Search Results (48)

The canonical quantization of gravity in general relativity is greatly simplified by the artificial decomposition of space time into a 3 + 1 formalism. Such a simplification appears to come at the cost of general covariance. This quantization procedure requires tangential and perpendicular infinitesimal diffeomorphisms generated by the symmetry group under the Legendre transformation of the given action. This gauge generator, along with the fact that Weyl curvature scalars may act as “intrinsic coordinates” (or a dynamical reference frame) that depend only on the spatial metric ($g_{ab}$) and the conjugate momenta ($p^{cd}$), allows for an alternative approach to canonical quantization of gravity. In this paper, we present the tensorial solution of the set of Weyl scalars in terms of canonical phase-space variables. Full article

Quantum channels define key objects in quantum information theory. They are represented by completely positive trace-preserving linear maps in matrix algebras. We analyze a family of quantum channels defined through the use of the Weyl operators. Such channels provide generalization of the celebrated qubit Pauli channels. Moreover, they are covariant with respective to the finite group generated by Weyl operators. In what follows, we study self-adjoint Weyl channels by providing a special Hermitian representation. For a prime dimension of the corresponding Hilbert space, the self-adjoint Weyl channels contain well-known generalized Pauli channels as a special case. We propose multipartite generalization of Weyl channels. In particular, we analyze the power of prime dimensions using finite fields and study the covariance properties of these objects. Full article

This editorial presents 24 research articles published in the Special Issue entitled Geometry of Manifolds and Applications of the MDPI Mathematics journal, which covers a wide range of topics from the geometry of (pseudo-)Riemannian manifolds and their submanifolds, providing some of the latest achievements in many branches of theoretical and applied mathematical studies, among which is counted: the geometry of differentiable manifolds with curvature restrictions such as complex space forms, metallic Riemannian space forms, Hessian manifolds of constant Hessian curvature; optimal inequalities for submanifolds, such as generalized Wintgen inequality, inequalities involving $\delta$-invariants; homogeneous spaces and Poisson–Lie groups; the geometry of biharmonic maps; solitons (Ricci solitons, Yamabe solitons, Einstein solitons) in different geometries such as contact and paracontact geometry, complex and metallic Riemannian geometry, statistical and Weyl geometry; perfect fluid spacetimes [...] Full article

For a Chevalley group $G$ over an algebraically closed field $K$ of characteristic $p>0$ with the irreducible root system $R,$ Lusztig’s character formula expresses the formal character of a simple $G$-module by the formal characters of the Weyl modules and the values of the Kazhdan–Lusztig polynomials at 1. It is known that, for a sufficiently large characteristic $p$ of the field $K,$ Lusztig’s character formula holds. The known lower bound of the characteristic $p$ is much larger than the Coxeter number $h$ of the root system $R.$ Observations show that for simple modules with restricted highest weights of small Chevalley groups such as those of types $A_1,A_2,A_3,B_2,B_3$, and $C_3$, Lusztig’s character formula holds for all $p\ge h$. For large Chevalley groups, no other examples are known. In this paper, for $G$ of type $A_l,$ we give some series of simple modules for which Lusztig’s character formula holds for all $p\ge h$. Using this result, we compute the cohomology of $G$ with coefficients in these simple modules. To prove the results, Jantzen’s filtration properties for Weyl modules and the properties of Kazhdan–Lusztig polynomials are used. Full article

The structure of the extended affine Weyl symmetry group of higher Painlevé equations of N periodicity depends on whether N is even or odd. We find that for even N, the symmetry group ${\widehat{A}}_{N-1}^{\left(1\right)}$ contains the conventional Bäcklund transformations $s_j,j=1,\dots ,N$, the group of automorphisms consisting of cycling permutations but also reflections on a periodic circle of N points, which is a novel feature uncovered in this paper. The presence of reflection automorphisms is connected to the existence of degenerated solutions, and for $N=4$, we explicitly show how even reflection automorphisms cause degeneracy of a class of rational solutions obtained on the orbit of the translation operators of ${\widehat{A}}_3^{\left(1\right)}$. We obtain the closed expressions for the solutions and their degenerated counterparts in terms of the determinants of the Kummer polynomials. Full article

NbAs is a Weyl semimetal and belongs to the group of topological phases that exhibit distinct quantum and topological attributes. Topological phases have a fundamentally different response to external perturbations, such as magnetic fields. To obtain insights into the response of such phases to pressure, we conducted a comprehensive study on the pressure-induced electronic and structural transitions in NbAs. We used micro-X-ray diffraction (XRD) and micro-X-ray spectroscopy (XAS) techniques to elucidate the changes at different atomic and electronic length scales (local, medium, and bulk) as combined with theoretical calculations. High-pressure XRD measurements revealed a rather common compression behavior up to ~12 GPa that could be fitted to an equation of state formalism with a bulk modulus of $⟨K_0⟩=$ 179.6 GPa. Complementary Nb K-edge XAS data unveiled anomalies at pressure intervals of ~12–15 and ~25–26 GPa in agreement with previous literature data from XRD studies. We attribute these anomalies to a previously reported topological Lifshitz transition and the tetragonal-to-hexagonal phase transition, respectively. Analysis of EXAFS results revealed slight changes in the mean next-nearest neighbor distance Nb–As₍₁₎ (~2.6 Å) at ~15 GPa, while the second nearest neighboring bond Nb–Nb₍₁₎ (~3.4 Å) shows a pronounced anomaly. This indicates that the electronic changes across the Lifshitz transition are accommodated first in the medium-range atomic structure and then at the local range and bulk. The variances of these bonds show anomalous but progressive evolutions close to the tetragonal-to-hexagonal transition at ~25 GPa, which allowed us to derive the evolution of vibration properties in this material. We suggest a prominent displacive character of the $I{4}_{1}md\to P\overline{6}m2$ transition facilitated by phonon modes. Full article

