Group-theoretical structure of quantum continuous measurements

The group-theoretical structure of continuous measurements is investigated in the framework of the path-integral phenomenological theory of quantum continuous measurements. The “transversal” group transforming alternative measurement results (outputs) into each other and the “longitudinal” semigroup describing the evolution of a quantum system subject to continuous measurement are introduced as well as their unification in a single semigroup. The resulting group-theoretical scheme generalizes the scheme describing the evolution of a nonrelativistic particle in an external field.     

Group-theoretical interpretation of the Korteweg-de Vries type equations

The Korteweg-de Vries equation is studied within the group-theoretical framework. Analogous equations are obtained for which the many-dimensional Schrödinger equation (with nonlocal potential) plays the same role as the one-dimensional Schrödinger equation does in the theory of the Korteweg-de Vries equation.     

Standard model and gravity from spinors

It is shown how the Lorentz and standard-model gauge groups can be unified by using algebraic spinors of the standard four-dimensional Clifford algebra, in left–right symmetric fashion. This defines a framework of unification with gravity and generates exactly a standard-model family of fermions, while a Pati–Salam unification group emerges, at the Planck scale, where (chiral) gravity decouples. We show that this low-energy broken phase emerges from the VEV of extended vierbein fields, which at this stage are assumed to be dynamically generated from a theory in the fully symmetric phase valid beyond the Planck scale (and whose consistency and dynamics is thus yet to be assessed) providing thus a geometrical and group-theoretical framework for the unification and breaking. At low energy, on the other hand, it is intriguing to find, as a remnant of this unification, new isospin-triplet spin-two particles that may naturally lie at the weak scale, providing a striking signal at the LHC.     

Patterns of symmetry breaking in supersymmetric gauge theories

We give a group-theoretical analysis of the spontaneous breaking of the gauge group in supersymmetric Yang-Mills theories with unbroken supersymmetry.     

Electronic structure and improper electric polarization of samarium orthoferrite

The band structure and distributions of the electron and spin densities of samarium orthoferrite have been calculated within the framework of the first-principles density functional theory in the LSDA + U approximation taking into account the collinear antiferromagnetic ordering of the magnetic moments of iron and samarium cations. The possibility of inducing a ferroelectric state at temperatures below the antiferromagnetic ordering temperature of the magnetic sublattice formed by samarium cations has been considered using the results of the group-theoretical analysis. In the high-temperature range, the formation of regions with a spontaneous electric polarization is possible in the presence of additional factors that reduce the symmetry of the crystal.     

The effective Hamiltonian for the coupling between inversion-torsion states of hydrazine

In the present paper, a procedure to calculate matrix elements of the group-theoretical Hamiltonian for hydrazine is described and explained in detail, and a complete set of rovibrational matrix elements responsible for interstate interactions is presented. The transformation properties of the framework functions under the symmetry operators of the double group are explained in a simple way using the MS Excel spreadsheet.     

C_（60k~2）π－能谱的群论解析计算

本文推导出正二十面体（Ｉ＿ｈ）对称的同系碳簇Ｃ＿（６０ｋ￣２）π－轨道的群约化通式;在紧束缚（Ｈｕｃｋｅｌ）方案下,应用Ｌａｎｃｚｏｓ三对用比方法,对每一类不可约表示仅需造出一个基函数作为起始态,便可使６０ｋ￣２×６０ｋ￣２久期方程达到理想的方块化,系统地求出了Ｃ＿（６０）—Ｃ＿（２１６０）（ｋ＝１—６）按不可约表示分类的方块本证多项式和能级的解析形式（但五次以上代数方程只能求数值报）。文中叙述本方法的梗概,示范Ｃ＿（６０）,Ｃ＿（２４０）及Ｃ＿（５４０）的计算结果。     

Contribution to the determination of the energy stability of structural types in crystals

Within the framework of the pair approximation the paper discusses how the interaction radius (measured in units of the coordination spheres of the lattice) must be taken into account so that a given superlattice in a lattice could be energetically stable. An original group-theoretical treatment permits this problem to be solved, even if the required number k of coordination spheres is fairly large (e.g., for the -W structural type on the basis of an auxiliary simple cubic lattice k=14).Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 72–77, June, 1985.     

