Realizability of greedy algorithms

We discuss new models of an “affine” theory of gravity in multidimensional space-times with symmetric connections. We use and develop ideas of Weyl, Eddington, and Einstein, in particular, Einstein’s proposal to specify the space-time geometry by the use of the Hamilton principle. More specifically, the connection coefficients are determined using a “geometric” Lagrangian that is an arbitrary function of the generalized (nonsymmetric) Ricci curvature tensor (and, possibly, of other fundamental tensors) expressed in terms of the connection coefficients regarded as independent variables. Such a theory supplements the standard Einstein gravity with dark energy (the cosmological constant, in the first approximation), a neutral massive (or tachyonic) vector field (vecton), and massive (or tachyonic) scalar fields. These fields couple only to gravity and can generate dark matter and/or inflation. The new field masses (real or imaginary) have a geometric origin and must appear in any concrete model. The concrete choice of the geometric Lagrangian determines further details of the theory, for example, the nature of the vector and scalar fields that can describe massive particles, tachyons, or even “phantoms.” In “natural” geometric theories, which are discussed here, dark energy must also arise. We mainly focus on intricate relations between geometry and dynamics while only very briefly considering approximate cosmological models inspired by the geometric approach.     

Effective Lagrangians of the Randall–Sundrum Model

We construct the second variation Lagrangian for the Randall*Sundrum model with two branes, study its gauge invariance, and introduce and decouple the corresponding equations of motion. For the physical degrees of freedom in this model, we find the effective four-dimensional Lagrangians describing the massless graviton, massive gravitons, and the massless scalar radion. We show that the masses of these fields and their matter coupling constants are different on the different branes.     

The BRST-BV approach to massless fields adapted for the AdS/CFT correspondence

In the framework of the BRST-BV approach to the formulation of relativistic mechanics, we consider massless and massive fields of arbitrary spin propagating in a flat space and massless fields propagating in the AdS space. For such fields, we obtain BRST-BV Lagrangians invariant under gauge transformations. The Lagrangians and gauge transformations are constructed in terms of traceless gauge fields and traceless parameters of the gauge transformations. We consider the fields in the AdS space using the Poincaré parameterization of this space, which leads to a simple form of the BRST-BV Lagrangian. We show that in the Siegel gauge, the Lagrangian of the massless AdS fields leads to a decoupling of the equations of motion, and this substantially simplifies the study of the AdS/CFT correspondence. In a conformal algebra basis, we find a realization of the relativistic symmetries of fields and antifields in the AdS space.     

Weyl-Eddington-Einstein affine gravity in the context of modern cosmology

We propose new models of the “affine” theory of gravity in multidimensional space-times with symmetric connections. We use and develop ideas of Weyl, Eddington, and Einstein, in particular, Einstein’s proposed method for obtaining the geometry using the Hamilton principle. More specifically, the connection coefficients are determined using a “geometric” Lagrangian that is an arbitrary function of the generalized (nonsymmetric) Ricci curvature tensor (and, possibly, other fundamental tensors) expressed in terms of the connection coefficients regarded as independent variables. Such a theory supplements the standard Einstein theory with dark energy (the cosmological constant, in the first approximation), a neutral massive (or tachyonic) meson, and massive (or tachyonic) scalar fields. These fields couple only to gravity and can generate dark matter and/or inflation. The new field masses (real or imaginary) have a geometric origin and must appear in any concrete model. The concrete choice of the Lagrangian determines further details of the theory, for example, the nature of the fields that can describe massive particles, tachyons, or even “phantoms.” In “natural” geometric theories, dark energy must also arise. The basic parameters of the theory (cosmological constant, mass, possible dimensionless constants) are theoretically indeterminate, but in the framework of modern “multiverse” ideas, this is more a virtue than a defect. We consider further extensions of the affine models and in more detail discuss approximate effective (“physical”) Lagrangians that can be applied to the cosmology of the early Universe.     

Spin c geometry of Kähler manifolds and the Hodge Laplacian on minimal Lagrangian submanifolds

From the existence of parallel spinor fields on Calabi-Yau, hyper-Kähler or complex flat manifolds, we deduce the existence of harmonic differential forms of different degrees on their minimal Lagrangian submanifolds. In particular, when the submanifolds are compact, we obtain sharp estimates on their Betti numbers which generalize those obtained by Smoczyk in [49]. When the ambient manifold is Kähler-Einstein with positive scalar curvature, and especially if it is a complex contact manifold or the complex projective space, we prove the existence of Kählerian Killing spinor fields for some particular spin ^c structures. Using these fields, we construct eigenforms for the Hodge Laplacian on certain minimal Lagrangian submanifolds and give some estimates for their spectra. These results also generalize some theorems by Smoczyk in [50]. Finally, applications on the Morse index of minimal Lagrangian submanifolds are obtained.     

