This study presents wavelets-computational aspects of Sterian-realistic approach to uncertainty principle in high energy physics. According to this approach, one cannot make a device for the simultaneous measuring of the canonical conjugate variables in reciprocal Fourier spaces. However, such aspects regarding the use of conjugate Fourier spaces can be also noticed in quantum field theory, where the position representation of a quantum wave is replaced by momentum representation before computing the interaction in a certain point of space, at a certain moment of time. For this reason, certain properties regarding the switch from one representation to another in these conjugate Fourier spaces should be established. It is shown that the best results can be obtained using wavelets aspects and support macroscopic functions for computing (i) wave-train nonlinear relativistic transformation, (ii) reflection/refraction with a constant shift, (iii) diffraction considered as interaction with a null phase shift without annihilation of associated wave, (iv) deflection by external electromagnetic fields without phase loss, and (v) annihilation of associated wave-train through fast and spatially extended phenomena according to uncertainty principle.
This study presents wavelets-computational aspects of Sterian-realistic approach to uncertainty principle in high energy physics. According to this approach, one cannot make a device for the simultaneous measuring of the canonical conjugate variables in reciprocal Fourier spaces. However, such aspects regarding the use of conjugate Fourier spaces can be also noticed in quantum field theory, where the position representation of a quantum wave is replaced by momentum representation before computing the interaction in a certain point of space, at a certain moment of time. For this reason, certain properties regarding the switch from one representation to another in these conjugate Fourier spaces should be established. It is shown that the best results can be obtained using wavelets aspects and support macroscopic functions for computing (i) wave-train nonlinear relativistic transformation, (ii) reflection/refraction with a constant shift, (iii) diffraction considered as interaction with a null phase shift without annihilation of associated wave, (iv) deflection by external electromagnetic fields without phase loss, and (v) annihilation of associated wave-train through fast and spatially extended phenomena according to uncertainty principle.
Due to the requirements of the principle of causality in the theory of relativity, one cannot make a device for the simultaneous measuring of the canonical conjugate variables in the conjugate Fourier spaces. Instead of admitting that a particle’s position and its conjugate momentum cannot be accurately measured at the same time, we consider the only probabilities which can be determined when working at subatomic level to be valid. On the other hand, based on Schwinger's action principle and using the quadridimensional form of the unitary transformation generator function of the quantum operators in the paper, the general form of the evolution equation for these operators is established. In the nonrelativistic case one obtains the Heisenberg's type evolution equations which can be particularized to derive Heisenberg's uncertainty relations. The analysis of the uncertainty relations as implicit evolution equations allows us to put into evidence the intrinsic nature of the correlation expressed by these equations in straight relations with the measuring process. The independence of the quantisation postulate from the causal evolution postulate of quantum mechanics is also put into discussion.
The starting point of this research is a representation of a quantum particle according to the Schrodinger equation of the conventional quantum mechanics. In this representation, a quantum particle is described by a wave packet in the coordinate space and the conjugated wave packet in the momentum space. The problem is that while in the coordinate space the group velocity is in agreement with one of the Hamilton equations, the group velocity in the momentum space is in contradiction with the other Hamilton equation - a minus sign is missing. Group velocities in agreement with the Hamilton equations are obtained only when in the time dependent phase of a quantum particle wave packet, instead of the Hamiltonian coming from the conventional Schrodinger equation, the Lagrangian is considered. This suggests us to consider the relativistic Lagrangian in the time dependent phase. In this way, the conventional relativistic principle of invariance of the time-space interval gets the more physically understandable form of the invariance of the time dependent phase of a quantum particle - the time dependent phase of a quantum particle is the same in any system of coordinates. Based on the relativistic time dependent phase invariance of a quantum particle, from the group velocities of this particle the relativistic kinematics and dynamics are obtained. The interaction with an electromagnetic field is described by a modification of the time dependent phase with a scalar potential conjugated to time and a vector potential conjugated to the space coordinates. In this paper, we find that, according to the formalism of the general relativity, any matter element in a field of forces is accelerated only perpendicularly to its velocity. This means that the matter propagation of a quantum particle can be conceived in planes perpendicular to velocity, while the mater distribution can be considered in a Fourier representation - quantum waves. We study the quantum particle dynamics in gravitational waves and derive the graviton spin. We consider a quantum particle in electromagnetic field and obtain a Schrödinger-Dirac-type equation, with additional terms describing the velocity dependence of the particle dynamics.
