在物理学中，特别是在位错的规范理论中，
Cartan's spiral staircase in physics and, in particular, in the gauge theory of dislocations
摘要：
在1922年，嘉当介绍了微分几何，除了黎曼曲率，新的概念挠。他通过“螺旋楼梯”可视化了三维（3D）中扭转的均匀和各向同性分布，他从3D欧几里得空间开始并通过螺旋运动定义了一个新的连接。我们详细描述了这个几何过程，并定义了相应的连接和扭转。这个问题的跨学科性质已经从嘉当的讨论中显而易见，因为他认为-但从未证明-螺旋楼梯应对应于一个连续的恒定压力和恒定的内部扭矩。我们讨论在物理学中实现螺旋楼梯的地方：（i）在Cosserat介质的连续介质力学中，（ii）在（相当投机）3d引力理论，即a）在3d爱因斯坦-嘉当引力中-这是嘉当的恒压和恒定内转矩的情况-和b）在3d庞加莱规范理论中与Mielke-Baekler拉格朗日量，并且，最终，（iii）在Lazar等人的位错规范场中，正如我们通过安排螺旋位错的适当分布首次证明的那样。我们的重点是位错场论的讨论。
英文摘要：
In 1922, Cartan introduced in differential geometry, besides the Riemannian curvature, the new concept of torsion. He visualized a homogeneous and isotropic distribution of torsion in three dimensions (3d) by the "helical staircase", which he constructed by starting from a 3d Euclidean space and by defining a new connection via helical motions. We describe this geometric procedure in detail and define the corresponding connection and the torsion. The interdisciplinary nature of this subject is already evident from Cartan's discussion, since he argued - but never proved - that the helical staircase should correspond to a continuum with constant pressure and constant internal torque. We discuss where in physics the helical staircase is realized: (i) In the continuum mechanics of Cosserat media, (ii) in (fairly speculative) 3d theories of gravity, namely a) in 3d Einstein-Cartan gravity - this is Cartan's case of constant pressure and constant intrinsic torque - and b) in 3d Poincare gauge theory with the Mielke-Baekler Lagrangian, and, eventually, (iii) in the gauge field theory of dislocations of Lazar et al., as we prove for the first time by arranging a suitable distribution of screw dislocations. Our main emphasis is on the discussion of dislocation field theory.
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