Topological field patterns in the Yang-Mills theory

We present a formalism where the topological configurations of pure Yang-Mills theory are characterised using gauge fields alone. Here, we obtain an expression for the charges of these topologicalSO(3) gauge field configurations in terms of the Abelian vector potentials. In this formalism we analyse the ’t Hooft-Polyakov monopole solution.     

Explicit solutions for relativistic acceleration and rotation

The Lorentz transformations are represented by Einstein velocity addition on the ball of relativistically admissible velocities. This representation is by projective maps. The Lie algebra of this representation defines the relativistic dynamic equation. If we introduce a new dynamic variable, called symmetric velocity, the above representation becomes a representation by conformal, instead of projective maps. In this variable the relativistic dynamic equation for systems with an invariant plane becomes a non-linear analytic equation in one complex variable. We obtained explicit solutions for the motion of a charge in uniform, mutually perpendicular electric and magnetic fields. By assuming the Clock hypothesis and using these solutions, we were able to describe the space-time transformations between two uniformly accelerated and rotating systems. Presented at the International Colloquium “Integrable Systems and Quantum Symmetries”, Prague, 16–18 June 2005.     

Instantaneous measurement of field quadrature moments and entanglement

We present a method of measuring expectation values of quadrature moments of a multimode field through two-level probe “homodyning”. Our approach is based on an integral transform formalism of measurable probe observables, where analytically derived kernels unravel efficiently the required field information at zero interaction time, minimizing decoherence effects. The proposed scheme is suitable for fields that, while inaccessible to a direct measurement, enjoy one and two-photon Jaynes-Cummings interactions with a two-level probe, like spin, phonon, or cavity fields. Available data from previous experiments are used to confirm our predictions.     

Quantum theory of dilaton gravity in 1+1 dimensions

We discuss the quantum theory of 1+1 dimensional dilaton gravity, which is an interesting model with features analogous to the spherically symmetric gravitational systems in 3+1 dimensions. The functional measures over the metrics and the dilaton field are explicitly evaluated and the diffeomorphism invariance is completely fixed in the conformal gauge by using the technique developed in two dimensional quantum gravity. We derive the Wheeler-DeWitt like equations as physical state conditions. In the ADM formalism the measures of fields are very ambiguous, but in our formalism they are explicitly defined. A singularity appears at ²=κ(>0), where and N is the number of matter fields. The final stage of the black hole evaporation corresponds to the region ²κ, where the Liouville term becomes important, which just comes from the measure of the metrics. If κ<0, the singularity disappears.     

Dynamics of coupled field solitons: A collective coordinate approach

In this paper we consider a class of systems of two coupled real scalar fields in bidimensional space-time, with the main motivation of studying classical stability of soliton solutions using collective coordinate approach. First, we present the class of systems of the collective coordinate equations which are derived using the presented method. After that, we follow the dynamics of the coupled fields with local inhomogeneity like a delta function potential wall as well as a delta function potential well. The results of the investigation of the two coupled fields are compared to each other and the differences are discussed. The method can predict most of the characters of the interaction.     

CP violation in the general two-Higgs-doublet model: a geometric view

We discuss the CP properties of the potential in the general two-Higgs-doublet model (THDM). This is done in a concise way using real gauge invariant functions built from the scalar products of the doublet fields. The space of these invariant functions, parametrising the gauge orbits of the Higgs fields, is isomorphic to the forward light cone and its interior. CP transformations are shown to correspond to reflections in the space of the gauge invariant functions. We consider CP transformations where no mixing of the Higgs doublets is taken into account as well as the general case where the Higgs basis is not fixed. We present basis independent conditions for explicit CP violation which may be checked easily for any THDM potential. Conditions for spontaneous CP violation, that is CP violation through the vacuum expectation values of the Higgs fields, are also derived in a basis independent way.     

Isoperiodic classical systems and their quantum counterparts

One-dimensional isoperiodic classical systems have been first analyzed by Abel. Abel’s characterization can be extended for singular potentials and potentials which are not defined on the whole real line. The standard shear equivalence of isoperiodic potentials can also be extended by using reflection and inversion transformations. We provide a full characterization of isoperiodic rational potentials showing that they are connected by translations, reflections or Joukowski transformations. Upon quantization many of these isoperiodic systems fail to exhibit identical quantum energy spectra. This anomaly occurs at order O(?²) because semiclassical corrections of energy levels of order O(?) are identical for all isoperiodic systems. We analyze families of systems where this quantum anomaly occurs and some special systems where the spectral identity is preserved by quantization. Conversely, we point out the existence of isospectral quantum systems which do not correspond to isoperiodic classical systems.     

Constraint algebra for interacting quantum systems

We consider relativistic constrained systems interacting with external fields. We provide physical arguments to support the idea that the quantum constraint algebra should be the same as in the free quantum case. For systems with ordering ambiguities this principle is essential to obtain a unique quantization. This is shown explicitly in the case of a relativistic spinning particle, where our assumption about the constraint algebra plus invariance under general coordinate transformations leads to a unique S-matrix.     

