scholarly journals Inevitability of the Poisson Bracket Structure of the Relativistic Constraints

2021 ◽  
Vol 51 (6) ◽  
Author(s):  
Jan Głowacki
Keyword(s):  
Poisson Bracket ◽  
Careful Analysis ◽  
Dynamical Theory ◽  
Lorentzian Geometry ◽  
Classical Literature ◽  
Discussion Section ◽  
Consistency Requirement ◽  
Carry Over ◽  
Conceptual Question ◽  
Differential Coupling

AbstractThe purpose of this paper is to shed some fresh light on the long-standing conceptual question of the origin of the well-known Poisson bracket structure of the constraints that govern the canonical dynamics of generally relativistic field theories, i.e. geometrodynamics. This structure has long been known to be the same for a wide class of fields that inhabit the space-time, namely those with non-differential coupling to gravity. It has also been noticed that an identical bracket structure can be derived independently of any dynamical theory, by purely geometrical considerations in Lorentzian geometry. Here we attempt to provide the missing link between the dynamics and geometry, which we understand to be the reason for this structure to be of the specific kind. We achieve this by a careful analysis of the geometrodynamical approach, which allows us to derive the structure in question and understand it as a consistency requirement for any such theory. In order to stay close to the classical literature on the subject we stick to the metric formulation of general relativity, but the reasoning should carry over to any other formulation as long as the non-metricity tensor vanishes. The discussion section is devoted to derive some interesting consequences of the presented result in the context of reconstructing the Arnowitt–Deser–Misner (ADM) framework, thus providing a precise sense to the inevitability of the Einstein’s theory under minimal assumptions.

scholarly journals Relativity quantum mechanics with an application to compton scattering

1926 ◽  
Vol 111 (758) ◽  
pp. 405-423 ◽  
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Poisson Bracket ◽  
Classical Theory ◽  
Analytic Functions ◽  
Algebraic Theory ◽  
Dynamical Theory ◽  
Canonical Variables ◽  
Algebraic Laws ◽  
Points Of View

The new quantum mechanics, introduced by Heisenberg and since developed from different points of view by various authors, takes its simplest form if one assumes merely that the dynamical variables are numbers of a special type (called q-numbers to distinguish them from ordinary or c-numbers) that obey all the ordinary algebraic laws except the commutative law of multiplication, and satisfy instead of this the relations q r q s – q s q r =0, p r p s – p s p r = 0 } q r q s – p s q r = 0 ( r ≠ s ) or ih ( r = s ) where the p' s and q' s are a set of canonical variables and h is a c-number euqal to (2π) -1 times the usual Planck’s constant. Equations (1) may be regarded as replacing the commutative law of the classical theory, as one can, with their help, build up a complete algebraic theory of quantities that are analytic functions of a set of canonical variables. Further, it may easily be seen that the quantity [ x, y ] defined by xy – yz = ih [ x, y ] is completely analogous to the Poisson bracket of the classical theory. By means of this analogy the whole of the classical dynamical theory, in so far as it can be expressed in terms of P. B.’s instead of differential coefficients, may be taken over immediately into the quantum theory.

Systematic Reflections and the Two Beam Concept of Extinction Distance

1967 ◽  
Vol 25 ◽  
pp. 332-333
Author(s):  
S. S. Sheinin ◽  
C. D. Cann
Keyword(s):  
Dynamical Theory ◽  
Beam Intensity

The effects of systematic reflections on the variation of diffracted beam intensity with depth in a crystal can only be taken into account by using the multi-beam dynamical theory. The results of calculations of this kind, which are presented here, indicate that the intensity profiles obtained are not periodic. Since extinction distance is a concept strictly applicable only when the diffracted beam intensity varies periodically with depth, its use as a parameter in describing multi-beam intensity profiles must be carefully considered.