A Weyl-Cartan space-time model based on the gauge principle

Based on the requirement that the gauge invariance principle for the Poincaré-Weyl group be satisfied for the space-time manifold, we construct a model of space-time with the geometric structure of a Weyl-Cartan space. We show that three types of fields must then be introduced as the gauge (“compensating”) fields: Lorentz, translational, and dilatational. Tetrad coefficients then become functions of these gauge fields. We propose a geometric interpretation of the Dirac scalar field. We obtain general equations for the gauge fields, whose sources can be the energy-momentum tensor, the total momentum, and the total dilatation current of an external field. We consider the example of a direct coupling of the gauge field to the orbital momentum of the spinor field. We propose a gravitational field Lagrangian with gauge-invariant transformations of the Poincaré-Weyl group. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 157, No. 1, pp. 64–78, October, 2008.     

A possibility to describe models of massive non-Abelian gauge fields in the framework of a renormalizable theory

We propose a renormalizable theory of massive non-Abelian gauge fields that does not require the existence of observable scalar fields.     

Calculation of the renormalized two-point function by adiabatic regularization

We calculate a renormalized two-point function using the adiabatic regularization method. We study the conformally and minimally coupled cases for massless and massive scalar fields in full detail. We reproduce previous results in a rigorous mathematical form and clarify some empirical approximations and bounds. We consider some applications to inflationary models.     

Quintessence scalar field in the relativistic theory of gravity

We propose a variant of the quintessence theory in order to obtain an accelerated expansion of the Friedmann universe in the framework of the relativistic theory of gravity. The scalar field of dark energy creates the substance of quintessence. We show that the function V(Φ) that factors the Lagrangian of the scalar field Φ does not influence the evolution of the universe. We find some relations that allow finding the explicit time-dependence of Φ if only the function V(Φ) is chosen. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 3, pp. 551–560, September, 2007.     

Non-Abelian tensor gauge fields

A recently proposed extension of Yang-Mills theory contains non-Abelian tensor gauge fields. The Lagrangian has quadratic kinetic terms, as well as cubic and quartic terms describing nonlinear interaction of tensor gauge fields with the dimensionless coupling constant. We analyze the particle content of non-Abelian tensor gauge fields. In four-dimensional space-time the rank-2 gauge field describes propagating modes of helicity 2 and 0. We introduce interaction of the non-Abelian tensor gauge field with fermions and demonstrate that the free equation of motion for the spinor-vector field correctly describes the propagation of massless modes of helicity 3/2. We have found a new metric-independent gauge invariant density which is a four-dimensional analog of the Chern-Simons density. The Lagrangian augmented by this Chern-Simons-like invariant describes the massive Yang-Mills boson, providing a gauge invariant mass gap for a four-dimensional gauge field theory.     

Two-dimensional systems that arise from the Noether classification of Lagrangians on the line

Noether-like operators play an essential role in writing down the first integrals for Euler-Lagrange systems of ordinary differential equations (ODEs). The classification of such operators is carried out with the help of analytic continuation of Lagrangians on the line. We obtain the classification of 5, 6 and 9 Noether-like operators for two-dimensional Lagrangian systems that arise from the submaximal and maximal dimensional Noether point symmetry classification of Lagrangians on the line. Cases in which the Noether-like operators are also Noether point symmetries for the systems of two ODEs are mentioned. In particular, the 8-dimensional maximal Noether algebra is remarkably obtained for the simplest system of the free particle equations in two dimensions from the 5-dimensional complex Noether algebra of the standard Lagrangian of the scalar free particle equation. We present the effectiveness of Noether-like operators for the determination of first integrals of systems of two nonlinear differential equations which arise from scalar complex Euler-Lagrange ODEs that admit Noether symmetry.     

Approximate KMS States for Scalar and Spinor Fields in Friedmann�CRobertson�CWalker Spacetimes

We construct and discuss Hadamard states for both scalar and Dirac spinor fields in a large class of spatially flat Friedmann–Robertson–Walker spacetimes characterised by an initial phase either of exponential or of power-law expansion. The states we obtain can be interpreted as being in thermal equilibrium at the time when the scale factor a has a specific value a = a ₀. In the case a ₀ = 0, these states fulfil a strict KMS condition on the boundary of the spacetime, which is either a cosmological horizon or a Big Bang hypersurface. Furthermore, in the conformally invariant case, they are conformal KMS states on the full spacetime. However, they provide a natural notion of an approximate KMS state also in the remaining cases, especially for massive fields. On the technical side, our results are based on a bulk-to-boundary reconstruction technique already successfully applied in the scalar case and here proven to be suitable also for spinor fields. The potential applications of the states we find range over a broad spectrum, but they appear to be suited to discuss in particular thermal phenomena such as the cosmic neutrino background or the quantum state of dark matter.     

