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arXiv 1204.2683gr-qcastro-ph.COhep-th
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Born-Infeld和其他非线性理论的实际线性类似物

Practically linear analogs of the Born-Infeld and other nonlinear theories

Mordehai Milgrom

备注：
20 pages, Version published in Phys. Rev. D
发表信息：
Phys. Rev. D 85 105018 (2012)
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摘要：

我讨论的理论描述完全非线性物理，而实际上是线性的（PL），因为它们只需要求解线性微分方程。这些理论本身作为可管理的非线性理论可能是有趣的。但是，它们也可以被选择来模拟真正的非线性理论的特殊利益，他们可以作为近似。这个想法可以应用到一个大类的非线性理论，举例说明这里与PL类似的标量理论，和玻恩-因菲尔德（BI）电动力学。这类电磁学的PL理论的一般类别由拉格朗日L=-（1/2）F_mnQ^mn+ S（Q_mn）支配，其中电磁场以标准方式耦合到电流，而Qmn是辅助场，从不直接耦合到电流的矢量势导出。通过选取S（Q_mn）的一种特殊形式，我们可以使这样的理论在某些方面类似于一个给定的完全非线性理论，由拉格朗日-U（F_mn）控制。一个特别恰当的选择是将S作为U的勒让德变换。对于BI理论，该勒让德变换具有与BI拉格朗日量本身相同的形式。关于这些理论，各种原则性问题仍有待回答。作为一个具体的例子，我更详细地讨论BI静电。作为一个旁白，BI，我推导出一个精确的表达式，在任何维度上的两个任意点电荷之间的短距离力的相同的符号。

英文摘要：

I discuss theories that describe fully nonlinear physics, while being practically linear (PL), in that they require solving only linear differential equations. These theories may be interesting in themselves as manageable nonlinear theories. But, they can also be chosen to emulate genuinely nonlinear theories of special interest, for which they can serve as approximations. The idea can be applied to a large class of nonlinear theories, exemplified here with a PL analogs of scalar theories, and of Born-Infeld (BI) electrodynamics. The general class of such PL theories of electromagnetism are governed by a Lagrangian L=-(1/2)F_mnQ^mn+ S(Q_mn), where the electromagnetic field couples to currents in the standard way, while Qmn is an auxiliary field, derived from a vector potential that does not couple directly to currents. By picking a special form of S(Q_mn), we can make such a theory similar in some regards to a given fully nonlinear theory, governed by the Lagrangian -U(F_mn). A particularly felicitous choice is to take S as the Legendre transform of U. For the BI theory, this Legendre transform has the same form as the BI Lagrangian itself. Various matter-of-principle questions remain to be answered regarding such theories. As a specific example, I discuss BI electrostatics in more detail. As an aside, for BI, I derive an exact expression for the short-distance force between two arbitrary point charges of the same sign, in any dimension.

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