各向异性狭义相对论（ASR）是一种在时空中具有优先方向的自然相对论理论。通过将时空的“完全各向同性”约束放宽为“偏好一个方向”，我们得到 Minkowski 度规的微扰修正，即对于小的微扰参数 ϕ，有 $\mathscr{g}_{μν}\simeqη_{μν}+2ϕε_{μν}$。ASR 的对称群被求得具有六个生成元，并满足完整的 Lorentz 群代数。然而，这些生成元会被微扰参数 ϕ 形变。因此，ASR 保留了狭义相对论（SR）的相同表示，但同时允许 Lorentz 不变性破缺。该理论中的任何不变量都是由 ${g}_{μν}$ 介导的两个逆变四维矢量的内积。粒子质量被修正为 $m^2=P^μ\mathscr{g}_{μν}P^ν$；在第一近似阶，与 SR 中的粒子质量相比，它多出一项 $(2ϕε_{μν})P^μP^ν$。因此，ASR 的一个应用例如是以自然方式解释中微子味振荡实验，而无需违反轻子数或添加惰性右手中微子。粒子质量并不是 ASR 中唯一被修正的量；由于各向异性度规 $\mathscr{g}_{μν}$ 被用于缩并任意一对协变—逆变指标，任何标量（例如场的 Lagrangian）也会被修正。作为 ASR 更一般的结果，任何量子场论（QFT）都会变成各向异性的，因为其 Lagrangian 必须包含各向异性度规。因此，我们提供了一种构造各向异性 QFT 的程序，其中 Lorentz 不变的 Lagrangian 被其 ASR 版本所替代。
英文摘要：
Anisotropic Special Relativity (ASR) is the relativistic theory of nature with a preferred direction in space-time. By relaxing the \textit{full-isotropy} constraint on space-time to the \textit{preference of one direction}, we obtain a perturbative modification of the Minkowski metric as $\mathscr{g}_{μν}\simeqη_{μν}+2ϕε_{μν}$ for a small perturbation parameter $ϕ$. The symmetry group of ASR is obtained to have six generators satisfying the full Lorentz group algebra. However, the generators are deformed by the perturbation parameter $ϕ$. So, ASR retains the same representations of Special Relativity (SR) but allows for Lorentz-invariant violation at the same time. Any invariant quantity of the theory is the inner product of two contravariant 4-vectors mediated by ${g}_{μν}$. The mass of a the particle is modified to $m^2=P^μ\mathscr{g}_{μν}P^ν$ which, in the first approximation level, has the extra term $(2ϕε_{μν})P^μP^ν$ compared to the mass of the particle in SR. So, one application of ASR is, for example, to explain the neutrino flavour oscillation experiments in a natural way without violating the lepton number or adding sterile right-handed neutrinos. The mass of a particle is not the only quantity that is modified in ASR; any scalar quantity such as the Lagrangian of fields are also modified since the anisotropic metric $\mathscr{g}_{μν}$ is used to contract any pair of covariant-contravariant indices. As a more general consequence of ASR, any Quantum Field Theory (QFT) becomes anisotropic since the Lagrangian must contain the anisotropic metric. So, we provide a procedure to make anisotropic QFTs where the Lorentz-invariant Lagrangians are replaced with their ASR version.
In the homogeneous and isotropic Friedmann-Robertson-Walker minisuperspace model, it is known that there are no Euclidean wormhole solutions in the pure gravity system. Here it is demonstrated explicitly that in Taub cosmology, which is one of the simplest anisotropic cosmology models, wormhole solutions do exist in pure general relativity in both classical and quantum contexts.
We show the first simple, systematic and direct approach to decoupling gravitational sources in general relativity. As a direct application, a robust and simple way to generate anisotropic solutions for self-gravitating systems from perfect fluid solutions is presented.
We show that if Yang's model as the earliest TSR is completed, it should contain both Snyder's quantized space-time model as the earliest DSR, the de Sitter SR and their duality.
We reformulate Special Relativity by a quaternionic algebra on reals. Using {\em real linear quaternions}, we show that previous difficulties, concerning the appropriate transformations on the $3+1$ space-time, may be overcome. This implies that a complexified quaternionic version of Special Relativity is a choice and not a necessity.
The current status of Doubly Special Relativity research program is shortly presented. I dedicate this paper to my teacher and friend Professor Jerzy Lukierski on occasion of his seventieth birthday.
I review the conceptual, algebraical, and geometrical structure of Doubly Special Relativity. I also speculate about the possible relevance of DSR for quantum gravity phenomenology.
We discuss the definition of particle velocity in doubly relativity theories. The general formula relating velocity and four-momentum of particle is given.
We show that depending on the direction of deformation of $κ$-Poincaré algebra (time-like, space-like, or light-like) the associated phase spaces of single particle in Doubly Special Relativity theories have the energy-momentum spaces of the form of de Sitter, anti-de Sitter, and flat space, respectively.