Microdiffraction Patterns of Small Metallic Particles II; Theoretical Analysis

1982 ◽  
Vol 40 ◽  
pp. 668-669
Author(s):  
A. Gómez ◽  
P. Schabes-Retchkiman ◽  
M. José-Yacamán ◽  
T. Ocaña
Keyword(s):  
Experimental Data ◽  
Electron Diffraction ◽  
Theoretical Analysis ◽  
Size Range ◽  
Dynamical Theory ◽  
Metallic Particles ◽  
Splitting Effect

The splitting effect that is observed in microdiffraction pat-terns of small metallic particles in the size range 50-500 Å can be understood using the dynamical theory of electron diffraction for the case of a crystal containing a finite wedge. For the experimental data we refer to part I of this work in these proceedings.

Analysis of Dark-Field Images of Disordered Materials

1972 ◽  
Vol 30 ◽  
pp. 560-561
Author(s):  
J. M. Cowley
Keyword(s):  
Amorphous Materials ◽  
Careful Analysis ◽  
Dark Field ◽  
Minimum Diameter ◽  
Statistical Fluctuations ◽  
Bright Spots ◽  
Sufficient Detail ◽  
Random Arrangement ◽  
Diffraction Patterns ◽  
Imaging Conditions

Recently a number of authors have reported detail in dark-field images obtained from diffuse-scattering regions of electron diffraction patterns. Bright spots in images from short-range order diffuse peaks of disordered binary alloys have been interpreted as evidence for the existence of microdomains of ordered lattice or of segragated clusters of one component. Spotty contrast in dark field images of near-amorphous materials has been interpreted as evidence for the existense of microcrystals. Without a careful analysis of the imaging conditions such conclusions may be invalid. Usually the conditions of the experiment have not been specified in sufficient detail to allow evaluation of the conclusions.Elementary considerations show that even for a completely random arrangement of atoms the statistical fluctuations of density will give a spotty contrast with spots of minimum diameter determined by the dark field aperture size and other factors influencing the minimum resolvable distance under darkfield imaging conditions, including fluctuations and drift over long exposure times (resolution usually 10Å or more).

Spacings and Intensities of Weak-Beam Dark Field Thickness Fringes

1981 ◽  
Vol 39 ◽  
pp. 222-223
Author(s):  
M. Avalos-Borja ◽  
K. Heinemann
Keyword(s):  
Selection Criterion ◽  
Dark Field ◽  
Dynamical Theory ◽  
Bragg Condition ◽  
Weak Beam ◽  
Kinematical Theory ◽  
Equal Thickness

Weak-beam dark field (WBDF) TEM produces narrowly spaced equal-thickness fringes in wedge-shaped crystals. Using non-systematic diffraction conditions, we have shown elsewhere that simple 2-beam kinematical theory (KT) calculations yield average fringe spacings that are for most practical purposes as satisfactorily accurate as the average spacings obtained from optimized multibeam dynamical theory (DT) calculations, As Fig. 1 shows, this result holds for deviations from the Bragg condition as low as 2x10-1 nm-1, and the differences between the results from the two calculational methods become increasingly insignificant for larger excitation errors. (Unless otherwise noted, all results reported here are for gold crystals, using the 200 beam at 100 KV; the DT calculations were made for 74 beams, using the selection criterion D as discussed in ref. [3]).

Towards quantitative simulations of inelastic electron diffraction patterns and images

1992 ◽  
Vol 50 (2) ◽  
pp. 1170-1171
Author(s):  
Z. L. Wang
Keyword(s):  
Phonon Scattering ◽  
Incident Beam ◽  
Dynamical Theory ◽  
Inelastic Electron ◽  
Additional Advantage ◽  
Coupled Equations ◽  
Atomic Vibrations ◽  
Diffraction Patterns ◽  
Primary Advantage ◽  
The One

A new dynamical theory has been developed based on Yoshioka's coupled equations for describing inelastic electron scattering in thin crystals. Compared to existing theories, the primary advantage of this theory is that the incoherent summation of the diffracted intensities contributed by electrons after exciting vast numbers of different excited states has been evaluated before any numerical calculation. An additional advantage is that the phase correlations of atomic vibrations are considered, so that full lattice dynamics can be combined in the phonon scattering calculation. The new theory has been proven to be equivalent to the inelastic multislice theory, and has been applied to calculate energy-filtered diffraction patterns and images formed by phonon, single electron and valence scattered electrons.A calculated diffraction pattern of elastic and phonon scattered electrons for a parallel incident beam case is in agreement with the one observed (Fig. 1), showing thermal diffuse scattering (TDS) streaks and Kikuchi pattern.

