Structure of the plane waves for a spin 3/2 particle, massive and massless cases, gauge symmetry

2021 ◽  
Vol 24 (4) ◽  
pp. 391-408
Author(s):  
A.V. Ivashkevich
Keyword(s):  
Wave Equation ◽  
Gauge Symmetry ◽  
General Solution ◽  
Explicit Form ◽  
Plane Waves ◽  
Massless Case ◽  
Independent Components ◽  
Relativistic Spin ◽  
Massless Equation

The structure of the plane waves solutions for a relativistic spin 3/2 particle described by 16-component vector-bispinor is studied. In massless case, two representations are used: Rarita – Schwinger basis, and a special second basis in which the wave equation contains the Levi-Civita tensor. In the second representation it becomes evident the existence of gauge solutions in the form of 4-gradient of an arbitrary bispinor. General solution of the massless equation consists of six independent components, it is proved in an explicit form that four of them may be identified with the gauge solutions, and therefore may be removed. This procedure is performed in the Rarita – Schwinger basis as well. For the massive case, in Rarita – Schwinger basis four independent solutions are constructed explicitly.

scholarly journals Zero mass field with the spin 3/2: solutions of the wave equation and the helicity operator

2019 ◽  
Vol 55 (3) ◽  
pp. 338-354
Author(s):  
A. V. Ivashkevich ◽  
E. M. Ovsiyuk ◽  
V. M. Red’kov
Keyword(s):  
Wave Equation ◽  
Gauge Symmetry ◽  
Explicit Form ◽  
Linear Combinations ◽  
Zero Mass ◽  
Mass Field

The wave equation for the vector bispinor Ψa(x), which describes a zero mass spin 3/2 particle in the Rarita – schwinger form, is transformed into a new basis of Ψa(x), in which the gauge symmetry in the theory becomes evident: there exist solutions in the form of the 4-gradient of an arbitrary bispinor Ψa0(x) = ∂аΨ(x), For 16-component equation in this new basis, two independent solutions are constructed in explicit form, which do not contain any gauge constituents. Zero mass solutions are transformed into linear combinations of helicity states, the derived formulas contain the terms with all helicities σ = ±1/2, ±3/2.

Time‐domain behavior of wide‐angle one‐way wave equations

Geophysics ◽  
1991 ◽  
Vol 56 (3) ◽  
pp. 382-384
Author(s):  
A. H. Kamel
Keyword(s):  
Wave Equation ◽  
Wave Equations ◽  
Source Function ◽  
Plane Waves ◽  
Wide Angle ◽  
Wave Processes ◽  
Inhomogeneous Wave ◽  
Dirac Delta ◽  
Inhomogeneous Wave Equation ◽  
Standard Tool

The constant‐coefficient inhomogeneous wave equation reads [Formula: see text], Eq. (1) where t is the time; x, z are Cartesian coordinates; c is the sound speed; and δ(.) is the Dirac delta source function located at the origin. The solution to the wave equation could be synthesized in terms of plane waves traveling in all directions. In several applications it is desirable to replace equation (1) by a one‐way wave equation, an equation that allows wave processes in a 180‐degree range of angles only. This idea has become a standard tool in geophysics (Berkhout, 1981; Claerbout, 1985). A “wide‐angle” one‐way wave equation is designed to be accurate over nearly the whole 180‐degree range of permitted angles. Such formulas can be systematically constructed by drawing upon the connection with the mathematical field of approximation theory (Halpern and Trefethen, 1988).