In this work we propose a novel device for controlling the flow of information using Weyl fermions. Based on a previous work by our group, we show that it is possible to fully control the flow of Weyl fermions on several different channels by applying an electric field perpendicular to the direction of motion of the particles on each channel. In this way, we can transmit information as logical bits, depending on the existence or not of a Weyl current on each channel. We also show that the response time of this device is exceptionally low, less than 1 ps, for typical values of its parameters, allowing for the control of the flow of information at extremely high rates of the order of 100 Petabits per second. Alternatively, this device could also operate as an electric field sensor. In addition, we demonstrate that Weyl fermions can be efficiently guided through the proposed device using appropriate magnetic fields. Finally, we discuss some particularly interesting remarks regarding the electromagnetic interactions of high-energy particles. Full article

Covariant integral quantization with rotational $\mathrm{SO}\left(3\right)$ symmetry is established for quantum motion on this group manifold. It can also be applied to Gabor signal analysis on this group. The corresponding phase space takes the form of a discrete-continuous hypercylinder. The central tool for implementing this procedure is the Weyl–Gabor operator, a non-unitary operator that operates on the Hilbert space of square-integrable functions on $\mathrm{SO}\left(3\right)$. This operator serves as the counterpart to the unitary Weyl or displacement operator used in constructing standard Schrödinger–Glauber–Sudarshan coherent states. We unveil a diverse range of properties associated with the quantizations and their corresponding semi-classical phase-space portraits, which are derived from different weight functions on the considered discrete-continuous hypercylinder. Certain classes of these weight functions lead to families of coherent states. Moreover, our approach allows us to define a Wigner distribution, satisfying the standard marginality conditions, along with its related Wigner transform. Full article

The automatic calculation of sediment maps from hydroacoustic data is of great importance for habitat and sediment mapping as well as monitoring tasks. For this reason, numerous papers have been published that are based on a variety of algorithms and different kinds of input data. However, the current literature lacks comparative studies that investigate the performance of different approaches in depth. Therefore, this study aims to provide recommendations for suitable approaches for the automatic classification of side-scan sonar data that can be applied by agencies and researchers. With random forests, support vector machines, and convolutional neural networks, both traditional machine-learning methods and novel deep learning techniques have been implemented to evaluate their performance regarding the classification of backscatter data from two study sites located in the Sylt Outer Reef in the German Bight. Simple statistical values, textural features, and Weyl coefficients were calculated for different patch sizes as well as levels of quantization and then utilized in the machine-learning algorithms. It is found that large image patches of 32 px size and the combined use of different feature groups lead to the best classification performances. Further, the neural network and support vector machines generated visually more appealing sediment maps than random forests, despite scoring lower overall accuracy. Based on these findings, we recommend classifying side-scan sonar data with image patches of 32 px size and 6-bit quantization either directly in neural networks or with the combined use of multiple feature groups in support vector machines. Full article

The canonical commutation relation, $[Q,P]=i\hslash$, stands at the foundation of quantum theory and the original Hilbert space. The interpretation of P and Q as observables has always relied on the analogies that exist between the unitary transformations of Hilbert space and the canonical (also known as contact) transformations of classical phase space. Now that the theory of quantum measurement is essentially complete (this took a while), it is possible to revisit the canonical commutation relation in a way that sets the foundation of quantum theory not on unitary transformations but on positive transformations. This paper shows how the concept of simultaneous measurement leads to a fundamental differential geometric problem whose solution shows us the following. The simultaneous P and Q measurement (SPQM) defines a universal measuring instrument, which takes the shape of a seven-dimensional manifold, a universal covering group we call the instrumental Weyl-Heisenberg ($\mathrm{IWH}$) group. The group $\mathrm{IWH}$ connects the identity to classical phase space in unexpected ways that are significant enough that the positive-operator-valued measure (POVM) offers a complete alternative to energy quantization. Five of the dimensions define processes that can be easily recognized and understood. The other two dimensions, the normalization and phase in the center of the $\mathrm{IWH}$ group, are less familiar. The normalization, in particular, requires special handling in order to describe and understand the SPQM instrument. Full article

The monomiality principle is based on an abstract definition of the concept of derivative and multiplicative operators. This allows to treat different families of special polynomials as ordinary monomials. The procedure underlines a generalization of the Heisenberg–Weyl group, along with the relevant technicalities and symmetry properties. In this article, we go deeply into the formulation and meaning of the monomiality principle and employ it to study the properties of a set of polynomials, which, asymptotically, reduce to the ordinary two-variable Kampè dè Fèrièt family. We derive the relevant differential equations and discuss the associated orthogonality properties, along with the relevant generalized forms. Full article

