Galilei Group Research Articles

Observables and Dynamics Quantum to Classical from a Relativity Symmetry and Noncommutative-Geometric Perspective

With approaching quantum/noncommutative models for the deep microscopic spacetime in mind, and inspired by our recent picture of the (projective) Hilbert space as the model of physical space behind basic quantum mechanics, we reformulate here the WWGM formalism starting from the canonical coherent states and taking wavefunctions as expansion coefficients in terms of this basis. This provides us with a transparent and coherent story of simple quantum dynamics where both the wavefunctions for the pure states and operators acting on them arise from the single space/algebra, which exactly includes the WWGM observable algebra. Altogether, putting the emphasis on building our theory out of the underlying relativity symmetry -- the centrally extended Galilean symmetry in the case at hand -- allows one to naturally derive both a kinematical and a dynamical description of a quantum particle, which moreover recovers the corresponding classical picture (understood in terms of the Koopman-von Neumann formalism) in the appropriate (relativity symmetry contraction) limit. Our formulation here is the most natural framework directly connecting all of the relevant mathematical notions and we hope it may help a general physicist better visualize and appreciate the noncommutative-geometric perspective behind quantum physics. It also helps to inspire and illustrate our perspective on looking at quantum mechanics and quantum physics in general in direct connection to the notion of quantum (deformed) relativity symmetries and the corresponding quantum/noncommutative models of spacetime as various levels of approximations all the way down to the Newtonian.

Galilean Transformations of Kinetic Energy, Work, and Potential Energy

Work, kinetic energy (KE), and potential energy (PE) are key physics concepts, taught in essentially every introductory physics course along with the physical laws such as the work-kinetic energy theorem and the conservation of mechanical energy, which describe how the values of those quantities change during various processes. Quite appropriately for introductory courses, little attention is paid to how those concepts play out when implemented via observations in reference frames moving with respect to each other.

Bulk-Surface Electrothermodynamics and Applications to Electrochemistry.

We propose a modeling framework for magnetizable, polarizable, elastic, viscous, heat conducting, reactive mixtures in contact with interfaces. To this end, we first introduce bulk and surface balance equations that contain several constitutive quantities. For further modeling of the constitutive quantities, we formulate constitutive principles. They are based on an axiomatic introduction of the entropy principle and the postulation of Galilean symmetry. We apply the proposed formalism to derive constitutive relations in a rather abstract setting. For illustration of the developed procedure, we state an explicit isothermal material model for liquid electrolyte|metal electrode interfaces in terms of free energy densities in the bulk and on the surface. Finally, we give a survey of recent advancements in the understanding of electrochemical interfaces that were based on this model.

Unification of Newtonian Physics with Einstein Relativity Theory by using Generalized Metrics of Complex Spacetime and application to the Motions of Planets and Stars, eliminating Dark Matter