Imaging diffractors using wave-equation migration

Geophysics ◽  
2016 ◽  
Vol 81 (6) ◽  
pp. S459-S468 ◽  
Author(s):  
Lu Liu ◽  
Etienne Vincent ◽  
Xu Ji ◽  
Fuhao Qin ◽  
Yi Luo
Keyword(s):  
Wave Equation ◽  
Field Data ◽  
Recursive Algorithm ◽  
Median Filter ◽  
Random Noise ◽  
Plane Waves ◽  
Computationally Efficient ◽  
Imaging Point ◽  
Local Plane ◽  
Dip Angle

We have developed a fast and practical wave-equation-based migration method to image subsurface diffractors. The method is composed of three steps in our implementation. First, it decomposes extrapolated receiver wavefields at every imaging point into local plane waves by a linear Radon transform; the transform is realized by a novel computationally efficient recursive algorithm. Second, the decomposed plane waves are zero lag-correlated with the incident source wavefields, where the incident angles are computed via the structure tensor approach. The resulting prestack images are binned into dip-angle gathers according to the directions of the decomposed plane waves and the calculated incident angles. Third, a windowed median filter is applied to the dip-angle gathers to suppress the focused reflection energy, and it produces the desired diffraction images. This method is tested on synthetic and field data. The results demonstrate that it is resistant to random noise, computationally efficient, and applicable to field data in practice. The results also indicate that the diffraction images are able to provide important discontinuous geologic features, such as scattering and faulting zones, and thus are helpful for seismic interpretation.

scholarly journals Plane waves in thermoelasticity with one relaxation time

2001 ◽  
Vol 26 (4) ◽  
pp. 225-232
Author(s):  
Jun Wang ◽  
Wen Dong Chang
Keyword(s):  
Relaxation Time ◽  
General Solution ◽  
Laplace Transform ◽  
Half Space ◽  
Plane Waves ◽  
Elastic Half Space ◽  
Second Sound ◽  
Boundary Temperature ◽  
Explicit Expressions

We apply the thermoelastic equations with one relaxation time developed by Lord and Shulman (1967) to solve some elastic half-space problems. Laplace transform is used to find the general solution. Problems concerning the ramp-type increase in boundary temperature and stress are studied in detail. Explicit expressions for temperature and stress are obtained for small values of time, where second sound phenomena are of relevance. Numerical values of stress and temperature are calculated and displayed graphically.

Crosshole seismics using vertical eigenstates

Geophysics ◽  
1990 ◽  
Vol 55 (7) ◽  
pp. 815-820 ◽  
Author(s):  
David M. Pai
Keyword(s):  
Wave Equation ◽  
Data Analysis ◽  
Phase Velocity ◽  
Linear Combination ◽  
Plane Waves ◽  
Basis Functions ◽  
Similar Role ◽  
Potential Applications ◽  
Data Expansion ◽  
And Inversion

This paper introduces the concept of vertical eigenstates to crosshole data analysis. In surface seismics, plane waves have played a fundamental role in many applications. This paper points out that for crosshole seismics, vertical eigenstates play a similar role. The vertical eigenstates separate the wave equation, and thus in terms of vertical eigenstate expansion the solution is a linear combination of modes, each mode traveling sideways with a distinct phase velocity. As a result, the vertical eigenstates form a natural set of basis functions for solution and data expansion, with potential applications to modeling, migration, and inversion.

Migration/inversion for incident waves synthesized from common-shot data gathers

Geophysics ◽  
2007 ◽  
Vol 72 (6) ◽  
pp. W17-W31 ◽  
Author(s):  
Norman Bleistein
Keyword(s):  
Wave Equation ◽  
Incidence Angle ◽  
Plane Waves ◽  
Data Sets ◽  
Amplitude Wave ◽  
Data Set ◽  
Data Synthesis ◽  
Single Wave ◽  
Incident Waves ◽  
Domain Source

