Faddeev Formulation Research Articles

First-order discrete Faddeev gravity at strongly varying fields

We consider the Faddeev formulation of general relativity (GR), which can be characterized by a kind of d-dimensional tetrad (typically d = 10) and a non-Riemannian connection. This theory is invariant w.r.t. the global, but not local, rotations in the d-dimensional space. There can be configurations with a smooth or flat metric, but with the tetrad that changes abruptly at small distances, a kind of “antiferromagnetic” structure. Previously, we discussed a first-order representation for the Faddeev gravity, which uses the orthogonal connection in the d-dimensional space as an independent variable. Using the discrete form of this formulation, we considered the spectrum of (elementary) area. This spectrum turns out to be physically reasonable just on a classical background with large connection like rotations by [Formula: see text], that is, with such an “antiferromagnetic” structure. In the discrete first-order Faddeev gravity, we consider such a structure with periodic cells and large connection and strongly changing tetrad field inside the cell. We show that this system in the continuum limit reduces to a generalization of the Faddeev system. The action is a sum of related actions of the Faddeev type and is still reduced to the GR action.

Spectrum of area in the Faddeev formulation of gravity

Faddeev formulation of general relativity (GR) is considered where the metric is composed of ten vector fields or a ten-dimensional tetrad. Upon partial use of the field equations, this theory results in the usual general relativity (GR). Earlier, we have proposed first-order representation of the minisuperspace model for the Faddeev formulation where the tetrad fields are piecewise constant on the polytopes like four-simplices or, say, cuboids into which [Formula: see text] can be decomposed, an analogue of the Cartan–Weyl connection-type form of the Hilbert–Einstein action in the usual continuum GR. In the Hamiltonian formalism, the tetrad bilinears are canonically conjugate to the orthogonal connection matrices. We evaluate the spectrum of the elementary areas, functions of the tetrad bilinears. The spectrum is discrete and proportional to the Faddeev analog [Formula: see text] of the Barbero–Immirzi parameter [Formula: see text]. The possibility of the tetrad and metric discontinuities in the Faddeev gravity allows to consider any surface as consisting of a set of virtually independent elementary areas and its spectrum being the sum of the elementary spectra. Requiring consistency of the black hole entropy calculations known in the literature we are able to estimate [Formula: see text].

First-order minisuperspace model for the Faddeev formulation of gravity

Faddeev formulation of general relativity (GR) is considered where the metric is composed of ten vector fields or a ten-dimensional tetrad. Upon partial use of the field equations, this theory results in the usual GR. Earlier we have proposed some minisuperspace model for the Faddeev formulation where the tetrad fields are piecewise constant on the polytopes like 4-simplices or, say, cuboids into which [Formula: see text] can be decomposed. Now we study some representation of this (discrete) theory, an analogue of the Cartan–Weyl connection-type form of the Hilbert–Einstein action in the usual continuum GR.

Some minisuperspace model for the Faddeev formulation of gravity

We consider Faddeev formulation of General Relativity (GR) in which the metric is composed of ten vector fields or a 4 ×10 tetrad. This formulation reduces to the usual GR upon partial use of the field equations. A distinctive feature of the Faddeev action is its finiteness on the discontinuous fields. This allows to introduce its minisuperspace formulation where the vector fields are constant everywhere on ℝ4 with exception of a measure zero set (the piecewise constant fields). The fields are parametrized by their constant values independently chosen in, e.g. the 4-simplices or, say, parallelepipeds into which ℝ4 can be decomposed. The form of the action for the vector fields of this type is found. We also consider the piecewise constant vector fields approximating the fixed smooth ones. We check that if the regions in which the vector fields are constant are made arbitrarily small, the minisuperspace action and equations of motion tend to the continuum Faddeev ones.

First-order representation of the Faddeev formulation of gravity

We study the Faddeev formulation of gravity in which the metric is composed of a ten-dimensional tetrad . Here, A = 1, …, 10 refers to an Euclidean (or Minkowsky) ten-dimensional spacetime and λ = 1, 2, 3, 4 is the usual world index. A remarkable unique feature of the Faddeev formulation is that the action remains finite for the discontinuous tetrad. If the spacetime is composed of flat microblocks (in the discrete version of gravity), this means that these microblocks in quantum theory do not coincide on their common faces, that is, they are independent. We propose a representation of the Faddeev formulation analogous to the Cartan–Weyl formalism. It is based on extending the set of variables by introducing an infinitesimal SO(10) connection. Excluding this connection via equations of motion we reproduce the original Faddeev action. A peculiar feature of this representation is the occurrence of a local SO(10) symmetry violating condition so that the SO(10) symmetry is only global one in full correspondence with the fact that the original Faddeev formulation possesses just the global SO(10) symmetry. We also consider an analogue of the Barbero–Immirzi parameter which can be naturally introduced in the considered representation.

