0000000000391279

AUTHOR

Erik Taflin

showing 4 related works from this author

On global solutions of the Maxwell-Dirac equations

1987

We prove, for the Maxwell-Dirac equations in 1+3 dimensions, that modified wave operators exist on a domain of small entire test functions of exponential type and that the Cauchy problem, inR+×R3, has a unique solution for each initial condition (att=0) which is in the image of the wave operator. The modification of the wave operator, which eliminates infrared divergences, is given by approximate solutions of the Hamilton-Jacobi equation, for a relativistic electron in an electromagnetic potential. The modified wave operator linearizes the Maxwell-Dirac equations to their linear part.

Momentum operatorElectromagnetic wave equationMathematical analysisStatistical and Nonlinear PhysicsInhomogeneous electromagnetic wave equationd'Alembert's formula35Q20Operator (computer programming)35L45Initial value problemD'Alembert operatorHyperbolic partial differential equation35P25Mathematical Physics81D25MathematicsCommunications in Mathematical Physics
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The cauchy problem for non-linear Klein-Gordon equations

1993

We consider in ℝ n+1,n≧2, the non-linear Klein-Gordon equation. We prove for such an equation that there is a neighbourhood of zero in a Hilbert space of initial conditions for which the Cauchy problem has global solutions and on which there is asymptotic completeness. The inverse of the wave operator linearizes the non-linear equation. If, moreover, the equation is manifestly Poincare covariant then the non-linear representation of the Poincare Lie algebra, associated with the non-linear Klein-Gordon equation is integrated to a non-linear representation of the Poincare group on an invariant neighbourhood of zero in the Hilbert space. This representation is linearized by the inverse of the …

Cauchy problemPure mathematicsMathematical analysisHilbert spaceStatistical and Nonlinear Physicssymbols.namesakeNorm (mathematics)Poincaré groupLie algebrasymbolsTrivial representationCovariant transformationKlein–Gordon equationMathematical PhysicsMathematicsCommunications in Mathematical Physics
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THE MAXWELL–DIRAC EQUATIONS: ASYMPTOTIC COMPLETENESS AND THE INFRARED PROBLEM

1994

In this article we present an announcement of results concerning: a) A solution to the Cauchy problem for the M-D equations, namely global existence, for small initial data at t = 0, of solutions for the M-D equations. b) Arguments from which asymptotic completeness for the M-D equations follows. c) Cohomological interpretation of the results in the spirit of nonlinear representation theory and its connection to the infrared tail of the electron in M-D classical field theory. The full detailed results will be published elsewhere.

Nonlinear systemCompleteness (order theory)Mathematical analysisDirac (software)Initial value problemClassical field theoryStatistical and Nonlinear PhysicsRepresentation theoryMathematical PhysicsMathematicsInterpretation (model theory)Mathematical physicsConnection (mathematics)Reviews in Mathematical Physics
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Initial Data for Non-Linear Evolution Equations and Differentiable Vectors of Group Representations

1995

Regularity properties of non-linear Lie algebra representations are defined. These properties are satisfied in examples given by evolution equations. We prove that this regularity implies that the set of C ∞ vectors for the non-linear group representation obtained by integration of the Lie algebra representation coincide with the set of C ∞ vectors of the linear part (the order one term) of this group representation.

AlgebraNonlinear systemLie algebra representationLie algebraDifferentiable functionWeak derivativeGroup representationMathematics
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