We explore the relationship between mechanical systems describing the motion of a particle with the mechanical systems describing a continuous medium. More specifically, we will study how the so-called intermediate integrals or fields of solutions of a finite dimensional mechanical system (a second order differential equation) are simultaneously Euler's equations of fluids and conversely. This will be done both in the classical and relativistic context. A direct relationship will be established by means of the so-called time constraint (classical unsteady case, static or not) and the relativistic correction (for arbitrary pseudo-Riemannian metrics).
We will prove that, in general, a system formed by several particles moving along relativistic trajectories can not be described by a mechanical system. The contradiction that leads to the previous assertion is due to the fact that a mechanical system defines a second order differential equation and this, in turn, induces an absolute time that will generally be incompatible with the proper times of the different particles.
It is shown that, under the Wentzel-Kramers-Brillouin approximation conditions, using the Foldy-Wouthuysen representation allows the problem of finding a classical limit of relativistic quantum mechanical equations to be reduced to the replacement of operators in the Hamiltonian and quantum mechanical equations of motion by the respective classical quantities.
This article elaborates on entanglement entropy and quantum information theory of geometric flows of (relativistic) Lagrange--Hamilton mechanical systems. A set of basic geometric and quantum mechanics and probability concepts together with methods of computation are developed in general covariant form for curved phase spaces modelled as cotangent Lorentz bundles. The constructions are based on ideas relating the Grigory Perelman's entropy for geometric flows and associated statistical thermodynamic systems to the quantum von Neumann entropy, classical and quantum relative and conditional entr…
A finite-dimensional pseudo-unitary framework is set up for describing the dynamics of free elementary particles in a purely relativistic quantum mechanical way. States of any individual particles or antiparticles are defined as suitably normalized vectors belonging to the two-complex-dimensional spaces that occur in local orthogonal decompositions of isomorphic copies of Cartan's space. The corresponding dynamical variables thus show up as bounded pseudo-Hermitian operator restrictions that possess real discrete spectra. Any measurement processes have to be performed locally in orthocronous p…
First, the properties of a classical model of spontaneous symmetry breakdown are analyzed. Then, the pros and cons of some pedagogical non-relativistic quantum-mechanical models, also used to illustrate spontaneous symmetry breakdown, are discussed. Finally, a simple quantum-mechanical toy model (a spinor on the line, with a spin-dependent interaction) is presented, that exhibits the spontaneous breaking of an internal symmetry.
The geodesic has a fundamental role in physics and in mathematics: roughly speaking, it represents the curve that minimizes the arc length between two points on a manifold. We analyze a basic but misinterpreted difference between the Lagrangian that gives the arc length of a curve and the one that describes the motion of a free particle in curved space. Although they provide the same formal equations of motion, they are not equivalent. We explore this difference from a geometrical point of view, where we observe that the non-equivalence is nothing more than a matter of symmetry. As application…
The advantages and disadvantages of some pedagogical non-relativistic quantum-mechanical models, used to illustrate spontaneous symmetry breakdown, are discussed. A simple quantum-mechanical toy model (a spinor on the line, subject to a magnetostatic interaction) is presented, that exhibits the spontaneous breakdown of an internal symmetry.
Being neither commutative nor associative, Einstein velocity addition of relativistically admissible velocities gives rise to gyrations. Gyrations, in turn, measure the extent to which Einstein addition deviates from commutativity and from associativity. Gyrations are geometric automorphisms abstracted from the relativistic mechanical effect known as Thomas precession.
After analyzing Dirac's equation, one can suggest that a well-known quantum-mechanical momentum operator is associated with relativistic momentum, rather than with non-relativistic one. Consideration of relativistic energy and momentum expressions allows us to define the non-relativistic, relativistic and pseudo-relativistic (present in Schrödinger equation) kinetic energy operators. Consequences of kinetic energy operator's correction for spectra of basic quantum Hamiltonians are investigated. In some cases this correction can produce remarkable spectra modifications.
A recent proposal to connect the loop quantization with the spin foam model for cosmology via the path integral is hereby adapted to the case of mechanical systems within the framework of the so called polymer quantum mechanics. The mechanical models we consider are deparametrized and thus the group averaging technique is used to deal with the corresponding constraints. The transition amplitudes are written in a vertex expansion form used in the spin foam models, where here a vertex is actually a jump in position. Polymer propagators previously obtained by spectral methods for a nonrelativisti…
The Feshbach-Villars equations, like the Klein-Gordon equation, are relativistic quantum mechanical equations for spin-$0$ particles. We write the Feshbach-Villars equations into an integral equation form and solve them by applying the Coulomb-Sturmian potential separable expansion method. We consider bound-state problems in a Coulomb plus short range potential. The corresponding Feshbach-Villars Coulomb Green's operator is represented by a matrix continued fraction.
The self adjoint operator of time in non-relativistic quantum mechanics is found within the approach where the ordinary Hamiltonian is not taken to be conjugate to time. The operator version of the reexpressed Liouville equation with the total Hamiltonian, consisting of the part that is a conventional function of coordinate and momentum and the part that is conjugate to time, is considered. The von Neumann equation with quantized time is found and discussed from the point of view of exact time measurement.
We show that the diagonal matrix elements $< Or^{p} >,$ where $O$ $={1,β,i\mathbf{αn}β}$ are the standard Dirac matrix operators and the angular brackets denote the quantum-mechanical average for the relativistic Coulomb problem, may be considered as difference analogs of the radial wave functions. Such structure provides an independent way of obtaining closed forms of these matrix elements by elementary methods of the theory of difference equations without explicit evaluation of the integrals. Three-term recurrence relations for each of these expectation values are derived as a by-product. Tr…
The classical tools which ensure the completeness of vector fields and second order differential equations for mechanical systems are revisited. Possible extensions in three directions are discussed: infinite dimensional Banach and Hilbert manifolds, Finsler metrics and pseudo-Riemannian spaces, including links with some relativistic spacetimes. Special emphasis is taken in the cleaning up of known techniques, the statement of open questions and the exploration of prospective frameworks.
An approximate quantum-mechanical two-body equation for spinless particles incorporating relativistic kinematics is derived. The derivation is based on the relativistic energy-momentum relation $mc^{2}+ε= \sqrt{m^{2}c^{4}+p^{2}c^{2}}+V$ for each single particle, where $mc^2$ is the particle rest mass energy, $p$ its linear momentum, $ε$ its dynamical energy, and $V$ being the time-like vector interaction potential. The resulting two-body equation assumes rapid wave oscillations in a single, slowly varying potential well. A Bohr-Sommerfeld-type quantization condition is obtained. The approximat…
In this paper, we consider two types of the scattering problems (relativistic case), namely, the stationary scattering problem, where the distance $r$ tends to infinity, and the dynamical scattering problem, where the time $t$ tends to infinity. Using our results on Lippmann-Schwinger equation in the relativistic case, we found the connection between the stationary scattering problem (the scattering amplitude) and the dynamical scattering problem (the scattering operator). This result is the quantum mechanical analog of the ergodic formulas in the classical mechanics.