In this paper we unify Theories of Physics (TPs) with real or infinite Universal Speed (cI), such as Einstein Relativity Theory (ERT) and Newtonian Physics (NPs). Generalized Special Relativity (SR) relates the frames of Relativistic Inertial observers (RIOs) where the spacetime has the metric gI=diag(gI00, gI11, gI11, gI11). The parameter ξI=sqrt(-gI11/gI00) is contained in the matrix (ΛI) of the Euclidean Closed Linear Transformation of complex spacetime (ECLSTT). Besides the elements of ΛI are complex numbers. So, the corresponding spacetime is necessarily complex and there exists real cI=c/ξI. In addition, the complex Cartesian Coordinates (CCs) of the theory, may be turned to the corresponding real CCs, in order to be perceived by human senses. The new real transformation is not closed (the corresponding real matrices ΛIR do not form a group) and the successive real transformations produce Generalized Thomas Rotation. The specific value ξI=1 gives Vossos transformation (VT) endowed with Lorentz metric (for gI11=1) of complex spacetime and invariant speed of light in vacuum (cI=c), which produce the Lorentzian version of Euclidean Complex Relativistic Mechanics (ECRMs). The corresponding real matrix (ΛIR) is the matrix of Lorentz Boost (ΛIR=ΛL). The specific value ξI→0 gives Galilean Transformation (GT) with invariant time, in which any other ECLSTT is reduced, if one RIO has small velocity wrt another RIO. Thus we unify TPs such as NPs and ERT, keeping the formalism of ERT. The generalized definition of Proper Time (r) gives us the possibility to compute four-velocity, four-momentum, Relativistic Doppler Shift etc, building the whole structure of the Generalized SR and General Relativity (GR). For instance the Generalized Relativistic Energy of Rest Mass (m) is Erest=mc2/ξI2. In case of NPs, the annihilation energy becomes infinite. Thus the Lomonosov-Lavoisier Law becomes clear theorem of NPs. In addition, the case of observers with variable metric of spacetime leads to GR. Thus we produce the 1st Generalized Schwarzschild metric (1GSM) and 2nd Generalized Schwarzschild metric (2GSM), which are in accordance with Einstein field equations for any TPs. In case of 1GSM, we compute the corresponding Lagrangian, geodesics, equations of motion, precession of planets’ orbits etc, resulting formulas which are referred to any TPs. We then combine the theoretical results to the experimental data of our Solar system, producing a set of valid values of ξI. In case of 2GSM, the combination of its Galilean version with Modified Newtonian Dynamics (MOND), leads to MOND relativization. After, we pass to RIOs with ordinary flat spacetime (Minkowski space), extending MOND methods to ERT. We use Simple and Standard Interpolating Function (μ) to the Lorentzian-Einsteinian version of 2GSM for the explanation of Rotation Curves in Galaxies as well as the Solar system, eliminating Dark Matter. Generally, this approach, in non rotating black hole, planetary and star system scale, coincides to the original Schwarzschild metric, while in galactic scale, it gives MONDian results. In universal scale, the gravitational field strength becomes negative, producing slight antigravity.

About the non-standard viewpoint on the dynamics of closed vortex filament

In this paper, we construct the Hamiltonian description of the closed vortex filament dynamics in terms of non-standard variables, phase space and constraints. The suggested approach makes obvious interpretation of the considered system as a structured particle that possesses certain external and internal degrees of the freedom. The constructed theory is invariant under the transformation of Galilei group. The appearance of this group allows for a new viewpoint on the energy of a closed vortex filament with zero thickness. The explicit formula for the effective mass of the structured particle “closed vortex filament” is suggested.

Solution of the dynamic frictional contact problem between a functionally graded coating and a moving cylindrical punch

This paper presents an analytical method developed to investigate the dynamic frictional contact mechanics between a functionally graded coating and a rigid moving cylindrical punch. Governing partial differential equations of elastodynamics are solved analytically by applying Galilean and Fourier transformations. Interface continuity and boundary conditions are written and contact problem is then reduced to a singular integral equation of the second kind. The singular integral equation is solved numerically by means of an expansion-collocation method. Developed solution procedures are verified through the comparisons made to the results available in the literature. Presented parametric analyses illustrate the effects of punch speed, coefficient of friction, material inhomogeneity and geometric parameters upon the contact stresses. It is shown that, especially at higher punch speeds, the difference between contact stresses obtained through elastodynamic and elastostatic solutions is rather significant. A formulation based on the elastodynamic theory, as presented in the current study, is required to compute more realistic contact stresses.

Letter: Galilean invariance of Rortex

A new vector named Rortex [C. Liu et al., “Rortex—A new vortex vector definition and vorticity tensor and vector decompositions,” Phys. Fluids 30, 035103 (2018)] was proposed to represent the local fluid rotation in our previous work. However, the Galilean invariance of Rortex is yet to be elaborated. In the present study, we prove that Rortex is invariant under the Galilean transformation and several examples are provided to confirm the conclusion.

Model of Mass Transfer Processes in a Mixture of Continua Consisting of One Deformable and Two Liquid Component

The proposed mathematical model is based on the theory of a mixture of interpenetrating continua: deformable (polymer) and two liquid continuums. The governing equations of the model are obtained as consequences of the laws of thermodynamics and the requirements of their invariance to Galilean transformations. Equations describing the motion of liquid components are formulated in coordinates related to the polymer component of the mixture. The need for such a choice arises as a result of the fact that only a polymer can be deformed. When solving problems, it is required to find polymer deformations and investigate the movement of solvents relative to it, including the release of solvents through the polymer boundary into the external environment.Material considered in this mathematical model is capable of working under conditions of finite deformations. The expression of the free energy of the mixture takes into account the energy of interaction of the molecules of the mixture with each other (polymer and two solvents).

