Phase space path-integral formulation of the above-threshold ionization

A new analytical expression for the energy eigenvalues of the Manning–Rosen potential for l-states, based on the path integral formalism, is derived by an improved approximation to the centrifugal term of the potential, in the framework of the Duru–Keinert method. Nonlinear space–time transformations in the radial path integral are applied. A transformation formula is derived that relates the original path integral to the Green function of a new quantum soluble system. The energy spectrum and the normalized eigenfunctions are both obtained for the application of this technique to the Manning–Rosen potential. Our results are in very good agreement with those found by using numerical and other approximation methods. Our solution applies also to the Hulthén potential.

The five laws of relativistic quantum mechanics, according to Feynman's path integral formulation, are concisely stated and applied to experiments. Reflection–diffraction-grating experiments for both photons and electrons are analysed, in particular, the Davisson–Germer experiment in which the wave-like property of electrons was first established. It is shown how classical, purely spatial, effective wave theories for both photons and electrons are predicted by the path integral formulation of quantum mechanics. The standard Copenhagen interpretation of wave mechanics is critically discussed in the light of the described experimental applications of the path integral formulation.

In two recent letters results for threshold ionization of hydrogen by electron impact (Rost, J.M. 1994 Phys. Rev. Lett. 72 1998) and positron impact (Rost, J.M. and Heller, E.J. 1994 Phys. Rev. A 49 R4289) were communicated. The results were obtained by calculating the S-matrix semiclassically in Feynman's path integral formalism. This paper gives a more complete account of the theoretical method. Moreover, it is shown how the ionization cross section of atomic targets other than hydrogen can be expressed with the hydrogen cross section through a scaling relation. This demonstrates the universality of the threshold behaviour.

Here we present a unified theory of above threshold ionization (ATI) phenomena. Based on this theory, we can describe ATI processes in realistic atomic screen potentials and can also elucidate experimental observations of ATI phenomena. The ATI phenomena mainly involve two successive procosses: (1) Quantum transition processe from an initial field-free state to final strong laser field dressed states. (2) in order to descibe experimental observations, one must consider post-interactions between the escaping photoelectron and the strong laser fields, which are classic processes owing to ponderomotive forces generated by non-uniform laser fields.

Quantisation in multiply connected spaces

A Path Integral (PI) formulation of linear elastostatics is first presented. For this, Navier equations are modified by adding a fictitious 'time' derivative of displacements so that equilibrium corresponds to the steady state of the resulting diffusion-like equations. The evolution of displacement is then represented as the propagation, through the fictitious time co-ordinate, of an initial displacement field corresponding to the unloaded state. The resulting procedure somehow mimics the well-known Feynman path integral of quantum mechanics, which is equivalent to the differential formulation embodied in Schrödinger equation. However, the path integral for elastostatics is formulated in terms of infinitesimal propagators of local support. In its simplest form, the formulation can be used as a relaxation method of solution, by updating displacements until convergence. This may be advantageous for problems involving a very large number of unknowns. On the other hand, by equating the updated displacement field to the actual one a direct method of solution is obtained, which leads to non-symmetric (but sparse and banded) discrete equations. Unlike variational principles this formulation does not require integration over the whole domain, effectively eliminating the need of a background mesh for integration. Also, it only requires continuity of the displacement field on the propagator's support. As a consequence, the formulation lends itself to very flexible meshless implementations. To demonstrate this we describe a simple numerical method in which displacements around each node are approximated by quadratic bivariate polynomials, which is the simplest approximation technique. The feasibility of the method is assessed through a number of numerical examples and comparisons with analytical solutions and other meshless methods. Copyright © 2000 John Wiley & Sons, Ltd.

