Based on a linear realization formulation of a quantum relativity -- the proposed relativity for quantum `space-time', we introduce the Poincaré-Snyder relativity and Snyder relativity as relativities in between the latter and the well known Galilean and Einstein cases. We discuss how the Poincaré-Snyder relativity may provide a stronger framework for the description of the usual (Einstein) relativistic quantum mechanics and beyond. In particular, we discuss a geometric quantization picture through the U(1) central extension of the relativity group, which had been establish to work well for the Galilean case but not for the Einstein case. We discuss similarities and differences between our Poincaré-Snyder picture with a still not fully understood $σ$ variable as the `evolution' parameter and some use of an invariant time or the proper time parameter in some earlier formulations with very similar mathematical structure. The study is a first step towards the investigation of physics of the $σ$ variable at the Poincaré-Snyder setting, plausible leading to experimental signatures to be studied.
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.
Poincaré, the Dynamics of the Electron, and Relativity
在1905年6月5日庞加莱提出了一份说明，科学学院题为“苏尔德拉dynamique德l'électron”（“论动力学的电子”）。在简要回顾了导致庞加莱写这篇文章的背景之后，我们对其内容进行了评论。我们强调，庞加莱的电子模型在于假设（空心）电子的世界管内部充满了正的宇宙学常数。然后，我们讨论了几个新的贡献，物理数学方面的狭义相对论这是勾勒在说明，虽然他们被淡化庞加莱谁形容为只有完成了1904年5月的结果洛伦兹“dans quelques points de detail”（“在几个点的细节”）。
Poincare and relativity: the logic of the 1905 Palermo Memoir
我们强调了四点，这已被忽视或低估了之前，并允许更好地理解“Sur la dynamique de l 'electron”：（i）使用庞加莱的积极洛伦兹变换（助推）;（ii）的必要性，力学所需的一组条件l=1消除膨胀;（iii）的关键作用的行动（电磁或不）和它的不变性;（四）作为例子或反例的电子模型的数学地位。
The Poincaré (inhomogeneous Lorentz) group underlies special relativity. In these lectures a consistent formalism is developed allowing an appropriate gauging of the Poincaré group. The physical laws are formulated in terms of points, orthonormal tetrad frames, and components of the matter fields with respect to these frames. The laws are postulated to be gauge invariant under local Poincaré transformations. This implies the existence of 4 translational gauge potentials ${e}^α$ (``gravitons") and 6 Lorentz gauge potentials $Γ^{αβ}$ (``rotons") and the coupling of the momentum current and the s…
Hidden Quantum Group Structure in Einstein's General Relativity
本文提出了一种新的形式方案，其中爱因斯坦的经典广义相对论作为一个单参数的不同理论家族的共同不变部分出现。这是通过用q变形的庞加莱群代替普通四分体形式的庞加莱群来实现的，通常的理论在q=1时恢复。虽然这两种理论都是用非对易的维尔拜因场和自旋连接场来表述的，但它们都有相同的度规扇区，导致了普通的爱因斯坦-希尔伯特作用量和相应的运动方程。克里斯塔费尔符号和黎曼张量的分量是普通的交换数，并且具有作为维因中适当的双线性建立的度量张量的通常形式。此外，我们展示了一个单参数家庭的广义相对论的哈密顿形式主义，通过显示一个典型的形式主义a` la Ashtekar可以建立任何值的q。约束仍然是多项式的，但是泊松括号对于q不等于1不是斜对称的。
During the last five decades, gravity, as one of the fundamental forces of nature, has been formulated as a gauge theory of the Weyl-Cartan-Yang-Mills type. The present text offers commentaries on the articles from the most prominent proponents of the theory. In the early 1960s, the gauge idea was successfully applied to the Poincaré group of spacetime symmetries and to the related conserved energy-momentum and angular momentum currents. The resulting theory, the Poincaré gauge theory, encompasses Einstein's general relativity as well as the teleparallel theory of gravity as subcases. The spac…
General relativity has two fundamental problems that render it unsuitable for tackling the gravitational field's quantization. The first problem is the lack of a genuine gravitational variable representing gravitation only, inertial forces apart. The second problem is its incompatibility with quantum mechanics, a problem inherited from the more fundamental conflict of special relativity with quantum mechanics. A procedure to overcome these difficulties is outlined, which amounts to replacing general relativity with its teleparallel equivalent and the Poincaré-invariant special relativity with …
According to Yang \& Mills (1954), a {\it conserved} current and a related rigid (`global') symmetry lie at the foundations of gauge theory. When the rigid symmetry is extended to a {\it local} one, a so-called gauge symmetry, a new interaction emerges as gauge potential $A$; its field strength is $F\sim {\rm curl} A$. In gravity, the conservation of the energy-momentum current of matter and the rigid translation symmetry in the Minkowski space of special relativity lie at the foundations of a gravitational gauge theory. If the translation invariance is made local, a gravitational potential $\…
