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.

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 CONFINING PHASE IN SUSY SO(12) GAUGE THEORY WITH ONE SPINOR AND SEVERAL VECTORS

1998 ◽  
Vol 13 (17) ◽  
pp. 1361-1370 ◽  
Author(s):  
NOBUHITO MARU
Keyword(s):  
Gauge Theory ◽  
Gauge Symmetry ◽  
Symmetry Breaking ◽  
Explicit Form ◽  
Phase Structure ◽  
Low Energy ◽  
Effective Theory ◽  
Gauge Invariant

We study the confining phase structure of [Formula: see text] supersymmetric SO(12) gauge theory with Nf≤7 vectors and one spinor. The explicit form of low-energy superpotentials for Nf≤7 are derived after gauge-invariant operators relevant in the effective theory are identified via gauge symmetry breaking pattern. The resulting confining phase structure is analogous to Nf≤Nc+1 SUSY QCD. Finally, we conclude with some comments on the search for duals to Nf≥8 SO(12) theory.

scholarly journals THE CAUCHY PROBLEM FOR THE WAVE EQUATION WITH LÉVY NOISE INITIAL DATA

2006 ◽  
Vol 09 (02) ◽  
pp. 249-270
Author(s):  
BERNT ØKSENDAL ◽  
FRANK PROSKE ◽  
MIKAEL SIGNAHL
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Initial Data ◽  
Explicit Form ◽  
Lévy Noise ◽  
Hermite Transform ◽  
The Cauchy Problem ◽  
Levy Noise

In this paper we study the Cauchy problem for the wave equation with spacetime Lévy noise initial data in the Kondratiev space of stochastic distributions. We prove that this problem has a strong and unique C2-solution, which takes an explicit form. Our approach is based on the use of the Hermite transform.

scholarly journals Spherical solutions of the wave equation for a spin 3/2 particle

2019 ◽  
Vol 63 (3) ◽  
pp. 282-290
Author(s):  
A. V. Ivashkevich ◽  
E. M. Ovsiyuk ◽  
V. V. Kisel ◽  
V. M. Red’kov
Keyword(s):  
Wave Equation ◽  
Bessel Functions ◽  
Algebraic Equations ◽  
Linear Combinations ◽  
First Order ◽  
Quantum Numbers ◽  
Main Equation ◽  
Double Degeneration ◽  
Simple Linear Equation ◽  
Additional Constraints

The wave equation for a spin 3/2 particle, described by 16-component vector-bispinor, is investigated in spherical coordinates. In the frame of the Pauli–Fierz approach, the complete equation is split into the main equation and two additional constraints, algebraic and differential. The solutions are constructed, on which 4 operators are diagonalized: energy, square and third projection of the total angular momentum, and spatial reflection, these correspond to quantum numbers {ε, j, m, P}. After separating the variables, we have derived the radial system of 8 first-order equations and 4 additional constraints. Solutions of the radial equations are constructed as linear combinations of the Bessel functions. With the use of the known properties of the Bessel functions, the system of differential equations is transformed to the form of purely algebraic equations with respect to three quantities a1, a2, a3. Its solutions may be chosen in various ways by solving the simple linear equation A1a1 + A2a2 + A3a3 = 0 where the coefficients Ai are expressed trough the quantum numbers ε, j. Two most simple and symmetric solutions have been chosen. Thus, at fixed quantum numbers {ε, j, m, P} there exists double-degeneration of the quantum states.

Light quanta and heavy bosons

1960 ◽  
Vol 257 (1290) ◽  
pp. 283-290 ◽  
Keyword(s):  
Wave Equation ◽  
Maxwell Equations ◽  
Wave Equations ◽  
Light Quantum ◽  
Zero Mass ◽  
Light Quanta ◽  
Spin Triplet ◽  
Spin Properties ◽  
K Mesons ◽  
New Treatment

The new treatment of space-time reflexions presented in a recent paper has led to wave equations whose solutions describe π- and K-mesons with correct isobaric spin properties. Generalization of this wave equation now leads to (i) an isobaric spin triplet (π-mesons), (ii) a singlet with zero mass (Maxwell equations, light quantum), (iii) an isobaric spin quadruplet with zero mechanical spin (K-mesons), and (iv) a particle with equal isobaric spin properties, but unit mechanical spin.

On Weyl’s solution for space-times with two commuting Killing fields

1988 ◽  
Vol 415 (1849) ◽  
pp. 329-345 ◽  
Keyword(s):  
Wave Equation ◽  
Differential Equations ◽  
Explicit Form ◽  
Physical Problem ◽  
Frequent Occurrence ◽  
Coordinate Frame ◽  
Physical Problems ◽  
Killing Fields ◽  
Metric Functions ◽  
Selection Of

The solution of the metric coefficients for space-times with diagonal metrics and two commuting Killing fields can be reduced to a Laplace or a wave equation in two variables and to a further pair of integrable differential equations. This reduction can be achieved in a variety of ways. The choice of a coordinate frame and the selection of the combination of metric functions that satisfies the Laplace or the wave equation depend on the physical problem that is considered. The resolution of the issues that arise is illustrated in the contexts of three physical problems; and the solution of the remaining pair of equations, of most frequent occurrence in these contexts, is obtained in explicit form.

scholarly journals NECESSARY NON-LOCAL CONDITIONS FOR A DIFFUSION-WAVE EQUATION

2017 ◽  
Vol 20 (7) ◽  
pp. 45-59 ◽  
Author(s):  
M.O. Mamchuev
Keyword(s):  
Wave Equation ◽  
Fractional Derivative ◽  
Explicit Form ◽  
Existence And Uniqueness ◽  
Integral Operators ◽  
Rectangular Domain ◽  
Uniqueness Of Solution ◽  
Local Conditions ◽  
Diffusion Wave ◽  
Non Local

In this article, diffusion-wave equation with fractional derivative in Rieman- n-Liouville sense is investigated. Integral operators with the Write function in the kernel associated with the investigational equation are introduced. In terms of these operators necessary non-local conditions binding traces of solution and its derivatives on the boundary of a rectangular domain are found. Necessary non-local conditions for the wave are obtained by using the limiting properties of Write function. By using the integral operator’s properties the theorem of existence and uniqueness of solution of the problem with integral Samarski’s condition for the diffusion-wave equation is proved. The solution is obtained in explicit form.

scholarly journals Internal boundary value problems with displacement for the mixed-wave equation

2021 ◽  
pp. 8-14
Author(s):  
Ж.А. Балкизов ◽  
В.А. Водахова
Keyword(s):  
Wave Equation ◽  
Boundary Value Problems ◽  
Explicit Form ◽  
Boundary Value ◽  
Goursat Problem ◽  
Internal Boundary ◽  
Boundary Displacement ◽  
Mixed Wave

В работе исследованы краевые задачи с внутреннекраевым смещением для модельного смешанно-волнового уравнения, которые являются обобщениями задачи Гурса и задач с данными на противоположных характеристиках. Показано, что при определенных условиях на заданные функции решение исследуемых задач существует, единственно и выписывается в явном виде. The paper investigates boundary value problems with an internal boundary displacement for a model mixed-wave equation, which are generalizations of the Goursat problem and problems with data on opposite characteristics. It is shown that, under certain conditions for given functions, the solution to the problems under study exists, is unique, and is written out in an explicit form.

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