Wavefield synthesis is a process for producing reflection responses from more general sources or from prescribed incident waves by combining common-shot data gathers. Synthesis can provide surveywide data sets, similar in that regard to common-offset data gathers, but with the added advantage that each synthesized data set is a solution to a single wave equation. A common-offset data set does not have this last feature. Thus, synthesized data sets can be processed by true-amplitude wave-equation migration. The output is then known to be true amplitude in the same sense as is the output of Kirchhoff inversion. That is, the peak amplitude is proportional to the ray-theoretic reflection coefficient at a determinable specular incidence angle multiplied bythe area under the frequency-domain source signature and scaled by [Formula: see text]. Alternatively, the Kirchhoff inversion of synthesized data has a Beylkin determinant that is expressed in terms of the ray-theoretic Green’s function amplitude. This is in contrast to 3D common-offset inversion, wherein the Beylkin determinant is most difficult to compute. We present a theory of data synthesis and true-amplitude migration/inversion based on the application of Green’s theorem to the ensemble of common-shot gathers and prescribed more general sources or prescribed incident waves. Specific examples include delayed-shot line sources and incident dipping plane waves at the upper surface. We also discuss two cases in which waves are prescribed at depth, back-projected to the upper surface, and then used to generate a synthesized data set.

On the mechanism of earthquakes

1964 ◽  
Vol 54 (5A) ◽  
pp. 1283-1289
Author(s):  
M. J. Randall
Keyword(s):  
Wave Equation ◽  
General Solution ◽  
Elastic Wave ◽  
Sudden Change ◽  
Elastic Equilibrium ◽  
Equilibrium Calculation ◽  
Elastic Wave Equation ◽  
The Earth ◽  
New Form ◽  
Elastostatic Problem

Abstract An earthquake may be regarded as resulting from a sudden change in the condition of elastic equilibrium in the Earth. A new form of the general solution of the elastic wave equation relates seismic radiation to displacement from equilibrium. Calculation of the radiation pattern for a proposed mechanism is thus reduced to an elastostatic problem.

Introduction to the Mathematics of Rays

2017 ◽  
Author(s):  
John A. Adam
Keyword(s):  
Differential Geometry ◽  
Wave Equation ◽  
General Solution ◽  
Jacobi Equation ◽  
Geometrical Optics ◽  
Linear Wave ◽  
Linear Wave Equation ◽  
The Family ◽  
Varying Environment ◽  
Hamilton Jacobi Equation

This chapter provides an overview of the mathematics of rays. It begins with a discussion of the theory of geometrical optics and how it can be formulated by means of the ray equations or the Hamilton-Jacobi equation. The two equations are of seemingly different types, but they are in fact equivalent. The ray equations are the characteristic equations of the Hamilton-Jacobi equation. This remark leads to the geometrical interpretation: the family of rays of geometrical optics is perpendicular to the wavefronts S = constant, if S denotes the appropriate solution of the Hamilton-Jacobi equation. The chapter considers the Hamilton-Jacobi theory in more detail, along with Hamilton's principle, ray differential geometry and the eikonal equation, and dispersion relations. It also presents the general solution of the linear wave equation before concluding with an analysis of the behavior of rays and waves in a slowly varying environment.

Some solutions of the Lanczos vacuum wave equation

1994 ◽  
Vol 447 (1931) ◽  
pp. 577-585 ◽  
Keyword(s):  
Wave Equation ◽  
Degrees Of Freedom ◽  
Vector Fields ◽  
Spin Coefficient ◽  
Independent Components ◽  
Conformal Curvature ◽  
Conformal Curvature Tensor ◽  
Non Local ◽  
Weyl Conformal Curvature Tensor ◽  
Gauge Conditions

In general relativity the non-local part of the gravitational field is described by the 10 degrees of freedom of the Weyl conformal curvature tensor C abcd . In every space-time the Weyl field C abcd is derivable from a potential L abc which has at most 16 algebraically independent components reducing to 10 degrees of freedom when the six gauge conditions L ab s ; s = 0 are imposed. The potential L abc discov­ered by Lanczos was shown by Illge to have an extremely simple vacuum wave equation, namely, □ L abc ≡ g sm L abc ; s ; m = 0. Using tensor, spinor and spin-coefficient methods we give some solutions of this new vacuum wave equation in some spacetimes containing one or more preferred vector fields.

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