After a brief introduction of Born’s reciprocal relativity theory is presented, we review the construction of the $deformed$ quaplectic group that is given by the semi-direct product of $U(1,3)$ with the $deformed$ (noncommutative) Weyl–Heisenberg group corresponding to $noncommutative$ fiber coordinates and momenta $[X_a,X_b]\ne 0$; $[P_a,P_b]\ne 0$. This construction leads to more general algebras given by a two-parameter family of deformations of the quaplectic algebra, and to further algebraic extensions involving antisymmetric tensor coordinates and momenta of higher ranks $[X_{a_1{a}_2\cdots a_n},X_{b_1{b}_2\cdots b_n}]\ne 0$; $[P_{a_1{a}_2\cdots a_n},P_{b_1{b}_2\cdots b_n}]\ne 0$. We continue by examining algebraic extensions of the Yang algebra in extended noncommutative phase spaces and compare them with the above extensions of the deformed quaplectic algebra. A solution is found for the exact analytical mapping of the noncommuting $x^{\mu },p^{\mu }$ operator variables (associated to an 8D curved phase space) to the canonical $Y^A,{\mathsf{\Pi }}^A$ operator variables of a flat 12D phase space. We explore the geometrical implications of this mapping which provides, in the $classical$ limit, the embedding functions $Y^A(x,p),{\mathsf{\Pi }}^A(x,p)$ of an 8D curved phase space into a flat 12D phase space background. The latter embedding functions determine the functional forms of the base spacetime metric $g_{\mu \nu }(x,p)$, the fiber metric of the vertical space $h^{ab}(x,p)$, and the nonlinear connection $N_{a\mu }(x,p)$ associated with the 8D cotangent space of the 4D spacetime. Consequently, we find a direct link between noncommutative curved phase spaces in lower dimensions and commutative flat phase spaces in higher dimensions. Full article

Six-dimensional spinors with $Spin(3,3)$ symmetry are utilized to efficiently encode three generations of matter. $E_{8(-24)}$ is shown to contain physically relevant subgroups with representations for GUT groups, spacetime symmetries, three generations of the standard model fermions, and Higgs bosons. Pati–Salam, $SU\left(5\right)$, and $Spin\left(10\right)$ grand unified theories are found when a single generation is isolated. For spacetime symmetries, $Spin(4,2)$ may be used for conformal symmetry, $Ad{S}_5\to d{S}_4$, or simply broken to $Spin(3,1)$ of a Minkowski space. Another class of representations finds $Spin(2,2)$ and can give $Ad{S}_3$ with various GUTs. An action for three generations of fermions in the Majorana–Weyl spinor $\mathbf{128}$ of $Spin(4,12)$ is found with $Spin\left(3\right)$ flavor symmetry inside $E_{8(-24)}$. The 128 of $Spin(12,4)$ can be regarded as the tangent space to a particular pseudo-Riemannian form of the octo-octonionic Rosenfeld projective plane $E_{8(-24)}/Spin(12,4)=\left(\mathbb{O}sx\mathbb{O}\right){\mathbb{P}}^2$. Full article

We develop a systematic approach to deriving rational solutions and obtaining classification of their parameters for dressing chains of even N periodicity or equivalent Painlevé equations invariant under $A_{N-1}^{\left(1\right)}$ symmetry. This formalism identifies rational solutions (as well as special function solutions) with points on orbits of fundamental shift operators of $A_{N-1}^{\left(1\right)}$ affine Weyl groups acting on seed configurations defined as first-order polynomial solutions of the underlying dressing chains. This approach clarifies the structure of rational solutions and establishes an explicit and systematic method towards their construction. For the special case of the $N=4$ dressing chain equations, the method yields all the known rational (and special function) solutions of the Painlevé V equation. The formalism naturally extends to $N=6$ and beyond as shown in the paper. Full article

In this study, we report the successful growth of single crystals of a magnetic Weyl semimetal candidate NdAlGe with the space group I4₁md. The crystals were grown using a floating-zone technique, which used five laser diodes, with a total power of 2 kW, as the heat source. To ensure that the molten zone was stably formed during the growth, we employed a bell-shaped distribution profile of the vertical irradiation intensity. After the nominal powder, crushed from an arc-melted ingot, was shaped under hydrostatic pressure, we sintered the feed and seed rods in an Ar atmosphere under ultra-low oxygen partial pressure (<10⁻²⁶ atm) generated by an oxygen pump made of yttria-stabilized zirconia heated at 873 K. Single crystals of NdAlGe were successfully grown to a length of 50 mm. The grown crystals showed magnetic order in bulk at 13.5 K. The fundamental physical properties were characterized by magnetic susceptibility, magnetization, specific heat, thermal expansion, and electrical resistivity measurements. This study demonstrates that the magnetic order induces anisotropic magnetoelasticity, magneto-entropy, and charge transport in NdAlGe. Full article