalized Fourier Transform Associated with Spherical Mean Operator

The aim of this paper is to establish an extension of qualitative and quantitative uncertainty principles for the Fourier transform connected with the spherical mean operator.

Uncertainty Principles on Clifford Modules

In this paper,we derive the optimal Cauchy–Schwarz inequalities on a class of Hilbert and Krein modules over a Clifford algebra,which heavily depend on the Clifford algebraic structure.The obtained inequalities further lead to very general uncertainty inequalities on these modules.Some new phenomena arise,due to the non-commutative nature,the Clifford-valued inner products and the Krein geometry.Taking into account applications,special attention is given to the Dirac operator and the Howe dual pair Pin(m)×osp(1|2).Moreover,it is surprisingly to find that the recent highly nontrivial uncertainty relation for triple observables is indeed a direct consequence of our Cauchy–Schwarz inequality.This new observation leads to refined uncertainty relations in terms of the Wigner–Yanase–Dyson skew information for mixed states and other generalizations.These show that the obtained uncertainty inequalities on Clifford modules can be considered as new uncertainty relations for multiple observables.

The effect of different generalized uncertainty principles on Jeans mass modification

Jeans mass is regarded as a crucial factor in the study of nebula collapse.Astronomical data shows that Jeans mass is larger in theory than it is in observation.Someone mentioned that Jeans mass can be modified by using the generalized uncertainty principle(GUP).However,different physical backgrounds lead to different forms of GUP expression.In order to make the theoretical values of Jeans mass and its observed values match better,we use three distinct types of GUPs to correct Jeans mass in this paper.We find that the corrected Jeans masses are smaller than the uncorrected ones,where the Pedram corrected Jeans mass is the minimum and is close to the observed value.In addition,we consider the impact of temperature T and the GUP parameters(η,βandγ)for the corrected Jeans mass.

Exploring the Implications of the Deformation Parameter and Minimal Length in the Generalized Uncertainty Principle

The breakdown of the Heisenberg Uncertainty Principle occurs when energies approach the Planck scale, and the corresponding Schwarzschild radius becomes similar to the Compton wavelength. Both of these quantities are approximately equal to the Planck length. In this context, we have introduced a model that utilizes a combination of Schwarzschild’s radius and Compton length to quantify the gravitational length of an object. This model has provided a novel perspective in generalizing the uncertainty principle. Furthermore, it has elucidated the significance of the deforming linear parameter β and its range of variation from unity to its maximum value.

Critical Analysis of the Origins of Heisenberg's Uncertainty Principle

Analysis of the initial stages of the logical process followed by Louis de Broglie in establishing the electron phase wave equation in his 1924 thesis, which triggered the development of Wave Mechanics when Erwin Schrödinger formalized this concept with his vectorial wave equation. This development was soon followed by Quantum Mechanics, when Schrödinger proved that the Matrix Mechanics independently developed by Werner Heisenberg was equivalent to Wave Mechanics, with both theories leaving room for some degree of uncertainty as to the physical localization of the moving electron. This is what led Heisenberg to also formalize the Uncertainty Principle to take this situation into account. This principle was soon regarded as a fundamental axiomatic principle that seemed to make further exploration of the subatomic level of magnitude appear impossible to most researchers. We will analyze in this article the reason why the phase-wave velocity established by de Broglie generated this uncertainty in the localization of the moving electron in light of the current state of knowledge on the behavior of the electron in motion, in view of establishing the relevance of maintaining the Uncertainty Principle in the study of the subatomic level of magnitude.

Quantitative Uncertainty Principles for the Canonical Fourier-Bessel Transform

The aim of this paper is to establish an extension of quantitative uncertainty principles and an algorithm for signal recovery about the essential supports related to a Bessel type of(LCT)so called canonical Fourier-Bessel transform.

Non-commutative Rényi entropic uncertainty principles

In this paper,we calculate the norm of the string Fourier transform on subfactor planar algebras and characterize the extremizers of the inequalities for parameters 0<p,q≤∞.Furthermore,we establish Renyi entropic uncertainty principles for subfactor planar algebras.

Generalized uncertainty principle from long-range kernel effects:The case of the Hawking black hole temperature

We prove the existence of an analogy between spatial long-range interactions,which are of the convolution-type introduced in non-relativistic quantum mechanics,and the generalized uncertainty principle predicted from quantum gravity theories.As an illustration,black hole temperature effects are discussed.It is observed that for specific choices of the moment's kernels,cold black holes may emerge in the theory.

Squaring the Circle Is Possible When Taking into Consideration the Heisenberg Uncertainty Principle

Squaring the circle is one of the oldest challenges in mathematical geometry. In 1882, it was proven that π was transcendental, and the task of squaring the circle was considered impossible. Demonstrating that squaring the circle was not possible took place before discovering quantum mechanics. The purpose of this paper is to show that it is actually possible to square the circle when taking into account the Heisenberg uncertainty principle. The conclusion is clear: it is possible to square the circle when taking into account the Heisenberg uncertainty principle.