Given a pair of conjugate variables x and y (time and frequency in signal theory, position and momentum in quantum mechanics) which cannot be fixed together, we have built a family of wavelets satisfying the following properties. Each wavelet is characterized by a pair of mean values xˉ, yˉ lying at a site of a lattice in the plane x, y (information plane or phase space); the wavelets are deduced from one another by the translation of this lattice; their family constitutes an orthonormal complete basis; each one is as localized as possible around xˉ, yˉ in both x and y. Such a paving of the plane x, y provides a representation of any wave, which specifies both conjugate variables at best. The orthogonality constraint implies however the divergence of the uncertainty Δx.Δy for each wavelet. The implications on measurement theory of this strong uncertainty principle are discussed. French abstract: Etant donne une paire de variables conjuguees x et y (temps et frequence en theorie du signal, position et impulsion en mecanique quantique) qui ne peuvent etre precisees a la fois, on a construit une famille d'ondelettes ayant les proprietes suivantes. Chacune est caracterisee par la paire des valeurs moyennes xˉ, yˉ, disposee en un nœud d'un reseau du plan x, y (plan d'information ou espace des phases); les ondelettes se deduisent l'une de l'autre par les translations de ce reseau; leur famille constitue une base complete orthonormee; chacune est aussi localisee que possible autour de xˉ, yˉ, a la fois en x et y. Le pavage du plan x, y ainsi constituee peut servir a representer toute onde en specifiant au mieux les deux variables conjuguees. La contrainte d'orthogonalite impose cependant que l'incertitude Δx.Δy relative a chaque ondelette diverge. On discute des implications de ce principe d'incertitude fort sur la theorie de la mesure.
Based on the hypothesis that the position-representation of a physical state <x|α> is the Fourier transform of its momentum-representation <P|α> and that the time-representation <t|α> is the inverse Fourier transform of its energy-representation <E|α>, we are able to find the Planck relation E = hv, the de Broglie relation P = λh. Afterward, utilizing the Dirac delta function δ(x) and the property FTxf(x) = -iℏ∂, pFTxf(x) = -iℏ∂, pF(p) we obtain the links between operators in ordinary and Hilbert spaces leading to the Dirac fundamental commutation relation [p^,x^] = -iℏI^, the Schrödinger equations, the Heisenberg uncertainty principle in quantum mechanics, the annihilation & creation of a photon from excitation & de-excitation of an atom according to Bohr.
We obtain both a global and local uncertainty principle for the short-time Fourier transform. Explicit expressions are obtained for the uncertainty product in terms of the uncertainty product of the signal and window. Although the uncertainty product of the spectrogram is not the sum of the uncertainty products of the window and signal, nonetheless, we show that the global uncertainty product has to be greater than one. We also derive local uncertainty relations which indicate how the local spread in time and frequency are related at a particlar time–frequency point. In addition, we obtain general results for the time, scale, and frequency moments of the scalogram in terms of the moments of the signal and mother wavelet. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
In the field of atomic and solid state physics, wavelet analysis has been applied so far in three different directions: (i) time-frequency analysis of harmonic generation in laser-atom interactions; (ii) ab initio electronic structure calculations in atoms and molecules; and (iii) construction of localized bases for the lowest Landau level of a 2-D electron gas submitted to a strong magnetic field. We survey these three types of applications, with more emphasis on methods than on precise results. Introduction There are two ways in which wavelets could play a role in atomic physics and possibly in solid state physics. First one may envisage them as physical objects , namely quantum states or wave functions. It is commonplace to remark that coherent states (CS) have a privileged role in atomic physics. Laser-atom interactions, revival phenomena, Rydberg wave packets and various semi-classical situations are all instances in which a coherent state description is clearly well-adapted. Of course, what is implied here are canonical CS, associated to the harmonic oscillator or the electromagnetic field [36]. But wavelets are also coherent states, namely those associated to the affine groups in various space dimensions, as we have seen in Chapter 2 (see [1] for a review on coherent states). Thus wavelets could well be thought of as convenient substitutes for canonical CS. However, this suggestion is still speculative at the present moment, very little has been achieved in this direction.