A field of random space-time events exhibiting complete spatial-temporal randomness appears statistically identical to all observers. Boost invariant lengths naturally emerge when we examine fluctuation scales of this field such as the nearest neighbor distance. If we interpret Planck's length as the characteristic fluctuation scale of quantum gravity, its boost invariance can then be understood without modifying Special Relativity.
The generalization of Cohen and Glashow's Very Special Relativity to curved space-times is considered. Gauging the SIM(2) symmetry does not, in general, provide the coupling to the gravitational background. However, locally SIM(2) invariant Lagrangians can always be constructed. For space-times with SIM(2) holonomy, they describe chiral fermions propagating freely as massive particles.
By means of Dirac procedure, we re-examine Yang's quantized space-time model, its relation to Snyder's model, the de Sitter special relativity and their UV-IR duality. Starting from a dimensionless dS_5-space in a 5+1-d Mink-space a complete Yang model at both classical and quantum level can be presented and there really exist Snyder's model, the dS special relativity and the duality.
Keeping the two fundamental postulates of the special theory of relativity, the principle of relativity and the constancy of the one-way velocity of light in all inertial frames of reference, and assuming two generalized Finslerian structures of gravity-free space and time in the usual inertial coordinate system, we can modify the special theory of relativity. The modified theory is still characterized by the localized Lorentz transformation between any two usual inertial coordinate systems. It together with the quantum mechanics theory features a convergent and invariant quantum field theory.…
In these notes, based on the lectures given at 40th Winter School on Theoretical Physics, I review some aspects of Doubly Special Relativity (DSR). In particular, I discuss relation between DSR and quantum gravity, the formal structure of DSR proposal based on $κ$-Poincaré algebra and non-commutative $κ$-Minkowski space-time, as well us some results and puzzles related to DSR phenomenology.
We reconsider in details the Dirac equation in the context of the Magueijo-Smolin approach to the Doubly Special Relativity. Starting from the deformed dispersion relation we obtain the Dirac equation in momentum space, allowing us to achieve a more in-depth study of its semiclassical approach. Finally by means of a deformed correspondence principle we gain access to an equation in the position space.
In searching for the essence of special relativity, we have been gradually accumulating ten arguments focusing on one fundamental postulate based on quantum mechanics.A particle is always not pure. It always contain two contradictory fields, $ϕ(\vec{x},t)$ and $χ(\vec{x},t)$,which are coupled together with the symmetry $ϕ(-\vec{x},-t)\longrightarrowχ(\vec{x},t)$ and $χ(-\vec{x},-t)\longrightarrowϕ(\vec{x},t)$.
Deformed Special Relativity (DSR) is obtained by imposing a maximal energy to Special Relativity and deforming the Lorentz symmetry (more exactly the Poincaré symmetry) to accommodate this requirement. One can apply the same procedure deforming the Galilean symmetry in order to impose a maximal speed (the speed of light). This leads to a non-commutative space structure, to the expected deformations of composition of speed and conservation of energy-momentum. In doing so, one runs into most of the ambiguities that one stumbles onto in the DSR context. However, this time, Special Relativity is t…
Locality of interactions is an essential ingredient of Special Relativity. Recently, a new framework under the name of relative locality \cite{AmelinoCamelia:2011bm} has been proposed as a way to consider Planckian modifications of the relativistic dynamics of particles. We note in this paper that the loss of absolute locality is a general feature of theories beyond Special Relativity with an implementation of a relativity principle. We give an explicit construction of such an implementation and compare it both with the previously mentioned framework of relative locality and the so-called Doub…
Some models (such as the Skyrme model, a low-energy effective field theory for QCD) suggest that the high-density matter prevailing in neutron star interiors may be significantly anisotropic. Anisotropy is known to affect the bulk properties of nonrotating neutron stars in General Relativity. In this paper we study the effects of anisotropy on slowly rotating stars in General Relativity. We also consider one of the most popular extensions of Einstein's theory, namely scalar-tensor theories allowing for spontaneous scalarization (a phase transition similar to spontaneous magnetization in ferrom…
The effects of a non-vanishing value for the cosmological constant in the scenario of Lorentz symmetry breaking recently proposed by Cohen and Glashow (which they denote as Very Special Relativity) are explored and observable consequences are pointed out.
It is plausible that quantum gravity effects may lead us to a description of Nature beyond the framework of special relativity. In this case, either the relativity principle is broken or it is maintained. These two scenarios (a violation or a deformation of special relativity) are very different, both conceptually and phenomenologically. We discuss some of their implications on the description of events for different observers and the notion of spacetime.
We show that some of the recent results reported in gr-qc/0308049 are based on assumptions which are in contrast with general properties of ``Doubly Special Relativity'' and/or with Planck-scale physics models.
