Weyl’s Gauge Argument

The standard $\mathbb{U}(1)$ “gauge principle” or “gauge argument” produces an exact potential A=dλ and a vanishing field F=d ² λ=0. Weyl (in Z. Phys. 56:330–352, 1929; Rice Inst. Pam. 16:280–295, 1929) has his own gauge argument, which is sketchy, archaic and hard to follow; but at least it produces an inexact potential A and a nonvanishing field F=dA≠0. I attempt a reconstruction.     

Quadratic Weyl points in metacrystals

The discovery of new topological states has become an important goal in the multidisciplinary fields of physics and material science [1,2]. In addition to topological insulators,topological semimetals without band gaps have been the focus of research efforts. Typical examples include Weyl and Dirac semimetals, which feature isolated 2-band and 4-band linear crossing points in their band structures.     

The Fresnel—Weyl complementary transformation

Based on the newly developed coherent-entangled state representation,we propose the so-called Fresnel-Weyl complementary transformation operator.The new operator plays the roles of both Fresnel transformation(for(a 1 a 2)/√ 2) and the Weyl transformation(for(a 1 + a 2)/√ 2).Physically,(a 1 a 2)/√ 2 and(a 1 + a 2)/√ 2 could be a symmetric beamsplitter’s two output fields for the incoming fields a 1 and a 2.We show that the two transformations are concisely expressed in the coherent-entangled state representation as a projective operator in the integration form.     

Spin waves in magnetic Weyl semimetals

Intrinsic magnetic topological materials,namely the stoichiometric magnetic compounds possessing both inherent magnetic order and topological electronic states,have attracted tremendous interest in the research of condensed matter physics and materials science[1].Such materials not only bring new opportunities to realize many exotic topological phenomena under time-reversal symmetry breaking,but also show great potential in applications of quantum technology[2].     

漫谈第二类Weyl半金属

近日,清华大学物理系研究人员及合作者在Weyl半金属研究中取得新进展:首次在实验上观测到第二类Weyl半金属(MoTe_2)上的拓扑费米弧,在真实材料中证实了破坏洛伦兹不变性的第二类Weyl费米子的存在。相关文章于2016年9月6日在Nature Physics上发表~([1])。这是继去年第一类Weyl费米子得到证实之后凝聚态物理研究上的又一重大突破。什么是第二类Weyl费米子?它与第一类Weyl费米子有何区别?作者通过本文来做一个简单的介绍。     

Hölder-Continuous Rough Paths by Fourier Normal Ordering

We construct in this article an explicit geometric rough path over arbitrary d-dimensional paths with finite 1/α-variation for any ${\alpha\in(0,1)}By a combination of asymptotic ODE estimates and numerical Evans function calculations, we establish stability of viscous shock solutions of the isentropic compressible Navier–Stokes equations with γ-law pressure (i) in the limit as Mach number M goes to infinity, for any γ ≥ 1 (proved analytically), and (ii) for M ≥ 2,500, ${\gamma\in [1,2.5]}We construct in this article an explicit geometric rough path over arbitrary d-dimensional paths with finite 1/α-variation for any a ? (0,1){\alpha\in(0,1)}. The method may be coined as ‘Fourier normal ordering’, since it consists in a regularization obtained after permuting the order of integration in iterated integrals so that innermost integrals have highest Fourier frequencies. In doing so, there appear non-trivial tree combinatorics, which are best understood by using the structure of the Hopf algebra of decorated rooted trees (in connection with the Chen or multiplicative property) and of the Hopf shuffle algebra (in connection with the shuffle or geometric property). H?lder continuity is proved by using Besov norms. The method is well-suited in particular in view of applications to probability theory (see the companion article [34] for the construction of a rough path over multidimensional fractional Brownian motion with Hurst index α < 1/4, or [35] for a short survey in that case).     

用编序算子简化Weyl规则

定义一编序算子，在通常条件下，对Weyl规则进行了简化，使其形式更筒洁，应用更方便．     

Massive mesons in Weyl–Dirac theory

In order to study the mass generation of the vector fields in the framework of a conformal invariant gravitational model, the Weyl–Dirac theory is considered. The mass of the Weyl’s meson fields plays a principal role in this theory, it connects basically the conformal and gauge symmetries. We estimate this mass by using the large-scale characteristics of the observed universe. To do this we firstly specify a preferred conformal frame as a cosmological frame, then in this frame, we introduce an exact possible solution of the theory. We also study the dynamical effect of the massive vector meson fields on the trajectories of an elementary particle. We show that a local change of the cosmological frame leads to a Hamilton–Jacobi equation describing a particle with an adjustable mass. The dynamical effect of the massive vector meson field presents itself in the form of a correction term for the mass of the particle.     

用编序算子简化Weyl规则

定义一编序算子，在通常条件下，对Ｗｅｙｌ规则进行了简化，使其形式更简洁，应用更方便．     

Weyl monopoles dance with the spin waves

In a Weyl semimetal(WSM),the conduction and valence bands cross each other near the Fermi energy,and the crossing points,called Weyl points,exhibit a monopole-like distribution of the Berry curvature.The Berry curvature is a fictitious magnetic field in the momentum-space and induces the anomalous velocity to the real-space electron motion.Therefore,Weyl monopoles play essential roles in the charge transport,for example,the anomalous Hall effect(AHE).The anomalous Hall conductivity(AHC)is nearly proportional to the distance between the Weyl point pairs with opposite chirality.The magnetic order and spin structure sensitively modify Weyl point positions and energies.     

经典函数与量子算符的Weyl对应

提出了通过定义一种算符编序来处理含有不对易因子的积分的思想,研究了经典函数与量子算符的Weyl对应问题,得到了计算简捷的Weyl对应规则之数学显式,并给出了一种证明量子力学表象完备性关系的新方法.以实例支持了Weyl对应规则的正确性,说明了Weyl对应规则理论是自洽的.     