Bound states in disclinated graphene with Coulomb impurities in the presence of a uniform magnetic field

In this contribution, we study the effects caused by an impurity on the quantum dynamics of massive excitations in a disclinated graphene in the presence of an external magnetic field. Within a continuum approach, the problem is mathematically modeled by the definition of a special vector potential containing all the information about the topology and the interacting fields. The presence of disclination is introduced by a term in the Dirac equation that translates the appearance of a phase associated with the transport of the spinor around the apex of the cone. We solve exactly the Dirac equation for this problem and the eigenvalues are obtained. We observe the influence of the disclination on the spectrum of energy and the allowed values of magnetic field.     

Finite size scale invariance

We develop a new approach to scale symmetry, which takes into account the possible finite cut-offs of the fields or the parameters. This new symmetry, called finite size scale symmetry: i) includes the traditional self-similarity as a limiting case, when the cut-offs are set to infinity (infinite size-system); ii) is consistent with the traditional finite size scaling approach already used in critical phenomena; iii) enables the computation of some of the universal functions appearing in the finite size scaling formulation; iv) allows scale transformations leaving the cut-offs invariant, like in the traditional renormalization approach; v) can be formulated to allow for positive or negative fields and parameters; vi) leads to new predictions about the shape of some distributions in critical phenomena or turbulence which are in very good agreement with the experimental or numerical findings. Received 26 January 1999 and Received in final form 25 October 1999     

On the Lense-Thirring test with the Mars Global Surveyor in the gravitational field of Mars

I discuss some aspects of a recent frame-dragging test performed by exploiting the Root-Mean-Square (RMS) orbit-overlap differences of the out-of-plane component (N) of the Mars Global Surveyor (MGS) spacecraft’s orbit in the gravitational field of Mars. A linear fit to the complete time series for the entire MGS data set (4 February 1999–14 January 2005) yields a normalized slope 1.03 ± 0.41 (with 95% confidence bounds). Other linear fits to different data sets confirm agreement with general relativity. Huge systematic effects induced by mismodeling the martian gravitational field which have been claimed by some authors are absent in the MGS out-of-plane record. The same level of effect is seen for both the classical non-gravitational and relativistic gravitomagnetic forces on the in-plane MGS orbital components; this is not the case for the out-of-plane components. Moreover, the non-conservative forces experience high-frequency variations which are not important in the present case where secular effects are relevant.     

Necessary and sufficient conditions for the existence of a Lagrangian in field theory. I. Variational approach to self-adjointness for tensorial field equations

In this paper we identify some of the most significant references on the inverse problem of the calculus of variations for single integrals and initiate the study of the generalization of the underlying methodology to classical field theories. We first classify Lorentz-covariant tensorial field equations into nonlinear, quasi-linear, and semilinear forms, and then introduce their systems of equations of variation and adjoint systems. The necessary and sufficient conditions for the self-adjointness of class $\text{C}$², regular, tensorial, nonlinear, quasi-linear and semilinear forms are worked out. We study the Lagrange equations, their system of equations of variations (Jacobi equations) and their adjoint system by proving that, for class $\text{C}$⁴ and regular Lagrangian densities, they are always self-adjoint. We then introduce a concept of analytic representation which occurs when the Lagrange equations coincide with the field equations up to equivalence transformations and refine the definition by particularizing it as direct or indirect and ordered or nonordered. Some of the conventional cases of tensorial fields are considered and we prove, in particular, that the conventional representation of the complex scalar field in interaction with the electromagnetic field is of the ordered indirect type. For the objective of identifying our program we recall the two classes of equivalence transformations of the Lagrangian densities which are primarily used nowadays, namely, the Lorentz (coordinate) transformations and the gauge transformations (transformations of fields within a fixed coordinate system), and postulate the existence of a third class, which we term isotopic transformations of the Lagrangian density and which consist of equivalence transformations within a fixed coordinate system and gauge. We finally outline the objectives of our program, which essentially consist of the identification of the necessary and sufficient conditions for the existence of a Lagrangian in field theories and their first application to the transformation theory within the framework of our variational approach to self-adjointness.     

Appearance of topological phases in superconducting nanocircuits

We construct non-Abelian geometric transformations in superconducting nanocircuits, which resemble in properties the Aharonov-Bohm phase for an electron transported around a magnetic flux line. The effective magnetic fields can be strongly localized, and the path is traversed in the region where the energy separation between the states involved is at maximum, so that the adiabaticity condition is weakened. In particular, we present a scheme of topological charge pumping.     

Discrete Symmetries as Automorphisms of the Proper Poincaré Group

We present a consistent approach to finding discrete transformations in representation spaces of the proper Poincaré group. To this end we establish a correspondence between involutory automorphisms of the group and the discrete transformations. Such a correspondence allows us to describe the action of discrete transformations on arbitrary spin-tensor fields without any use of relativistic wave equations. Extending the proper Poincaré group by the discrete transformations, we construct explicitly fields carrying corresponding irreps.     

