Conserved quantity partition for Dirac’s equation

Author:

Branson Thomas P.

Abstract

Let M M be the ( n + 1 ) (n + 1) -dimensional Minkowski space, n 3 n \ge 3 . The energy of a solution ψ \psi to Dirac’s equation in M M is a sum of n n terms, the j j th term depending on ψ \psi and the space derivative ψ / x j \partial \psi /\partial {x_j} . We show that if the Cauchy datum for ψ \psi is compactly supported, then each of these terms is eventually constant. Specifically, if ψ \psi is initially supported in the closed ball of radius b b about the origin in space ( R n ) \left ( {{R^n}} \right ) , then for times | t | b \left | t \right | \ge b , the j j th term is equal to the energy of the j j th Riesz transform ( Δ ) 1 / 2 ( / x j ) ψ {( - \Delta )^{ - 1/2}}(\partial /\partial {x_j})\psi , which also solves Dirac’s equation.

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics

Reference9 articles.

1. Eventual partition of conserved quantities in wave motion;Branson, Thomas P.;J. Math. Anal. Appl.,1983

2. Eventual partition of conserved quantities for Maxwell’s equations;Branson, Thomas P.;Arch. Rational Mech. Anal.,1984

3. Energy splitting;Costa, David G.;Quart. Appl. Math.,1981

4. Equipartition of energy for Maxwell’s equations;Dassios, George;Quart. Appl. Math.,1979

5. Equipartition of energy in wave motion;Duffin, R. J.;J. Math. Anal. Appl.,1970

Cited by 1 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Equipartition of energy for waves in symmetric space;Journal of Functional Analysis;1991-05

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