On sliding interface contact in layered smart structures

• A dynamic, rough contact model is developed for smart structures. • Partial contact conditions and limiting cases are concerned. • The electric field and magnetic field are given explicitly. • The critical loading to achieve the full contact is determined. • Influences of the velocity and the gap are demonstrated. Based on the linear piezoelectricity and magnetoelectroelasticity theories, this study presents an exact solution for a frictionless, dynamic periodic contact problem of smart structures. Anisotropic piezoelectric materials and anisotropic magneto-electro-elastic materials both with wavy surfaces moves smoothly. The Galilean transformation and the dual series equation methods are used to derive exact solutions for dynamic wavy surface contact problem. The stresses, displacements, electric field and magnetic field are given explicitly for partial contact conditions. The influences of the velocity and the gap at the crests of the wavy surfaces on various physical quantities are detailed for the selected material coefficients of piezoelectric materials PVDF and magneto-electro-elastic materials BaTiO 3 − CoFe 2 O 4 . The illustrative examples imply that a higher critical loading leads to a full contact under two conditions: (1) at a higher velocity or (2) with an increased gap at the crests of the wavy surfaces. The largest magnitude of the surface normal stresses can be relieved by either lowering the velocity or reducing the gap at the crests of the wavy surfaces. The largest magnitudes of the surface electrical displacements and magnetic inductions occur at the edges of the contact region with spikes occurring, which suggests that the edges of the contact region are the likely locations of potential crack initiation and propagation.

Effective field theory of a vortex lattice in a bosonic superfluid

Using boson-vortex duality, we formulate a low-energy effective theory of a two-dimensional vortex lattice in a bosonic Galilean-invariant compressible superfluid. The excitation spectrum contains a gapped Kohn mode and an elliptically polarized Tkachenko mode that has quadratic dispersion relation at low momenta. External rotation breaks parity and time-reversal symmetries and gives rise to Hall responses. We extract the particle number current and stress tensor linear responses and investigate the relations between them that follow from Galilean symmetry. We argue that elementary particles and vortices do not couple to the spin connection which suggests that the Hall viscosity at zero frequency and momentum vanishes in a vortex lattice.

Confined dynamical systems with Carroll and Galilei symmetries

We introduce a general method to construct classes of dynamical systems invariant under generalizations of the Carroll and of the Galilei groups. The method consists in starting from a space-time in $D+1$ dimensions and partitioning it in two parts, the first Minkowskian and the second Euclidean. Tha action consist of two terms that are separately invariant the Minkwoskian and Euclidean partitioning. One of those contains a system of lagrangian multiplies that confine the system to a subspace. The other term defines the dynamics of the system. The total lagrangian is invariant under the Carroll or the Galilei groups with zero central charge.

Non-classical laws with conserved quantities from the arbitrary functions included in general solution to elasticity and application

• Noether symmetries are discussed by standard techniques not for the Lagrangian density of an elastic body. • The deep-level symmetries of an elastic field from arbitrary functions included in general solution. • Conserved quantities with arbitrary integer order spatial derivative or integral. • An invariant finite physical parameter can be obtained for arbitrary integer order singularity field. The general solution to static and/or dynamic linear elasticity is a transformation between the displacements and new arbitrary functions, whose conservativeness depends on some independent partial differential equations (PDEs) satisfied by the new arbitrary functions. Zhang's general solutions are mathematically appropriate since the displacements are expressed in terms of two new arbitrary functions, and the sum of the highest order derivative added together from the independent PDEs satisfied by the two new arbitrary functions is the same as that of Navier–Cauchy equations. Therefore, the following points should be emphasized: (i) the independent PDEs come from the Laplace and D'Alembert operators acting on the two new arbitrary functions in static and dynamic general solutions, respectively, and it is found that the two new arbitrary functions are related to the rotations, first strain invariant and distortion; (ii) especially, conservation laws constructed from the equations satisfied by the spatial integrals of functions hold true, although some arbitrary functions of the spatial integrals have been canceled. Based on these facts, since Noether's identity not only can be applied to a Lagrangian but also can be used to construct a functional for widespread PDEs, the functionals relating to the rotations, first strain invariant and distortion are constructed with arbitrary integer order spatial derivative or integral, and the conservation laws follow. This kind of non-classical conservation laws does not come from the Lagrangian density of an elastic body and belongs to the deep-level natures of symmetries of elastic field derived by standard techniques. Availability is shown by two examples, from which the field intensity of a vertical load applied to the surface of an elastic half-space and the path-independent integrals in a coordinate system moving with Galilean transformation are presented for comparison.

