修正无质量Klein-Gordon方程的非线性理论

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"这篇论文提出了一种新的非线性修正无质量Klein-Gordon方程，解决了原始方程在任意弯曲时空下光锥内部形成‘尾巴’的问题。新方程来源于拉格朗日力学，展现出电流-电流的相互作用，并包含量子力学Bohm电位相关的自耦合项。该研究由Asenjo和Hojman合作完成，发表于Eur.Phys.J.C(2017)77:732，文章编号10.1140/epjc/s10052-017-5330-7，是一个开放获取的学术资源。" 无质量Klein-Gordon方程是量子场论中的基本方程之一，它描述了无质量粒子的波动行为。在平坦时空中，这个方程是线性的，能够正确地预测粒子的行为。然而，在弯曲时空中，无质量Klein-Gordon方程的解可能出现不期望的行为，如在光锥内部形成“尾巴”，这并不符合零大地测量学（零波沿光锥传播）的预期。DeWitt和Brehme在1960年代初就发现了这个问题。 Asenjo和Hojman提出的修正方案旨在克服这一挑战，他们通过引入非线性项来修改方程，以确保波动总是沿着光锥传播，即沿零大地测量路径。这种修正源于一个基于拉格朗日的框架，这意味着它是从动力学系统的完整描述出发自然导出的。新方程包含了一个与量子力学Bohm电位相关的自耦合项，这使得方程在数学上和物理上都更加复杂，但同时也更准确地反映了粒子在弯曲时空中的行为。 Bohm电位是量子力学的隐变量理论——Bohm诠释中的一个重要概念，它提供了对粒子运动轨迹的量子力学解释。在这里，Bohm电位的引入使得非线性项的出现成为可能，进而导致粒子波动的传播更符合经典几何的预期。论文作者通过数值模拟和解析分析探讨了新方程的性质，展示了它如何改进原始无质量Klein-Gordon方程的预测能力。这种修正对于理解和模拟高能量物理过程，特别是在强引力场或宇宙尺度的背景下，可能具有重要意义，因为它能更准确地描述无质量粒子的传播。此外，由于该研究是开放获取的，这意味着全球的科研人员都能够免费访问并利用这些成果进行进一步的研究，这将有助于促进科学知识的共享和进步。这项工作不仅提出了一个新的物理模型，而且为解决量子场论在非平坦时空中的问题提供了一个有希望的方向。

Eur. Phys. J. C (2017) 77:732

https://doi.org/10.1140/epjc/s10052-017-5330-7

Regular Article - Theoretical Physics

New non-linear modiﬁed massless Klein–Gordon equation

Felipe A. Asenjo

1,2,a

, Sergio A. Hojman

1,3,4,5,b

UAI Physics Center, Universidad Adolfo Ibáñez, Santiago, Chile

Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez, Santiago, Chile

Departamento de Ciencias, Facultad de Artes Liberales, Universidad Adolfo Ibáñez, Santiago, Chile

Departamento de Física, Facultad de Ciencias, Universidad de Chile, Santiago, Chile

Centro de Recursos Educativos Avanzados, CREA, Santiago, Chile

Received: 29 June 2017 / Accepted: 24 October 2017 / Published online: 2 November 2017

Abstract The massless Klein–Gordon equation on arbi-

trary curved backgrounds allows for solutions which develop

“tails” inside the light cone and, therefore, do not strictly

follow null geodesics as discovered by DeWitt and Brehme

almost 60 years ago. A modiﬁcation of the massless Klein–

Gordon equation is presented, which always exhibits null

geodesic propagation of waves on arbitrary curved space-

times. This new equation is derived from a Lagrangian which

exhibits current–current interaction. Its non-linearity is due to

a self-coupling term which is related to the quantum mechan-

ical Bohm potential.

1 Introduction

It is well known that point structureless particles which move

on arbitrary background gravitational ﬁelds follow geodesics

according to the Equivalence Principle (EP) of General Rel-

ativity. In particular, massless particles move along null

geodesics. For example, the crucial light deﬂection obser-

vational results were predicted and matched by considering

photons as massless particles moving along null geodesics.

However, these results are valid under the assumption of

weak and smooth gravitational ﬁelds and light behaving in the

geometrical optics limit (where light propagation is treated

as the motion of massless spinless particles).

The caveat is that point structureless particles do not exist

in Nature. If the particle has structure, then its motion may

be completely different from the one followed by point par-

ticles. This is the case of spinning massive particles mov-

ing on arbitrary curved backgrounds that do not, in general,

follow geodesics [1–7]. On the other hand, Nature is bet-

ter described in terms of ﬁelds which are neither point-like

e-mail: felipe.asenjo@uai.cl

e-mail: sergio.hojman@uai.cl

nor structureless entities, in the sense that they in general

correspond to extended propagating objects that in addition

can carry spin. Therefore, the dynamics of spinning particles

and ﬁelds differs from the one for spinless point particles,

and one should expect that the EP may not be applied in

those cases. The reason is that, for extended (not point-like)

and/or structured (such as spinning) objects, several geodesic

curves cross a sufﬁciently extended region of the body, mak-

ing it subject to tidal forces. Thus, the geodesic path does

not make sense for physical objects different from point-like

ones.

In the spirit of the discussion presented above, one can

expect new features in the dynamics of any ﬁeld coupled to

gravity. The study of the propagation of massless ﬁelds on

arbitrary curved backgrounds has been the subject of research

for a long time [8–22]. In 1960, DeWitt and Brehme [8]

found solutions both for the Klein–Gordon and the Maxwell

ﬁelds on a curved background which exhibit “tails” inside

the light cone, meaning that the propagators for these ﬁelds

do not vanish inside the light cone. These “tails” are a pure

general relativistic effect, showing how the spreading of the

wave ﬁeld over spacetime can modify its dynamics. These

tails can be obtained through the propagators of the ﬁelds, the

Green function [23,24], for linear wave equations in different

backgrounds.

Recently, several new results for the propagation of

(classical and quantum) electromagnetic waves on arbitrary

curved backgrounds suggest non-null geodesic and polariza-

tion dependent propagation of waves [25–36]. Similar results

arise in the study of the propagation of spin ﬁeld waves

[37,38]. Up to now, these new results seem to be neither

conﬁrmed nor refuted by experiments or observations.

One can study many of the “tails” effects for ﬁeld dynam-

ics even at the simplest level. That is the case of the Klein–

Gordon equation for a massless charged ﬁeld  = (x

)

on a background curved spacetime

123

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