A review of super riemannian space-time using differential forms

We analyse the content of curved super space-time, which consists of points labelled by four ordinary space-time variables and a set of anti-commuting quantities. A definition of a curved manifold including such variables is given, and a coordinate free formalism reviewed using the super space-time extended version of modern differential geometric techniques. Such a formalism allows the incorporation of internal symmetries in a geometric fashion. Finally we analyse the extended version of Einstein's source-free equation on this manifold, particularly in the linear approximation.     

A Three-Dimensional Treatment of the Three-Nucleon Bound State

Recently a formalism for a direct treatment of the Faddeev equation for the three-nucleon bound state in three dimensions has been proposed. It relies on an operator representation of the Faddeev component in the momentum space and leads to a finite set of coupled equations for scalar functions which depend only on three variables. In this paper we provide further elements of this formalism and show the first numerical results for chiral NNLO nuclear forces.     

Solitons in the statistical mechanics of the Toda lattice

Using proper canonical variables of the Toda soliton we develop a formalism to describe a gas of independent solitons. We find that this model gives the exact thermodynamic properties for high temperatures.     

Nonequilibrium thermodynamics of stochastic systems with odd and even variables

The total entropy production of stochastic systems can be divided into three quantities. The first corresponds to the excess heat, while the second two comprise the housekeeping heat. We denote these two components the transient and generalized housekeeping heat and we obtain an integral fluctuation theorem for the latter, valid for all Markovian stochastic dynamics. A previously reported formalism is obtained when the stationary probability distribution is symmetric for all variables that are odd under time reversal, which restricts consideration of directional variables such as velocity.     

Alternative adiabatic elimination schemes for fast variables in stochastic processes

We explore an alternative adiabatic elimination scheme for fast variables in stochastic processes, recently proposed by Haake. For the example of a Brownian particle in an external field we determine the reduced evolution operator, and the initial condition that should be used with it, by means of the Chapman-Enskog formalism. We exploit the close analogy between this formalism and the familiar perturbation theory for degenerate energy levels. We conclude that Haake's scheme is less suitable than other schemes already available in the literature for systems close to equilibrium; it may well be preferable far from equilibrium. We briefly discuss a still broader class of elimination procedures and a criterion for choosing between them.     

Deformation quantization of fermi fields

Deformation quantization for any Grassmann scalar free field is described via the Weyl-Wigner-Moyal formalism. The Stratonovich-Weyl quantizer, the Moyal -product and the Wigner functional are obtained by extending the formalism proposed recently in [I. Galaviz, H. García-Compeán, M. Przanowski, F.J. Turrubiates, Weyl-Wigner-Moyal Formalism for Fermi Classical Systems, arXiv:hep-th/0612245] to the fermionic systems of infinite number of degrees of freedom. In particular, this formalism is applied to quantize the Dirac free field. It is observed that the use of suitable oscillator variables facilitates considerably the procedure. The Stratonovich-Weyl quantizer, the Moyal -product, the Wigner functional, the normal ordering operator, and finally, the Dirac propagator have been found with the use of these variables.     

The Clifford bundle and the nature of the gravitational field

In this paper we formulate Einstein's gravitational theory with the Clifford bundle formalism. The formalism suggests interpreting the gravitational field in the sense of Faraday, i.e., with the field residing in Minkowski spacetime. We succeeded in discovering the condition for this interpretation to hold. For the variables that play the role of the gravitational field in our theory, the Lagrangian density turns out to be of the Yang-Mills type (with an auto-interaction plus gauge-fixing terms). We give a brief comparison of our theory with other field theories of the gravitational field in the flat Minkowski spacetime.     

Generalized characteristic functions for a single-mode radiation field

We derive simple formulas for the higher-order moments of the quadrature operators operators of a two-field superposition by introducing a new generalized characteristic function depending on two complex variables. Using this formalism we investigate the preservation of higher-order squeezing during weak thermalization of an even coherent state.     

A note on unparticle in lower dimensions

Using the gauge-invariant but path-dependent variables formalism, we examine the effect of the space–time dimensionality on a physical observable in the unparticle scenario. We explicitly show that long-range forces between particles mediated by unparticles are still present whenever we go over into lower dimensions.     

Nonlinear rheological models for structured interfaces

The GENERIC formalism is a formulation of nonequilibrium thermodynamics ideally suited to develop nonlinear constitutive equations for the stress-deformation behavior of complex interfaces. Here we develop a GENERIC model for multiphase systems with interfaces displaying nonlinear viscoelastic stress-deformation behavior. The link of this behavior to the microstructure of the interface is described by including a scalar and a tensorial structural variable in the set of independent surface variables. We derive an expression for the surface stress tensor in terms of these structural variables, and a set of general nonlinear time evolution equations for these variables, coupling them to the deformation field. We use these general equations to develop a number of specific models, valid for application near equilibrium, or valid for application far beyond equilibrium.     