The braneworld model of Dvali-Gabadadze-Porrati (DGP) provides an interesting alternative to a positive cosmological constant by modifying gravity at large distances. We investigate the asymptotic behavior of homogeneous and anisotropic cosmologies on the DGP brane. It is shown that all Bianchi models except type IX isotropize, as in general relativity, if the so called $E_{μν}$ term satisfies some energy condition. Isotropization can proceed slower in DGP gravity than in general relativity.
Based on the principle of relativity and the postulate on universal invariant constants ($c,l$) as well as Einstein's isotropy conditions, three kinds of special relativity form a triple with a common Lorentz group as isotropy group under full Umov-Weyl-Fock-Lorentz transformations among inertial motions.
In the context of departures from Special Relativity written as a momentum power expansion in the inverse of an ultraviolet energy scale M, we derive the constraints that the relativity principle imposes between coefficients of a deformed composition law, dispersion relation, and transformation laws, at first order in the power expansion. In particular, we find that, at that order, the consistency of a modification of the energy-momentum composition law fixes the modification in the dispersion relation. We therefore obtain the most generic modification of Special Relativity that preserves the …
We introduce a Lorentz-covariant description of tachyons, free of inconsistencies. Our approach is based on an appropriate extension of the special relativity beyond the light barrier, owing to the freedom of synchronization of distant clocks.
We demonstrate that Einstein's general relativity theory arises as a special case in the framework of the Poincaré gauge theory of gravity under the assumption of a suitable nonminimal coupling of matter to the Riemann-Cartan geometry of spacetime.
We consider the Doubly Special Relativity (DSR) generalization of Dirac equation in an external potential in the Magueijo-Smolin base. The particles obey a modified energy-momentum dispersion relation. The semiclassical diagonalization of the Dirac Hamiltonian reveals the intrinsic Berry phase effects in the particle dynamics.
By studying the notion of a fundamentally minimal length scale in asymptotically safe gravity we find that a specific version of deformed special relativity (DSR) naturally arises in this approach. We then consider two thought experiments to examine the interpretation of the scenario and discuss similarities and differences to other approaches to DSR.
We analyse the quantum geometry of 3-dimensional deformed special relativity (DSR) and the notion of spacetime points in such a context, identified with coherent states that minimize the uncertainty relations among spacetime coordinates operators. We construct this system of coherent states in both the Riemannian and Lorentzian case, and study their properties and their geometric interpretation.
This is a write-up of two lectures delivered at COST CA18108 First Training School. They cover the motivations and some basic technical results in the field of Doubly Special Relativity and Relative Locality. The energy-dependent speed of light is a recurring theme here. The soccer ball problem is also briefly described. The video recording of these and other lectures can be found at https://www.youtube.com/channel/UChRfzgjTkO9sOkwq358kHZw/videos.
We consider a single free spin 1/2 particle. The reduced density matrix for its spin is not covariant under Lorentz transformations. The spin entropy is not a relativistic scalar and has no invariant meaning.
Double Special Relativity theories are the relativistic theories in which the transformations between inertial observers are characterized by two observer-independent scales of the light speed and the Planck length. We study two main examples of these theories and want to show that these theories are not the new theories of relativity, but only are re-descriptions of Einstein's special relativity in the non-conventional coordinates.
From the quantum analog of the Iwasawa decomposition of $SL(2,C)$ group and the correspondence between quantum $SL(2,C)$ and Lorentz groups we deduce the different properties of the Hopf algebra representing the boost of particles in noncommutative special relativity. The representation of the boost in the Hilbert space states is investigated and the addition rules of the velocities are established from the coaction. The q-deformed Clebsch-Gordon coefficients descibing the transformed states of the evolution of particles in noncommutative special relativity are introduced and their explicit ca…
We examine the transformation of particle trajectories in models with deformations of Special Relativity that have an energy-dependent and observer-independent speed of light. These transformations necessarily imply that the notion of what constitutes the same space-time event becomes dependent on the observer's inertial frame. To preserve observer-independence, the such arising nonlocality should not be in conflict with our knowledge of particle interactions. This requirement allows us to derive strong bounds on deformations of Special Relativity and rule out a modification to first order in …
We argue that existing doubly special relativities may not be operationally distinguishable from the special relativity. In the process we point out that some of the phenomenologically motivated modifications of dispersion relations, and arrived conclusions, must be reconsidered. Finally, we reflect on the possible conceptual issues that arise in quest for a theory of spacetime with two invariant scales.
Based on the doubly special relativity we find a new type of generalized uncertainty principle (GUP) where the coordinate remain unaltered at the high energy while the momentum is deformed at the high energy so that it may be bounded from the above. For this GUP, we discuss some quantum mechanical problems in one dimension such as box problem, momentum wave function, and harmonic oscillator problem.