Then+n+αsystem in a continuum Faddeev formulation

The continuum Faddeev equations for the neutron-neutron-alpha (n-n-$\alpha$) system are formulated for a general interaction as well as for finite rank forces. In addition, the capture process n+n+$\alpha \rightarrow ^6$He+$\gamma$ is derived.

Continuum discretized coupled-channels method as a truncation of a connected-kernel formulation of three-body problems.

The widely used continuum discretized coupled-channels (CDCC) method for approximate calculations of three-body systems is discussed as a truncation of a Faddeev formulation in angular momentum space. A set of coupled equations is presented for converting a CDCC solution into a full solution of the original Schr\"odinger equation. A practical iterative procedure for solving the equations is outlined, based on the ``distorted Faddeev equations'' of Birse and Redish.

Unified theory of rearrangement scattering

We present a unified microscopic theory of rearrangement scattering which satisfies the Pauli principle. The equations for the transition amplitudes and wave functions are reduced to effective two-body equations with the complexity put into the distortion potentials and the transition potentials. The central theme in the development of this theory is the importance of elastic scattering in initial and final channel, although this is not a prerequisite for its validity as the theory is exact. The elastic degrees of freedom are absorbed into the distorted waves using projection techniques in combination with the Faddeev formulation. The resulting transition amplitude can be expanded in the transition potential, leading to a modified coupled channel series. The transition potential, which has a constructive nature (i.e., no counter terms), can also be expanded. We provide arguments for the convergence of both expansions. Various contributions to the transition potential are discussed, and we put special emphasis on a rescattering term which has not been considered before in the context of coupled channel equations. Various inconsistence which may hamper standard coupled channel calculations are discussed as well.

Comparison of multiple scattering approximations in a three body problem

An analysis of the nonrelativistic quantum mechanical problem of three particles of equal mass interacting via attractive delta function potentials in one dimension is carried out. Following the methods of McGuire and Dodd with two body potentials of equal strength, exact matrix elements for the process of elastic and rearrangement scattering are rederived and some mistakes occurring in their papers have been corrected. Then expressions are obtained for transition matrix elements for these processes in the first order using Newton, Born, Newton–Faddeev, and Lovelace–Faddeev formulation. Numerical results obtained are compared with the exact results. It is found that above breakup threshold the Newton–Faddeev scheme gives better results in comparison to other schemes for elastic as well as rearrangement scattering. Thus, this paper provides us with a comparative analysis of several multiple scattering expansions available in the literature.

Gravitational-scalar field coupling II: James H. Higbie. Department of Physics, Brandeis University, Waltham, Massachusetts and Technion-Israel Institute of Technology, Haifa, Israel

This chapter discusses the dynamical equations and approximation method. The integral equations approach to the three-body problem, decisively stimulated by Faddeev's formulation, provides the most powerful tool for studying the internal structure of this system. An essential step toward a detailed understanding of composite particle dynamics has been done in this way. The search for adequate extensions to the general N-body situation represents a natural challenge. Emphasizing different aspects of the three-body theory, different generalizations have been found. It was the concept of connectedness of the integral kernel that represented the guiding idea of almost all proposals. The four-body problem shows already the typical difficulties of the general case. In the Faddeev case, it can be shown that after a sufficient number of iterations, the kernel becomes even completely continuous. The advantage of such a procedure is that the completeness does not have to be assumed but is directly proved.

Review of the theoretical study of resonances in atomic collisions

The various theoretical methods for the investigation of resonances in atomic collisions are reviewed together with a discussion of the merits and limitations of these methods as well as of their inter-relationships. The methods which have been explicitly considered are the Siegert and Kapur-Peierls methods, the Feshbach projection-operator approach, the Faddeev formulation, the pseudo-state expansion, and the Harris-Nesbet algebraic expansion. A new method which proves to converge rapidly and has a decisive computational advantage over the conventional coupled-equation method is presented.