Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Conserved quantity partition for Dirac's equation


Author: Thomas P. Branson
Journal: Quart. Appl. Math. 42 (1984), 179-191
MSC: Primary 35Q20; Secondary 35L45, 81D25
DOI: https://doi.org/10.1090/qam/745098
MathSciNet review: 745098
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Abstract: Let $ M$ be the $ (n + 1)$-dimensional Minkowski space, $ n \ge 3$. The energy of a solution $ \psi $ to Dirac's equation in $ M$ is a sum of $ n$ terms, the $ j$ th term depending on $ \psi $ and the space derivative $ \partial \psi /\partial {x_j}$. We show that if the Cauchy datum for $ \psi $ is compactly supported, then each of these terms is eventually constant. Specifically, if $ \psi $ is initially supported in the closed ball of radius $ b$ about the origin in space $ \left( {{R^n}} \right)$, then for times $ \left\vert t \right\vert \ge b$, the $ j$th term is equal to the energy of the $ j$th Riesz transform $ {( - \Delta )^{ - 1/2}}(\partial /\partial {x_j})\psi $, which also solves Dirac's equation.


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DOI: https://doi.org/10.1090/qam/745098
Article copyright: © Copyright 1984 American Mathematical Society


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