De Broglie Wave Analysis of the Heisenberg Uncertainty Minimum Limit under the Lorentz Transformation

A simple analysis using differential calculus has been done to consider the minimum limit of the Heisenberg uncertainty principle in the relativistic domain. An analysis is made by expressing the form of and based on the Lorentz transformation, and their corresponding relation according to the de Broglie wave packet modification. The result shows that in the relativistic domain, the minimum limit of the Heisenberg uncertainty is p x ?/2 and/or E t ?/2, with is the Lorentz factor which depend on the average/group velocity of relativistic de Broglie wave packet. While, the minimum limit according to p x ?/2 or E t ?/2, is the special case, which is consistent with Galilean transformation. The existence of the correction factor signifies the difference in the minimum limit of the Heisenberg uncertainty between relativistic and non-relativistic quantum. It is also shown in this work that the Heisenberg uncertainty principle is not invariant under the Lorentz transformation. The form p x ?/2 and/or E t ?/2 are properly obeyed by the Klein-Gordon and the Dirac solution. 
 Key words: De Broglie wave packet, Heisenberg uncertainty, Lorentz transformation, and minimum limit.

Moment analysis for reaction kinetics of intermolecular interactions.

Moment equations were developed on the basis of the principle of relativity for analyzing elution peak profiles measured by ACE to analytically determine the association (ka ) and dissociation (kd ) rate constants of intermolecular interactions. Basic equations representing the mass balance, mass transfer rate, and reaction kinetics in ACE system in a Galilean coordinate system S were transformed to those in another coordinate system S', which imaginarily moved with respect to S. Moment equations for ACE peaks in S' in the time domain were derived from the analytical solution of the modified basic equations in the Laplace domain. Moment equations for ACE peaks in S were derived from those in S' by the inverse Galilean transformation. The moment equations were used for analyzing some ACE data previously published to determine ka and kd values. It was demonstrated that the moment equations were effective for extracting the information about affinity kinetics of intermolecular interactions from the elution peak profiles measured by ACE. The moment equations were also used to discuss the influence of mass transfer and reaction kinetics on ACE peak profiles. Some results of the numerical calculations are also indicated in Supporting Information.

Moment analysis for mass transfer kinetics at the interface of spherical molecular aggregates

New moment equations were developed on the basis of the principle of relativity for explaining some characteristics of elution peaks measured by electrokinetic chromatography (EKC) using spherical molecular aggregates. Basic equations representing mass balance and mass transfer kinetics in EKC system in a Galilean coordinate system S were transformed to those in another coordinate system S’, which imaginarily moved with respect to S. Moment equations for EKC peaks in S’ in the time domain were derived from the analytical solution of the modified basic equations in the Laplace domain. Moment equations for EKC peaks in S were derived from those in S’ by the inverse Galilean transformation. The moment equations were used to the re-analysis of EKC data previously measured. The values of permeation rate constants of thymol at the interface of sodium dodecylsulfate micelles were fairly in agreement with those determined in a previous study. The moment equations were also used to the numerical simulation of elution peaks in EKC systems. The influence of some experimental parameters on elution peak profiles was quantitatively analyzed. The moment equations are useful for determining the rate constants of interfacial solute permeation from elution peak profiles measured by EKC.