Thermodynamics in a quantum corrected Reissner-Nordstr?m-AdS black hole and its GUP-corrections

We calculate the thermodynamic quantities in the quantum corrected Reissner-Nordstr?m-AdS(RN-AdS)black hole,and examine their quantum corrections.By analyzing the mass and heat capacity,we give the critical state and the remnant state,respectively,and discuss their consistency.Then,we investigate the quantum tunneling from the event horizon of massless scalar particle by using the null geodesic method,and charged massive boson W^(±)and fermions by using the Hamilton-Jacob method.It is shown that the same Hawking temperature can be obtained from these tunneling processes of different particles and methods.Next,by using the generalized uncertainty principle(GUP),we study the quantum corrections to the tunneling and the temperature.Then the logarithmic correction to the black hole entropy is obtained.

Heuristic Estimation of the Vacuum Energy Density of the Universe: Part II-Analysis Based on Frequency Domain Electromagnetic Radiation

In Part I of this paper, an inequality satisfied by the vacuum energy density of the universe was derived using an indirect and heuristic procedure. The derivation is based on a proposed thought experiment, according to which an electron is accelerated to a constant and relativistic speed at a distance L from a perfectly conducting plane. The charge of the electron was represented by a spherical charge distribution located within the Compton wavelength of the electron. Subsequently, the electron is incident on the perfect conductor giving rise to transition radiation. The energy associated with the transition radiation depends on the parameter L. It was shown that an inequality satisfied by the vacuum energy density will emerge when the length L is pushed to cosmological dimensions and the product of the radiated energy, and the time duration of emission is constrained by Heisenberg’s uncertainty principle. In this paper, a similar analysis is conducted with a chain of electrons oscillating sinusoidally and located above a conducting plane. In the thought experiment presented in this paper, the behavior of the energy radiated by the chain of oscillating electrons is studied in the frequency domain as a function of the length L of the chain. It is shown that when the length L is pushed to cosmological dimensions and the energy radiated within a single burst of duration of half a period of oscillation is constrained by the fact that electromagnetic energy consists of photons, an inequality satisfied by the vacuum energy density emerges as a result. The derived inequality is given by where is the vacuum energy density. This result is consistent with the measured value of the vacuum energy density, which is 5.38 × 10-10 J/m. The result obtained here is in better agreement with experimental data than the one obtained in Part I of this paper with time domain radiation.

Unruh Metric Tensor HUP via Planckian Space-Time Compared to HUP Based Complexity of Measured System Results to Obtain Inflaton Potential Magnitude

First of all, we restate a proof of a highly localized special case of a metric tensor uncertainty principle first written up by Unruh. Unruh did not use the Roberson-Walker geometry which we do, and it so happens that the dominant metric tensor we will be examining, is variation in δgtt. The metric tensor variations given by δgrr, δgθθand δgϕϕare negligible, as compared to the variation δgtt. Afterwards, what is referred to by Barbour as emergent duration of time δtis from the Heisenberg Uncertainty principle (HUP) applied to δgttin such a way as to be compared with ΔxΔp≥ℏ2+γ˜∂C∂Vwith V here a volume spatial term and γ˜a complexification strength term and ∂C∂Vinfluence of complexity of physical system being measured in order to obtain a parameterized value for the initial value of an inflaton which we call V0.

Uncertainty Principle for the Quaternion Linear Canonical Transform in Terms of Covariance

An uncertainty principle(UP),which offers information about a signal and its Fourier transform in the time-frequency plane,is particularly powerful in mathematics,physics and signal processing community.Under the polar coordinate form of quaternion-valued signals,the UP of the two-sided quaternion linear canonical transform(QLCT)is strengthened in terms of covariance.The condition giving rise to the equal relation of the derived result is obtained as well.The novel UP with covariance can be regarded as one in a tighter form related to the QLCT.It states that the product of spreads of a quaternion-valued signal in the spatial domain and the QLCT domain is bounded by a larger lower bound.

Generalized Uncertainty Principle and Black Hole Entropy of Higher-Dimensional de Sitter Spacetime

Recently, there has been much attention devoted to resolving the quantum corrections to the Bekenstein-- Hawking black hole entropy. In particular, many researchers have expressed a vested interest in the coetticient of the logarithmic term of the black hole entropy correction term. In this paper, we calculate the correction value of the black hole entropy by utilizing the generalized uncertainty prlnciple and obtain the correction term caused by the generalized uncertainty principle. Because in our calculation we think that the Bekenstein-Hawking area theorem is still valid after considering the generalized uncertainty principle, we derive that the coefficient of the logarithmic term of the black hole entropy correction term is positive. This result is different from the known result at present. Our method is valid not only for four-dimensional spacetimes but also for higher-dimensional spacetimes. In the whole process, the physics idea is clear and calculation is simple. It offers a new way for studying the entropy correction of the complicated spacetime.

