论文标题
从弹性四元到矩形Vielbein
From elasticity tetrads to rectangular vielbein
论文作者
论文摘要
该论文致力于记忆Igor E. Dzyaloshinsky。在我们的普通论文中，即Dzyaloshinskii和G.E.沃洛维克（Volovick），凝结物质的泊松支架，安。物理。 {\ bf 125} 67--97（1980），我们讨论了用重力场变量来描述的弹性理论 - 弹性Vielbein $e_μ^a $。它们来自相位场，描述了晶体的变形。弹性的重要属性Vielbein $ e^a_μ$通常不是方形的。虽然时空索引$μ$取值$μ=（0,1,2,3）$，在晶体中，索引$ a =（1,2,3）$，在涡旋晶格$ a =（1,2）$，而在锡克克液晶中只有一个相位场，$ a = 1 $。这些相位场可以视为自旋量规场，与标准模型（SM）或大统一（GUT）中的量规场相似。 另一方面，矩形vielbein $ e^μ_a$可能会在狄拉克材料的狄拉克点附近出现。特别是，在Spin-Triplet超级流体$^3 $的平面阶段中，时空指数$μ=（0,1,2,3）$，而旋转指数$ a $ take值$ a =（0,1,2,3,4）$。尽管这些$（4 \ times 5）$ vielbein描述了dirac fermions是矩形，但狄拉克果皮粒子的有效度量$ g^{μν} $仍然存在（3+1）二维。所有这些都表明，通过引入矩形vielbein，旋转场属于较高基团的矩形Vielbein，可能会扩展爱因斯坦 - 卡丹的重力，其中可能包括SM甚至肠子。
The paper is devoted to the memory of Igor E. Dzyaloshinsky. In our common paper I.E. Dzyaloshinskii and G.E. Volovick, Poisson brackets in condensed matter, Ann. Phys. {\bf 125} 67--97 (1980), we discussed the elasticity theory described in terms of the gravitational field variables -- the elasticity vielbein $E_μ^a$. They come from the phase fields, which describe the deformations of crystal. The important property of the elasticity vielbein $E^a_μ$ is that in general they are not the square mstrices. While the spacetime index $μ$ takes the values $μ=(0,1,2,3)$, in crystals the index $a=(1,2,3)$, in vortex lattices $a=(1,2)$, and in smectic liquid crystals there is only one phase field, $a=1$. These phase fields can be considered as the spin gauge fields, which are similar to the gauge fields in Standard Model (SM) or in Grand Unification (GUT). On the other hand, the rectangular vielbein $e^μ_a$ may emerge in the vicinity of Dirac points in Dirac materials. In particular, in the planar phase of the spin-triplet superfluid $^3$He the spacetime index $μ=(0,1,2,3)$, while the spin index $a$ takes values $a=(0,1,2,3,4)$. Although these $(4 \times 5)$ vielbein describing the Dirac fermions are rectangular, the effective metric $g^{μν}$of Dirac quasiparticles remains (3+1)-dimensional. All this suggests the possible extension of the Einstein-Cartan gravity by introducing the rectangular vielbein, where the spin fields belong to the higher groups, which may include SM or even GUT groups.