Negativity Bounds for Weyl–Heisenberg Quasiprobability Representations

The appearance of negative terms in quasiprobability representations of quantum theory is known to be inevitable, and, due to its equivalence with the onset of contextuality, of central interest in quantum computation and information. Until recently, however, nothing has been known about how much negativity is necessary in a quasiprobability representation. Zhu (Phys Rev Lett 117 (12):120404, 2016) proved that the upper and lower bounds with respect to one type of negativity measure are saturated by quasiprobability representations which are in one-to-one correspondence with the elusive symmetric informationally complete quantum measurements (SICs). We define a family of negativity measures which includes Zhu’s as a special case and consider another member of the family which we call “sum negativity.” We prove a sufficient condition for local maxima in sum negativity and find exact global maxima in dimensions 3 and 4. Notably, we find that Zhu’s result on the SICs does not generally extend to sum negativity, although the analogous result does hold in dimension 4. Finally, the Hoggar lines in dimension 8 make an appearance in a conjecture on sum negativity.     

On Measuring Standards in Weyl’s Geometry

In Weyl’s geometry the nonintegrability problem and difficulties in defining measuring standards are reconsidered. Approaches removing the nonintegrability of length in the interior of atoms are given, so that atoms can serve as measuring standards. The Weyl space becomes a well founded framework for classical theories of electromagnetism and gravitation.     

Noncrossing Normal Ordering for Functions of Boson Operators

Normally ordered forms of functions of boson operators are important in many contexts in particular concerning Quantum Field Theory and Quantum Optics. Beginning with the seminal work of Katriel (Lett. Nuovo Cimento 10(13):565–567, 1974), in the last few years, normally ordered forms have been shown to have a rich combinatorial structure, mainly in virtue of a link with the theory of partitions. In this paper, we attempt to enrich this link. By considering linear representations of noncrossing partitions, we define the notion of noncrossing normal ordering. Given the growing interest in noncrossing partitions, because of their many unexpected connections (like, for example, with free probability), noncrossing normal ordering appears to be an intriguing notion. We explicitly give the noncrossing normally ordered form of the functions (a ^r (a ^† ) ^s ) ⁿ ) and (a ^r +(a ^† ) ^s ) ⁿ , plus various special cases. We are able to establish for the first time bijections between noncrossing contractions of these functions, k-ary trees and sets of lattice paths.     

Weyl规则的三维表述及其若干应用

对于形如ｘｍｐｎ的力学量，用Ｗｅｙｌ规则构造的量子力学算符，计及了ｍ个ｘ和ｎ个ｐ的全部不尽相异排列方式，这是与 Ｂｏｈｍ规则不同的．本文将Ｗｅｙｌ规则推广到三维情况，得出了与力学量 Ｐ（ ｒ）Ｐ， Ｆ（ ｒ）· ｐ， Ｆ（ ｒ） ｘ ｐ，Ｆ（ ｒ） ｐ２和 Ｆ（ｒ）ｐ２相对应的厄密算符的 Ｗｅｙｌ表示，对于后两种力学量，Ｗｅｙｌ表示和 Ｂｏｈｍ表示之间存在差异．     

关于光子相算符的Weyl对应理论

本文用湮灭算符的极分解的思想来研究单模相算符,指出相算符可以作为量子谐振子海森堡方程的解而被引入。我们还审视量子光学的相算符的经典对应,即用算符的Weyl编序公式和Wigner算符的Weyl排序形式我们直接导出光子相算符的经典Weyl对应,发现它确实对应一个经典相。     

电流密度算子、对应原理和Weyl规则

一、问题的提出: 人们熟知的量子力学中单体的电流密度为另外,人们又知道量子力学中的对应原理:如果经典力学中的力学量为F(x,p),则对应量子力学中的力学量可用下列算子表示(1.2)在经典电动力学中,电流为可测量的量,其密度表式为 j=pv(1.3)p为电荷密度,v为速度它对应的量子力学表式应为 式(1.1)来自电荷守恒与Schrodinger方程,式(1.4)来自经典表式与“对应原理”,仿佛都有道理,但结果不同.那么观念问题出在哪里呢? 鉴于电流密度是个常见的物理量,它在实际中很有用处,特别是在超导物理的London方程、Ginzburg-Landau方程、Josephson方程、…     

A generalized Weyl—Wigner quantization scheme unifying P-Q and Q-P ordering and Weyl ordering of operators

By extending the usual Wigner operator to the s-parameterized one as 1/4π2 integral (dyduexp [iu(q-Q)+iy(p-P)+is/2yu]) from n=- ∞ to ∞ with s beng a,real parameter,we propose a generalized Weyl quantization scheme which accompanies a new generalized s-parameterized ordering rule.This rule recovers P-Q ordering,Q-P ordering,and Weyl ordering of operators in s = 1,1,0 respectively.Hence it differs from the Cahill-Glaubers’ ordering rule which unifies normal ordering,antinormal ordering,and Weyl ordering.We also show that in this scheme the s-parameter plays the role of correlation between two quadratures Q and P.The formula that can rearrange a given operator into its new s-parameterized ordering is presented.     

L'asymptotique de Weyl pour les bouteilles magnétiques

Nous prouvons une formule pour le comportement asymptotique de la fonctionN() de dénombrement des valeurs propres de l'opérateur de Schrödinger avec un champ magnétique qui tend vers l'infini `a l'infini de ^d . La preuve utilise un résultat précis sur l'estimation des valeurs propres pour un champ magnétique constant dans un cube de ^d.