Entanglement of Positive Definite Functions on Compact Groups

We define and study entanglement of continuous positive definite functions on products of compact groups. We formulate and prove an infinite-dimensional analog of the Horodecki Theorem, giving a necessary and sufficient criterion for separability of such functions. The resulting characterisation is given in terms of mappings of the space of continuous functions, preserving positive definiteness. A relation between the developed group-theoretical formalism and the conventional one, given in terms of density matrices, is established through the non-commutative Fourier analysis. It shows that the presented method plays the role of a “generating function” formalism for the theory of entanglement.     

Theory of neutrino masses and mixing

We motivate the usage of finite groups as symmetries of the Lagrangian. After a presentation of basic group-theoretical concepts, we introduce the notion of characters and character tables in the context of irreducible representations and discuss their applications. We exemplify these theoretical concepts with the groups S ₄ and A ₄. Finally, we discuss the relation between tensor products of irreducible representations and Yukawa couplings and describe a model for tri-bimaximal lepton mixing based on A ₄.     

Violation of rotational invariance and mesomorphic phase transitions characterized by an order parameter

By group-theoretical arguments we predict all the possible anisotropic liquid phases which can appear as a result of the violation of the rotational invariance of the isotropic liquid. We discuss the mathematical nature of the order parameters which are found to be irreducible tensors, the order of the corresponding phase transitions and the stability of the phases against small spatial fluctuations of the order parameter.     

Gauge hierarchies in the E8 × E′8 superstring theory

We present a systematic approach to the problem of gauge hierarchy structure in E₈ × E′₈ superstring theory. It is shown that only very limited E₆ breaking schemes are allowed from the group-theoretical and phenomenological point of view. The ordinary GUTs are not permitted and moreover the unification of SU(3), and SU(2)_w cannot occur at the energy scale below M_PI. We give four promising models and add some comments on each of them.     

Algebraic properties of extended supergravity in DE sitter space

General features of extended supergravity with a gauge internal symmetry are discussed from a group-theoretical point of view.     

The wigner-lubanski first integral in the Papapetrou-Corinaldesi equations of motion

Some years ago Papapetrou and Corinaldesi applied Papapetrou's equation's of motion of spinning particles to the case of motion in the Schwarzschild field. For the particular case of motion in the equatorial plane they found an extra integral of motion (in addition to the constants of energy and total angular momentum). We here give a group-theoretical interpretation to the origin of this constant by relating it to the Wigner-Lubanski constant known from the theory of representations of the Poincaré group.Work supported by the Israel National Academy of Sciences, Grant No. 197 (B)P.     

The complete N = 3 matter coupled supergravity

The complete N = 3 matter coupling to supergravity is obtained in a geometrical framework. This coupling always exists if the 3n complex scalars of the n vector multiplets are co-ordinates of the Kähler-grassmannian manifold SU(3, n)/SU(3) × SU(n) × U(1). Subgroups of SO(3, n) ⊂ SU(3, n) of dimension 3 + n can be gauged and give rise to a non-trivial scalar potential. The techniques used in this paper allow for the calculation of scalar potentials of extended supergravities in any dimension without explicit construction of the lagrangian. This opens the possibility of discussing patterns of partial supersymmetry and gauge symmetry breaking on a purely group-theoretical ground.     

Low-temperature specific heat of the Rb<Subscript>2</Subscript>KScF<Subscript>6</Subscript> elpasolite

The specific heat of single-crystal Rb₂KScF₆ is measured using ac calorimetry in the range 4–280 K. The results are discussed in the context of a group-theoretical analysis of possible distortions of the elpasolite structure and of experimental data obtained earlier with an adiabatic calorimeter in a narrower temperature region.     

Mathematical Theory of Non-Equilibrium Quantum Statistical Mechanics

We review and further develop a mathematical framework for non-equilibrium quantum statistical mechanics recently proposed in refs. 1–7. In the algebraic formalism of quantum statistical mechanics we introduce notions of non-equilibrium steady states, entropy production and heat fluxes, and study their properties. Our basic paradigm is a model of a small (finite) quantum system coupled to several independent thermal reservoirs. We exhibit examples of such systems which have strictly positive entropy production.     

On the Poincaré-invariant equations for particles with variable spin and mass

The Poincaré-invariant equations without redundant components, describing the motion of a particle which can be in different spin and mass states are obtained. The quasi-relativistic equation for a particle with arbitrary spin in external electromagnetic field is found. The group-theoretical analysis of these equations in carried out.     

Note on Clock Synchronization and Edwards Transformations

Edwards transformations relating inertial frames with arbitrary clock synchronization are reviewed and put in more general setting. Their group-theoretical context is described.