Quantum Physics in Phase Space: an Analysis of Simple Pendulum

In this work we present a brief review about quantum mechanics in phase space. The approach discussed is based in the notion of symplectic structure and star-operators. In this sense, unitary representations for the Galilei group are construct, and the Schrodinger equation in phase space is derived. The connection between phase space amplitudes and Wigner function is presented. As a new result we solved the Schrodinger equation in phase space for simple pendulum. PACS Numbers: 11.10.Nx, 11.30.Cp, 05.20.Dd

Covariant Galilean versus Carrollian hydrodynamics from relativistic fluids

We provide the set of equations for non-relativistic fluid dynamics on arbitrary, possibly time-dependent spaces, in general coordinates. These equations are fully covariant under either local Galilean or local Carrollian transformations, and are obtained from standard relativistic hydrodynamics in the limit of infinite or vanishing velocity of light. All dissipative phenomena such as friction and heat conduction are included in our description. Part of our work consists in designing the appropriate coordinate frames for relativistic spacetimes, invariant under Galilean or Carrollian diffeomorphisms. The guide for the former is the dynamics of relativistic point particles, and leads to the Zermelo frame. For the latter, the relevant objects are relativistic instantonic space-filling branes in Randers–Papapetrou backgrounds. We apply our results for obtaining the general first-derivative-order Galilean fluid equations, in particular for incompressible fluids (Navier–Stokes equations) and further illustrate our findings with two applications: Galilean fluids in rotating frames or inflating surfaces and Carrollian conformal fluids on two-dimensional time-dependent geometries. The first is useful in atmospheric physics, while the dynamics emerging in the second is governed by the Robinson–Trautman equation, describing a Calabi flow on the surface, and known to appear when solving Einstein’s equations for algebraically special Ricci-flat or Einstein spacetimes.

Person running in the rain with an umbrella: invariance of distances and angles under a Galilean transformation

It is a common experience that raindrops falling vertically appear to fall at an angle to a moving person. When trying to describe the sheltered area provided by an umbrella from the viewpoints of both moving and stationary observers, at first glance it may appear that the descriptions for the sheltered area in these two frames are not identical. It is shown as well as illustrated by simulation that the sheltered areas are indeed identical in both these frames as expected. The physics behind this is explained and in the process confirming distances, angles, and consequently areas, remain invariant under a Galilean transformation. The detailed treatment of the problem given in this paper may help undergraduate students gain a robust understanding of the important concepts of frame of reference and Galilean transformation.

Dynamic stress concentration in pre-stressed poroelastic media due to moving punch influenced by shear wave

During the occurrence of earthquake, the shear wave propagates in the rocks present inside/at the Earth’s crust. The propagation of shear wave may lead to the progression of punch present inside the rock medium. As a result of this, substantial stress accumulated at the vicinity of propagating punch inside rock medium which significantly affects the stability of various geological and human-made structure and, hence, may cause failure of structure. Therefore, the analysis of stress concentration at the vicinity of punch moving due to shear wave propagation has become prominent in the area of seismology. In the present paper, an analytical perspective has been employed to discuss the influence of velocity of moving punch associated with the propagation of shear wave on developed dynamic stress concentration (DSC) in three types of pre-stressed vertical transversely isotropic (VTI) poroelastic media viz. granite (an igneous rock); sandstone (a sedimentary rock); and marble (a metamorphic rock). The closed form expression of DSC for the force of constant intensity has been derived with the aid of Weiner-Hopf technique along with Galilean and two-sided Fourier integral transformations. The noticeable influence of different affecting parameters (viz. velocity of moving punch associated with the shear wave propagation, horizontal compressive/tensile initial stresses, vertical compressive/tensile initial stress, porosity, and anisotropy parameter) on dynamic stress concentration has also been reported. Numerical computation and graphical illustrations have been carried out for the aforementioned three different types of porous rocks to investigate the profound impact of affecting parameters on DSC. Moreover, some noteworthy peculiarities have also been derived from the obtained expression of dynamic stress concentration.

Special relativity: Its inconsistency with the standard wave equation

By means of the Lorentz Transformation, Einstein’s Special Theory of Relativity purports invariance of the standard wave equation. Counter-examples, satisfying the Lorentz Transformation, and hence Lorentz Invariance, prove that the Lorentz Transformation does not in fact produce invariance of the standard wave equation. Systems of clock-synchronised stationary observers are Galilean and necessarily transform by the Galilean Transformation. Einstein’s insistence that inertial (i.e. Galilean) systems of clock-synchronised stationary observers transform, not by the Galilean Transformation, but by the non-Galilean Lorentz Transformation, is logically inconsistent. The Special Theory of Relativity is therefore logically inconsistent. Therefore, it is false. The Lorentz Transformation is meaningless.