Generalized Uncertainty Principle and Thermodynamic Quantities of SAdS_5 Black Hole

Recently, there has been much attention devoted to resolving the quantum corrections to the Bekenstein- Hawking black hole entropy. The different correction leading terms are obtained by the different methods. In this paper, we calculate the correction to SAdS5 black hole thermodynamic quantity due to the generalized uncertainty principle. Furthermore we derive that the black hole entropy obeys Bekenstein Hawking area theorem. The entropy has infinite correction terms. And every term is finite and calculable. The corrected Cardy-Vedinde formula is derived. In our calculation, Bekenstein Hawking area theorem still holds after considering the generalized uncertainty principle. We have not introduced any hypothesis. The calculation is simple. Physics meaning is clear. We note that our results are quite general. It is not only valid for four-dimensional spacetime but also for higher-dimensional SAdS spacetime.

Generalized uncertainty principle and tunneling radiation of the SAdS_5 black hole

After considering the generalized uncertainty principle, we discuss the quantum tunneling radiation of a fivedimensional Sehwarzschild anti de Sitter black hole. The radiation spectrum and the correction value of the Bekenstein-- Hawking entropy are derived. In a five-dimensional black hole the one order correction term in the Bekenstein-Hawking entropy correction term is proportional to the third power of the area, and the logarithmic correction term is a twoorder small quantity. The correction term is related to the dimension constant introduced in the generalized uncertainty principle. Because the black hole entropy is not divergent, the lowest value of the five-dimensional Schwarzschild anti de Sitter black hole horizon radius is obtained. After considering the generalized uncertainty principle, the radiation spectrum is still consistent with normalization theory.

Novel uncertainty relations associated with fractional Fourier transform

In this paper the relations between two spreads, between two group delays, and between one spread and one group delay in fractional Fourier transform （FRFT） domains, are presented and three theorems on the uncertainty principle in FRFT domains are also developed. Theorem 1 gives the bounds of two spreads in two FRFT domains. Theorem 2 shows the uncertainty relation between two group delays in two FRFT domains. Theorem 3 presents the crossed uncertainty relation between one group delay and one spread in two FRFT domains. The novelty of their results lies in connecting the products of different physical measures and giving their physical interpretations. The existing uncertainty principle in the FRFT domain is only a special ease of theorem 1, and the conventional uncertainty principle in time-frequency domains is a special case of their results. Therefore, three theorems develop the relations of two spreads in time-frequency domains into the relations between two spreads, between two group delays, and between one spread and one group delay in FRFT domains.

Generalized Uncertainty Inequalities on Fisher Information Associated with LCT

Uncertainty principle plays an important role in multiple fields such as physics,mathem-atics,signal processing,etc.The linear canonical transform(LCT)has been used widely in optics and information processing and so on.In this paper,a few novel uncertainty inequalities on Fisher information associated with linear canonical transform are deduced.These newly deduced uncer-tainty relations not only introduce new physical interpretation in signal processing,but also build the relations between the uncertainty lower bounds and the LCT transform parameters a,b,c and d for the first time,which give us the new ideas for the analysis and potential applications.In addi-tion,these new uncertainty inequalities have sharper and tighter bounds which are the generalized versions of the traditional counterparts.Furthermore,some numeric examples are given to demon-strate the efficiency of these newly deduced uncertainty inequalities.

Generating Real Random Numbers with Uncertainty Principle

The real random number generation is a critical problem in computer science.The current generation methods are either too dangerous or too expensive,such as using decays of some radioactive elements.They are also hard to control.By the declaration of uncertainty principles in quantum mechanics,real probabilistic events can be substituted by easier and safer processes,such as electron diffraction,photon diffraction and qubits.The key to solve the problem of Schr?dinger’s cat is to identify that the atom stays in different states after and before the decay,and the result of the decay is probabilistic according to the wave packet co llapse hypothesis.Same matter is able to possess different kinds of properties such as wave-particle duality due to that it can stay in various states,and which state will the matter stay is determined by the chosen set of physical quantities(or mechanical quantities).One eigenstate of a set of physical quantities can be a superpos ition of other eigenstates of different sets of physical quantities,and the collapse from a superposition to an eigenstate it contains is really random.Using this randomness,real random number can be generated more easily.

Addition to the Article with Stepan Moskaliuk on the Inter Relationship of General Relativity and (Quantum) Geometrodynamics, via Use of Metric Uncertainty Principle

We take note of the material offered in [1] as to Geometrodynamics as a way to quantify an inter relationship between a quantum style Heisenberg uncertainty principle for a metric tensor and conditions postulated as to a barotropic fluid, i.e. dust for early universe conditions. By looking at the onset of processes at/shorter than a Planck Length, in terms of initial expansion of the universe, we use inputs from the metric tensor as a starting point for the variables used in Geometrodynamics.