A Stationary Solution of High-Energy Electron Reflection (HEER) for An Arbitrary Surface

1990 ◽  
Vol 48 (2) ◽  
pp. 374-375
Author(s):  
YIQUN MA
Keyword(s):  
Stationary Solution ◽  
High Energy ◽  
Bloch Wave ◽  
Dynamical Theory ◽  
High Energy Electron ◽  
Bulk Materials ◽  
Periodic Surface ◽  
Electron Transmission ◽  
Electron Reflection ◽  
Long Time

For a long time, the development of dynamical theory for HEER has been stagnated for several reasons. Although the Bloch wave method is powerful for the understanding of physical insights of electron diffraction, particularly electron transmission diffraction, it is not readily available for the simulation of various surface imperfection in electron reflection diffraction since it is basically a method for bulk materials and perfect surface. When the multislice method due to Cowley & Moodie is used for electron reflection, the “edge effects” stand firmly in the way of reaching a stationary solution for HEER. The multislice method due to Maksym & Beeby is valid only for an 2-D periodic surface.Now, a method for solving stationary solution of HEER for an arbitrary surface is available, which is called the Edge Patching method in Multislice-Only mode (the EPMO method). The analytical basis for this method can be attributed to two important characters of HEER: 1) 2-D dependence of the wave fields and 2) the Picard iteractionlike character of multislice calculation due to Cowley and Moodie in the Bragg case.

The mensuration of interfaces

1992 ◽  
Vol 50 (1) ◽  
pp. 216-217
Author(s):  
W.M. Stobbs
Keyword(s):  
Electron Microscopy ◽  
50Th Anniversary ◽  
Numerical Data ◽  
Industrial Applications ◽  
Dynamical Theory ◽  
Large Strains ◽  
Reasonable Approach ◽  
Lattice Resolution ◽  
Analogue To Digital ◽  
The Way

I do not have access to the abstracts of the first meeting of EMSA but at this, the 50th Anniversary meeting of the Electron Microscopy Society of America, I have an excuse to consider the historical origins of the approaches we take to the use of electron microscopy for the characterisation of materials. I have myself been actively involved in the use of TEM for the characterisation of heterogeneities for little more than half of that period. My own view is that it was between the 3rd International Meeting at London, and the 1956 Stockholm meeting, the first of the European series , that the foundations of the approaches we now take to the characterisation of a material using the TEM were laid down. (This was 10 years before I took dynamical theory to be etched in stone.) It was at the 1956 meeting that Menter showed lattice resolution images of sodium faujasite and Hirsch, Home and Whelan showed images of dislocations in the XlVth session on “metallography and other industrial applications”. I have always incidentally been delighted by the way the latter authors misinterpreted astonishingly clear thickness fringes in a beaten (”) foil of Al as being contrast due to “large strains”, an error which they corrected with admirable rapidity as the theory developed. At the London meeting the research described covered a broad range of approaches, including many that are only now being rediscovered as worth further effort: however such is the power of “the image” to persuade that the above two papers set trends which influence, perhaps too strongly, the approaches we take now. Menter was clear that the way the planes in his image tended to be curved was associated with the imaging conditions rather than with lattice strains, and yet it now seems to be common practice to assume that the dots in an “atomic resolution image” can faithfully represent the variations in atomic spacing at a localised defect. Even when the more reasonable approach is taken of matching the image details with a computed simulation for an assumed model, the non-uniqueness of the interpreted fit seems to be rather rarely appreciated. Hirsch et al., on the other hand, made a point of using their images to get numerical data on characteristics of the specimen they examined, such as its dislocation density, which would not be expected to be influenced by uncertainties in the contrast. Nonetheless the trends were set with microscope manufacturers producing higher and higher resolution microscopes, while the blind faith of the users in the image produced as being a near directly interpretable representation of reality seems to have increased rather than been generally questioned. But if we want to test structural models we need numbers and it is the analogue to digital conversion of the information in the image which is required.