The existence of Hamiltonian stationary Lagrangian tori in K?hler manifolds of any dimension

Hamiltonian stationary Lagrangians are Lagrangian submanifolds that are critical points of the volume functional under Hamiltonian deformations. They are natural generalizations of special Lagrangians or Lagrangian and minimal submanifolds. In this paper, we obtain a local condition that gives the existence of a smooth family of Hamiltonian stationary Lagrangian tori in K?hler manifolds. This criterion involves a weighted sum of holomorphic sectional curvatures. It can be considered as a complex analogue of the scalar curvature when the weighting are the same. The problem is also studied by Butscher and Corvino (Hamiltonian stationary tori in Kahler manifolds, 2008).     

Dimensional Regularization and the n-Wave Procedure for Scalar Fields in Many-Dimensional Quasi-Euclidean Spaces

We obtain the vacuum expectation values of the energy–momentum tensor for a scalar field arbitrarily coupled to a curvature in the case of an N-dimensional quasi-Euclidean space–time; the vacuum is defined in accordance with the Hamiltonian diagonalization method. We extend the n-wave procedure to the many-dimensional case. We find all the counterterms in the case N=5 and the counterterms for the conformal scalar field in the cases N=6,7. We determine the geometric structure of the first three counterterms in the N-dimensional case. We show that all the subtractions in the four-dimensional case and the first three subtractions in the many-dimensional case correspond to the renormalization of the parameters in the bare gravitational Lagrangian. We discuss the geometric structure of the other counterterms in the many-dimensional case and the problem of eliminating the conformal anomaly in the four-dimensional case.     

Lagrangian Submanifolds with Zero Scalar Curvature in Complex Euclidean Space

In this paper we show how to embed a time slice of the Schwarzchild spacetime that models the outer space around a massive star, as a Lagrangian submanifold invariant under the standard action of the special orthogonal group on complex Euclidean space. This result is generalized for rotationally invariant metrics that can be considered as higher-dimensional versions of Schwarzchild's and these submanifolds are locally characterized as the only ones with zero scalar curvature inside the above family.     

Revisiting the Concentration Problem of Vector Fields within a Spherical Cap: A Commuting Differential Operator Solution

We propose a novel basis of vector functions, the mixed vector spherical harmonics that are closely related to the functions of Sheppard and Török and help us reduce the concentration problem of tangential vector fields within a spherical cap to an equivalent scalar problem. Exploiting an analogy with previous results published by Grünbaum, Longhi and Perlstadt, we construct a differential operator that commutes with the concentration operator of this scalar problem and propose a stable and convenient method to obtain its eigenfunctions. Having obtained the scalar eigenfunctions, the calculation of tangential vector Slepian functions is straightforward.     

Integral Equation for the Spinor Amplitude of a Dirac Particle in a Curved Space–Time

We obtain a system of integral equations for the spinor amplitude of a wave packet describing a massive neutral Dirac particle in a curved space–time with an arbitrary geometry. This equation permits describing the spin dynamics of fermions in gravitational fields adequately to the quantum nature of spin. We consider a specific example of the Kerr–Schild metric. We also discuss the problem of massive neutrino oscillations in an external gravitational field.     

The <Emphasis Type="Italic">p</Emphasis>-adic sector of the adelic string

We study the construction of Lagrangians that can be considered the Lagrangians of the p-adic sector of an adelic open scalar string. Such Lagrangians are closely related to the Lagrangian for a single p-adic string and contain the Riemann zeta function with the d’Alembertian in its argument. In particular, we present a new Lagrangian obtained by an additive approach that combines all p-adic Lagrangians. This new Lagrangian is attractive because it is an analytic function of the d’Alembertian. Investigating the field theory with the Riemann zeta function is also interesting in itself.     

Lagrangian formulation of irreducible massive fields of arbitrary spin in (2+1) dimensions

The Lagrangian formulation for free massive fields corresponding to arbitrary integer and half-integer spin-irreducible representations of the Poincaré group in three-dimensional space-time is presented. The relationship of the theory under consideration to the self-duality equations in four dimensions is discussed. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 113, No. 1, pp. 45–57, October, 1997.     

Gravity in the stabilized brane world model in the five-dimensional Brans-Dicke theory

We obtain and solve the linearized equations of motion for gravity and scalar fields in the world stabilized brane model in the five-dimensional Brans-Dicke theory. We segregate the physical degrees of freedom and find the mass spectrum of the Kaluza-Klein excitations and the coupling constants of Kaluza-Klein modes to matter on the brane with negative tension.