Bäcklund transformations for finite-dimensional integrable systems: a geometric approach

We present a geometric construction of Bäcklund transformations and discretizations for a large class of algebraic completely integrable systems. To be more precise, we construct families of Bäcklund transformations, which are naturally parameterized by the points on the spectral curve(s) of the system. The key idea is that a point on the curve determines, through the Abel–Jacobi map, a vector on its Jacobian which determines a translation on the corresponding level set of the integrals (the generic level set of an algebraic completely integrable systems has a group structure). Globalizing this construction we find (possibly multi-valued, as is very common for Bäcklund transformations) maps which preserve the integrals of the system, they map solutions to solutions and they are symplectic maps (or, more generally, Poisson maps). We show that these have the spectrality property, a property of Bäcklund transformations that was recently introduced. Moreover, we recover Bäcklund transformations and discretizations which have up to now been constructed by ad hoc methods, and we find Bäcklund transformations and discretizations for other integrable systems. We also introduce another approach, using pairs of normalizations of eigenvectors of Lax operators and we explain how our two methods are related through the method of separation of variables.     

INVARIANT LINEARIZATION CRITERIA FOR SYSTEMS OF CUBICALLY NONLINEAR SECOND-ORDER ORDINARY DIFFERENTIAL EQUATIONS

Invariant linearization criteria for square systems of second-order quadratically nonlinear ordinary differential equations (ODEs) that can be represented as geodesic equations are extended to square systems of ODEs cubically nonlinear in the first derivatives. It is shown that there are two branches for the linearization problem via point transformations for an arbitrary system of second-order ODEs and its reduction to the simplest system. One is when the system is at most cubic in the first derivatives. One obtains the equivalent of the Lie conditions for such systems. We explicitly solve this branch of the linearization problem by point transformations in the case of a square system of two second-order ODEs. Necessary and sufficient conditions for linearization to the simplest system by means of point transformations are given in terms of coefficient functions of the system of two second-order ODEs cubically nonlinear in the first derivatives. A consequence of our geometric approach of projection is a rederivation of Lie's linearization conditions for a single second-order ODE and sheds light on more recent results for them. In particular we show here how one can construct point transformations for reduction to the simplest linear equation by going to the higher space and just utilizing the coefficients of the original ODE. We also obtain invariant criteria for the reduction of a linear square system to the simplest system. Moreover these results contain the quadratic case as a special case. Examples are given to illustrate our results.     

Mapping quantum many-body system to decoupled harmonic oscillators: General discussions and examples

We present a class of quantum systems that can be mapped to decoupled harmonic oscillators through appropriate similarity transformations. We will take advantage of these similarity transformations to discover hidden ladder operators, such that the eigenstates of the system can be constructed like those of harmonic oscillator. We also provide five systems belonging to this family as examples.     

Tomography Methods for Vector Field Study Using Space-Distributed Fiber Optic Sensors with Integral Sensitivity

The problem of tomographic reconstruction of vector physical fields is studied. This problem can be solved by using fiber optic measuring lines (MLs) of special shape. In the case that the ML output signal is proportional to the vector's projection, the ML must be shaped like a narrow loop. This problem can be solved by means of the integral theorem. If an ML output signal is proportional to projection of a vector derivative with respect to ML direction, the ML with a step shape can be used. In this case the potential component of a vector field can be reconstructed. This approach can be applied to research on distributions of electromagnetic, deforming, and other vector fields and can be used for developing systems to monitor vector physical fields.     

Topological insulators and C-algebras: Theory and numerical practice

We apply ideas from C^∗-algebra to the study of disordered topological insulators. We extract certain almost commuting matrices from the free Fermi Hamiltonian, describing band projected coordinate matrices. By considering topological obstructions to approximating these matrices by exactly commuting matrices, we are able to compute invariants quantifying different topological phases. We generalize previous two dimensional results to higher dimensions; we give a general expression for the topological invariants for arbitrary dimension and several symmetry classes, including chiral symmetry classes, and we present a detailed K-theory treatment of this expression for time reversal invariant three dimensional systems. We can use these results to show non-existence of localized Wannier functions for these systems.We use this approach to calculate the index for time-reversal invariant systems with spin–orbit scattering in three dimensions, on sizes up to 12³, averaging over a large number of samples. The results show an interesting separation between the localization transition and the point at which the average index (which can be viewed as an “order parameter” for the topological insulator) begins to fluctuate from sample to sample, implying the existence of an unsuspected quantum phase transition separating two different delocalized phases in this system. One of the particular advantages of the C^∗-algebraic technique that we present is that it is significantly faster in practice than other methods of computing the index, allowing the study of larger systems. In this paper, we present a detailed discussion of numerical implementation of our method.