Euler-Poincaré Formalism of (Two Component) Degasperis-Procesi and Holm-Staley type Systems

Abstract

In this paper we propose an Euler-Poincaré formalism of the Degasperis and Procesi (DP) equation. This is a second member of a one-parameter family of partial differential equations, known as b-field equations. This one-parameter family of pdes includes the integrable Camassa-Holm equation as a first member. We show that our Euler-Poincaré formalism exactly coincides with the Degasperis-Holm-Hone (DHH) Hamiltonian framework. We obtain the DHH Hamiltonian structues of the DP equation from our method. Recently this new equation has been generalized by Holm and Staley by adding viscosity term. We also discuss Euler-Poincaré formalism of the Holm-Staley equation. In the second half of the paper we consider a generalization of the Degasperis and Procesi (DP) equation with two dependent variables. we study the Euler-Poincaré framework of the 2-component Degasperis-Procesi equation. We also mention about the b-family equation.     

Spin versus helicity: Processes involving transversity

We construct the spin formalism in order to deal naturally with processes involving transversity which are now of increasing popularity. The helicity formalism which is more appropriate for collision processes of definite helicity has been so far used to also manage processes with transversity, but at the price of computing numerous helicity amplitudes which generally involve unnecessary kinematical variables.     

Contextualist viewpoint to Greenberger–Horne–Zeilinger paradox

We present probabilistic analysis of the Greenberger–Horne–Zeilinger (GHZ) scheme in the contextualist framework, namely under the assumption that distributions of hidden variables depend on settings of measurement devices. On one hand, we found classes of probability distributions of hidden variables for that the GHZ scheme does not imply a contradiction between the local realism and quantum formalism. On the other hand, we found classes of probability distributions of hidden variables for that the GHZ scheme still induce such a contradiction (despite variations of distributions). It is also demonstrated that (well known in probability theory) singularity/absolute continuity dichotomy for probability distributions is closely related to the GHZ paradox. Our conjecture is that this GHZ coupling between singularity/absolute continuity dichotomy and incompatible/compatible measurements might be a general feature of quantum theory.     

Bäcklund transformation for the first flows of the relativistic Toda hierarchy and associated properties

In this communication we study a class of one parameter dependent auto-Bäcklund transformations for the first flow of the relativistic Toda lattice and also a variant of the usual Toda lattice equation. It is shown that starting from the Hamiltonian formalism such transformations are canonical in nature with a well defined generating function. The notion of spectrality is also analyzed and the separation variables are explicitly constructed.     

Nonlinear,nondispersive wave equations: Lagrangian and Hamiltonian functions in the hodograph transformation

The hodograph transformation is generally used in order to associate a system of linear partial differential equations to a system of nonlinear (quasilinear) differential equations by interchanging dependent and independent variables. Here we consider the case when the nonlinear differential system can be derived from a Lagrangian density and revisit the hodograph transformation within the formalism of the Lagrangian-Hamiltonian continuous dynamical systems.Restricting to the case of nondissipative, nondispersive one-dimensional waves, we show that the hodograph transformation leads to a linear partial differential equation for an unknown function that plays the role of the Lagrangian in the hodograph variables. We then define the corresponding hodograph Hamiltonian and show that it turns out to coincide with the wave amplitude. i.e., with the unknown function of the independent variables to be solved for in the initial nonlinear wave equation.     

Geometrical approach to stationary waves in a shallow grating

A geometrical analysis of the monochromatic solutions near the first band gap for a shallow Kerr grating is presented.The analysis is based on the coupled mode formalism and on Stokes variables. We investigate the electric field for nonzero energy flow, in particular we consider the phase difference between the counter-propagating coupled modes. Phase portraits for zero and nonzero energy flow are topologically different, and we clarify the way in which they are connected, thus identifying families of trajectories for nonzero flow that disappear when the flow goes to zero.     

Finding the Hamiltonian for Cosmological Models in Fourth-Order Gravity Theories Without Resorting to the Ostrogradski or Dirac Formalism

The Hamilton formalism of cosmological models in fourth-order theories of gravity is considered. An approach to constructing the Hamilton function is presented which starts by replacing the second order derivatives of configuration space coordinates by functions depending on these coordinates, its first order derivatives, and additional variables playing the role of configuration space coordinates. This formalism, which does not resort to the Ostrogradski or Dirac formalism, is elucidated and applied to examples. For a special class of Lagrange functions, it is demonstrated that the canonical coordinates of the considered formalism and of the Ostrogradski formalism are related via a canonical transformation. The canonical transformation is a transformation of the configuration space coordinates and a transformation of momentum components induced by the transformation of the configuration space coordinates for a special element of the class of Lagrange functions mentioned. The Wheeler-DeWitt equations belonging to this Lagrange function are related via minisuperspace coordinate transformations.     

Collective group coordinates: quantization in the neighborhood of classical solutions

A new relativistic form of Bogolioubov’s group transformation in canonical formalism is proposed. The variables that play the role of the symmetry group parameters have been constructed. It is shown that the operators of conserved quantities are represented by canonically conjugated momenta for these variables.     

Gauge Formalism for General Relativity and Fermionic Matter

A new formulation for General Relativity is developed; it is a canonical, global and geometrically well posed formalism in which gravity is described using only variables related to spin structures. It does not require any background metric fixing and it applies to quite general manifolds, i.e. it does not need particular symmetries requirement or global frames. A global Lagrangian framework for Dirac spinors is also provided; conserved quantities and superpotentials are given. The interaction between gravity and spinors is described in a minimal coupling fashion with respect to the new variables and the Hilbert stress tensor of spinor fields is computed, providing the gravitational field generated by spinors. Finally differences and analogies between this formalism and gauge theories are discussed.