The principle of least action is known to apply to many domains of Physics. Yet its justifications and derivations usually come from archaic classical principles like the Fermat or Maupertuis principles, or mathematical analytical frameworks used for classical mechanics like the Lagrange, Hamilton and D’Alembert formalisms. As is well known, it is widely applicable to other domains, without necessarily a clear justification for it, especially when applied to Physics where Lagrangians or Hamiltonians may not be defined or do not exist. In addition, the Feynman Path Integral formulation of Quantum Mechanics gave it a second life, and another way to justify its validity not only in classical, but also in quantum Physics. However, the conventional logic to reach these conclusions is at time circular, and the principle is applied without proof that it is applicable: it just works and everybody does it. It is this applicability that led us in the past to treat the Action path integral as God’s equation.The paper justifies the principle of least action and Feynman’s path integrals through different physical reasoning instead of the traditional mathematical formalism, and on that basis we confirm its validity in most situations, even if Hamiltonians or Lagrangians may be shaky to define, or the system may be dissipative. We also explain when, and why, sometimes, it may not conventionally apply. The derivation uncovers different links between Entropy and Action, which also opens a different way to look at the origin of the probability distribution of even just a single lone particle; especially when complemented with the W-type multi-fold hypothesis, and makes some considerations on Feynman diagrams, instantons, renormalons and resurgence. Also discontinuities in the paths of the path integrals imply that spacetime is discrete, and non-commutative, and supersymmetry in flat spacetime is unphysical. Surfaceology, as optimized computation of the Feynman diagram is shown to trivially predict that all gauge theories double copy dual to gravity share the same forbidden scatterings. Relying on a recently encountered paper, we show how the results of the multi-fold theory, in particular the multi-fold space time matter induction and scattering, built explicitly on the path integral formalism, its underlying principles, and the principle of least action, allows recovering almost everything in Today’s physics. It leaves out essentially only some of the coupling constants and mixing angles/parameters. This goes a long way towards the ambition of a Theory Of Everything (TOE), even if not presented as usually expected. It maybe the best that can be achieved considering Gödel's incompleteness theorems. Our justification of the principle of least action, and path integrals, and our work on multi-fold theory encounters relationships between Entropy and Action. We review these encounters, including how our early prediction that entanglement might be irreversible match recent published results that argue against the existence of a (second) Law of entanglement analogous to the second Law of Thermodynamics. Also, from our approach, we obtain a formulation of the evolution of entropy, including for irreversible systems, which matches Onsager and Prigogine’s famous models and principles.

It is a well known result that the Feynman's path integral (FPI) approach to quantum mechanics is equivalent to Schrodinger's equation when we use as integration measure the Wiener-Lebesgue measure. This results in little practical applicability due to the great algebraic complexibity involved, and the fact is that almost all applications of (FPI) “practical calculations” — are done using a Riemann measure. In this paper we present an expansion to all orders in time of FPI in a quest for a representation of the latter solely in terms of differentiable trajetories and Riemann measure. We show that this expansion agrees with a similar expansion obtained from Schrodinger's equation only up to first order in a Riemann integral context, although by chance both expansions referred to above agree for the free particle and harmonic oscillator cases. Our results permit, from the mathematical point of view, to estimate the many errors done in “practical” calculations of the FPI appearing in the literature and, from the physical point of view, our results supports the stochastic approach to the problem.

In this paper, the Dirac equation with the q-deformed Scarf potential for spin symmetry is solved for an arbitrary spin-orbit quantum number κ, in the presence of Coulomb-like potential tensor. Using the Feynman path integral formalism and the Pekeris approximation of the centrifugal term, we obtain the bound state energy eigenvalues and the associated spinor of the Dirac particle. Furthermore, this method is used to determine the spectrum of two diatomic molecules Li 2(61Π u ) and KRb(B −1Π). The obtained results are compared to the experimental and numerical ones.