Direct measurement of electron-diffraction-pattern intensities using an energy loss spectrometer

1989 ◽  
Vol 47 ◽  
pp. 668-669
Author(s):  
A. G. Jackson ◽  
M. Rowe
Keyword(s):  
Thermal Effects ◽  
Dynamic Range ◽  
Scattering Amplitude ◽  
Dynamical Theory ◽  
Site Symmetry ◽  
Proof Of Concept ◽  
Parallel Acquisition ◽  
Kinematic Approximation ◽  
Speed Up ◽  
Structure Calculations

Diffraction intensities from intermetallic compounds are, in the kinematic approximation, proportional to the scattering amplitude from the element doing the scattering. More detailed calculations have shown that site symmetry and occupation by various atom species also affects the intensity in a diffracted beam. [1] Hence, by measuring the intensities of beams, or their ratios, the occupancy can be estimated. Measurement of the intensity values also allows structure calculations to be made to determine the spatial distribution of the potentials doing the scattering. Thermal effects are also present as a background contribution. Inelastic effects such as loss or absorption/excitation complicate the intensity behavior, and dynamical theory is required to estimate the intensity value.The dynamic range of currents in diffracted beams can be 104or 105:1. Hence, detection of such information requires a means for collecting the intensity over a signal-to-noise range beyond that obtainable with a single film plate, which has a S/N of about 103:1. Although such a collection system is not available currently, a simple system consisting of instrumentation on an existing STEM can be used as a proof of concept which has a S/N of about 255:1, limited by the 8 bit pixel attributes used in the electronics. Use of 24 bit pixel attributes would easily allowthe desired noise range to be attained in the processing instrumentation. The S/N of the scintillator used by the photoelectron sensor is about 106 to 1, well beyond the S/N goal. The trade-off that must be made is the time for acquiring the signal, since the pattern can be obtained in seconds using film plates, compared to 10 to 20 minutes for a pattern to be acquired using the digital scan. Parallel acquisition would, of course, speed up this process immensely.

Effects of Higher-Order Laue Zone reflections on structure images

1993 ◽  
Vol 51 ◽  
pp. 450-451
Author(s):  
H. S. Kim ◽  
S. S. Sheinin
Keyword(s):  
Wave Vector ◽  
Theoretical Calculations ◽  
Reciprocal Lattice ◽  
Image Plane ◽  
Bloch Wave ◽  
Dynamical Theory ◽  
Higher Order ◽  
Lattice Image ◽  
Lattice Vector ◽  
Image Position

The importance of image simulation in interpreting experimental lattice images is well established. Normally, in carrying out the required theoretical calculations, only zero order Laue zone reflections are taken into account. In this paper we assess the conditions for which this procedure is valid and indicate circumstances in which higher order Laue zone reflections may be important. Our work is based on an analysis of the requirements for obtaining structure images i.e. images directly related to the projected potential. In the considerations to follow, the Bloch wave formulation of the dynamical theory has been used.The intensity in a lattice image can be obtained from the total wave function at the image plane is given by: where ϕg(z) is the diffracted beam amplitide given by In these equations,the z direction is perpendicular to the entrance surface, g is a reciprocal lattice vector, the Cg(i) are Fourier coefficients in the expression for a Bloch wave, b(i), X(i) is the Bloch wave excitation coefficient, ϒ(i)=k(i)-K, k(i) is a Bloch wave vector, K is the electron wave vector after correction for the mean inner potential of the crystal, T(q) and D(q) are the transfer function and damping function respectively, q is a scattering vector and the summation is over i=l,N where N is the number of beams taken into account.

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