非常特殊相对论与洛伦兹违反理论探索

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"PhysicsLettersB734(2014)239–244" 在《非常狭义相对论和洛伦兹违反理论》这篇文章中，作者Jorge Alfaro和Victor O. Rivelles探讨了一种特殊理论，该理论保留了狭义相对论的核心概念，但同时打破了旋转不变性，即不遵循传统的洛伦兹对称性。这种理论被称为非常狭义相对论（Very Special Relativity, VSR）。VSR的关键在于引入了一个固定的空向量，这使得理论具有非均匀的对称性。文章首先介绍了VSR的基本概念，通过一种包含VSR的纺丝颗粒公式来建立理论基础。纺丝颗粒是物理学中描述具有内在旋转性质的粒子的模型，它们在VSR框架下可以呈现出独特的动态行为。通过这个公式，作者能够推导出VSR对麦克斯韦方程的修正。麦克斯韦方程是描述电磁场变化的基本方程，在VSR下，这些方程可能会受到新的物理效应的影响。接下来，作者转向了VSR对托马斯进动的修正。托马斯进动是粒子在相对论性运动中由于相对论效应产生的自旋进动，它是经典电动力学中的一个重要现象。在VSR理论中，托马斯进动的计算需要考虑到那个固定的空向量，这会导致与标准相对论不同的校正项。为了进一步阐述这一点，文章引入了旋转粒子与电磁场的耦合，其中包含了一个旋磁因子，它导致粒子产生磁矩。这与量子力学中的自旋磁矩相呼应，自旋磁矩是粒子自旋与其电荷的乘积，决定了粒子在磁场中的行为。作者随后定义了一个旋转粒子的自旋向量，这个向量在VSR环境下同样需要满足特定的动力学方程。他们证明了这个自旋向量遵循BMT（ Bargmann-Michel-Telegdi）方程，这是描述带电粒子自旋在电磁场中进动的经典方程。最后，作者将所有这些概念归纳到VSR的上下文中，分析了VSR对BMT方程的具体贡献。这意味着在VSR理论中，旋转粒子的自旋进动将受到那个固定空向量的影响，从而可能导致与传统相对论预测的不同实验结果。这篇文章通过深入研究VSR，揭示了其对基础物理理论如相对论和电磁学的潜在影响，提供了理解可能存在的洛伦兹对称性违反的新视角。这项工作对于未来实验探索物理定律的潜在局限性和寻找新的物理现象具有重要意义。

Physics Letters B 734 (2014) 239–244

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Very special relativity and Lorentz violating theories

Jorge Alfaro

, Victor O. Rivelles

Facultad de Física, Pontiﬁcia Universidad Católica de Chile, Casilla 306, Santiago 22, Chile

Instituto de Física, Universidade de São Paulo, C. Postal 66318, São Paulo, SP, Brazil

a r t i c l e i n f o a b s t r a c t

Article history:

Received

13 January 2014

Received

in revised form 16 May 2014

Accepted

22 May 2014

Available

online 2 June 2014

Editor: M.

Cveti

Very special relativity (VSR) keeps the main features of special relativity but breaks rotational invariance.

We will show how VSR like terms which depend on a ﬁxed null vector can be generated systematically.

We start with a formulation for a spinning particle which incorporates VSR. We then use this formulation

to derive the VSR modiﬁcations to the Maxwell equations. Next we consider VSR corrections to Thomas

precession. We start with the coupling of the spinning particle to the electromagnetic ﬁeld adding a

gyromagnetic factor which gives rise to a magnetic moment. We then propose a spin vector in terms of

the spinning particle variables and show that it obeys the BMT equation. All this is generalized to the

VSR context and we ﬁnd the VSR contributions to the BMT equation.

(http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP

1. Introduction

The standard model of particle physics is a well established and

experimentally conﬁrmed theory but it needs to be extended in

order to incorporate some known phenomena like, for instance,

neutrino masses and dark matter. Possibly the standard model is

the low energy limit of a larger theory which hopefully includes

gravity. Since the present experimental data are not enough to

point out how to extend the standard model we must seek for

small deviations of it which could be detected at low energies. One

possibility is that ﬁelds from a more complete theory couple to

the standard model ﬁelds like constant background ﬁelds causing

deviations of Lorentz symmetry [1]. This is a very active line of in-

vestigation

with many theoretical results awaiting for experimental

conﬁrmation (for a review see [2]). Usually such proposals have as

a consequence that the dispersion relation for light is modiﬁed.

Amore conservative alternative would keep the essential features

of special relativity, like the constancy of the velocity of light, but

leave aside rotation symmetry for instance. This can be achieved

by taking subgroups of the Lorentz group which preserve the con-

stancy

of the velocity of light. Such subgroups were identiﬁed and

used to built what is called very special relativity (VSR) [3]. One of

its main features is that the inclusion of P , T or CP symmetries

enlarges VSR to the full Lorentz group so that VSR is only rele-

vant

in theories where one of the discrete symmetries is broken.

Two subgroups of the Lorentz group, SIM(2) and HOM(2), have the

E-mail addresses: jalfaro@puc.cl (J. Alfaro), rivelles@fma.if.usp.br (V.O. Rivelles).

property of rescaling a ﬁxed null vector n

. Then terms contain-

ing

ratios of contractions of n

with other kinematic vectors will

be invariant under transformations of these subgroups. A proposal

to generate mass for neutrinos along these lines was presented in

[4] where an equation for a left-handed fermion incorporating VSR

was given



/p −



=0, (1)

where m sets the VSR scale. When the equation of motion is

squared we ﬁnd that it describes a free fermion of mass m. The

price to be paid is the presence of non-local operators as well as

the lack of rotational symmetry. In this way it is possible to save

some of the important effects of special relativity and consider

possible violations of space isotropy. Several aspects of VSR have

been considered, like the inclusion of supersymmetry [5,6], curved

spaces [7,8], noncommutativity [9,10], dark matter [11] and also in

cosmology [12].

can take this speciﬁc realization of VSR and consider the

addition of interaction terms in the context of the usual Lorentz

violating theories. We can regard the inclusion of operators con-

taining

a constant and null vector n

as determining a preferred

direction in space. It breaks Lorentz symmetry to ISO(2) but allow-

ing

a scale transformation on n

the symmetry can be enlarged

to SIM(2). So the inclusion of terms containing ratios of n

con-

tracted

with other kinematic vectors will lead to the consideration

of VSR like terms as that in (1). When a Lorentz invariant action is

extended by the addition of Lorentz violating operators the coeﬃ-

cients

of such operators are in general arbitrary and unrelated to

http://dx.doi.org/10.1016/j.physletb.2014.05.068

0370-2693/

SCOAP

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