The GEM (Gravity EM) unification theory builds on the Sakharov and Kaluza-Klein approaches to unify all four force fields of nature: Gravity, EM, Strong, and Weak using Feynman Path integral formalism. The model is rudimentary, and can be called a “Bohr Model” of unification. It is basically found that Gravity and the other forces can be understood as quantum electrodynamics. In particular the proton emerges as a fundamental particle despite being composed of quarks and is the principle interaction vertex of the Higgs boson, which is seen here as direct consequence of a hidden 5th dimension, where the Higgs mass is due to 5th dimensional compactification. Collapse and compactification of a 5th dimension is argued as the triggering event for the Big Bang. A particle mass formula based on Feynman Path Integrals including paths across the hidden 5th dimension gives the proton and electron masses to high accuracy and finds the charged bosons responsible for the short range nuclear forces. The masses calculated for the particles are as follows: the charged pion m = 2 me / 140.0 MeV and W boson: mw =2 mp = 80.4 GeV. The c meson m =2985 GeV is identified with the 5th dimension compactification force mediated by the Radion field. The Higgs boson associated with this mass inducing field is the most general EM+Radion scattering quanta off the hidden dimension size with a mass mp/α  127.7 GeV. This results in a structural resonance Higgs = rp where Higgs is the Compton radius of the Higgs boson and Higgs =c/mHiggsc2 the rp electro-dynamic length of the proton rp=e2/mpc2. Vacuum decay results from this value of the Higgs boson mass, which is a key parameter in a mass system that creates the proton-electron mass system and thus hydrogen in the Big Bang. The vacuum is found to decay into proton-electron pairs and to give an “eternally inflating” cosmos of Hubble Time TH  ( 9/ )1/3 /2 e2 /(4oGmemp ) re/c = 1.2 x1010 years. The predicted value of the CBR(Cosmic Background Radiation) temperature from this vacuum decay is TCBR  (4cGme2/(3Th2Stefan-Boltz.)1/3 = 2.65K Support for this consists of observation of the failure of the Sunaeyev-Zeldovich effect, where distant galaxies are not observed to cast shadows in the CBR indicating a source for the CBR in the vacuum itself in the line of sight to more distant galaxies.

The overcompleteness of the coherent states basis leads to a multiplicity of representations of Feynman's path integral. These different representations, although equivalent quantum mechanically, lead to different semiclassical limits. Two such semiclassical formulas were derived in [1] for the two corresponding path integral forms suggested by Klauder and Skagerstan in [2]. Each of these formulas involve trajectories governed by a different classical representation of the Hamiltonian operator: the P representation in one case and the Q representation in other. In this paper we construct a third representation of the path integral whose semiclassical limit involves directly the Weyl representation of the Hamiltonian operator, i.e., the classical Hamiltonian itself.

As a first step towards meeting the recent demand for new computational tools capable of reproducing molecular-ionization continua in a wide energy range, we introduce a hybrid Gaussian--$B$-spline basis (GABS) that combines short-range Gaussian functions, compatible with standard quantum-chemistry computational codes, with $B$ splines, a basis appropriate to represent electronic continua. We illustrate the performance of the GABS hybrid basis for the hydrogen atom by solving both the time-independent and the time-dependent Schr\odinger equation for a few representative cases. The results are in excellent agreement with those obtained with a purely $B$-spline basis, with analytical results, when available, and with recent above-threshold ionization spectra from the literature. In the latter case, we report fully differential photoelectron distributions which offer further insight into the process of above-threshold ionization at different wavelengths.

We describe in detail a physical situation in which instantons are necessarily complex, not just Wick rotations of classical solutions to Euclidean spacetime. These complex instantons arise in the semiclassical evaluation of vacuum pair production rates, based on Feynman's worldline path integral formulation. Even though the path integral is a sum over all real closed trajectories in spacetime, the semiclassical description of non-perturbative pair production is dominated by closed classical trajectories that are generically complex. These closed trajectories contain segments associated with nonperturbative instanton suppression factors as well as segments producing phase factors that incorporate quantum interference effects. For a class of time-dependent electric fields we implement this procedure and demonstrate excellent quantitative agreement with alternative methods.

A mapping of a finite directed graph onto a curve in space-time is considered. The mapping induces the dynamics of a free particle moving along the curve. The distinction between the Lagrangian and the Hamiltonian formulation of particle mechanics is expressed in terms of the distinction between referring to a particle in space and time and referring to the points in space which the particle occupies, respectively. These elements are combined to yield an interpretation of Feynman's path integral formulation of quantum mechanics. Describing a bound state of a system as a particle is discussed.

Resonant tunneling is studied by use of Feynman's path integrals. The semiclassical Green's function is obtained for a one-dimensional double-barrier structure. This approach leads to a clearer and deeper understanding of the physical process of resonant tunneling. It is shown that sequential tunneling never gives rise to negative differential resistance.