Linear derivative Cartan formulation of general relativity

Beside diffeomorphism invariance also manifest SO(3,1) local Lorentz invariance is implemented in a formulation of Einstein gravity (with or without cosmological term) in terms of initially completely independent vielbein and spin connection variables and auxiliary two-form fields. In the systematic study of all possible embeddings of Einstein gravity into that formulation with auxiliary fields, the introduction of a “bi-complex” algebra possesses crucial technical advantages. Certain components of the new two-form fields directly provide canonical momenta for spatial components of all Cartan variables, whereas the remaining ones act as Lagrange multipliers for a large number of constraints, some of which have been proposed already in different, less radical approaches. The time-like components of the Cartan variables play that role for the Lorentz constraints and others associated to the vierbein fields. Although also some ternary ones appear, we show that relations exist between these constraints, and how the Lagrange multipliers are to be determined to take care of second class ones. We believe that our formulation of standard Einstein gravity as a gauge theory with consistent local Poincaré algebra is superior to earlier similar attempts.Received: 24 January 2005, Published online: 8 June 2005     

Holonomies of gauge fields in twistor space 2: Hecke algebra, diffeomorphism, and graviton amplitudes

We define a theory of gravity by constructing a gravitational holonomy operator in twistor space. The theory is a gauge theory whose Chan–Paton factor is given by a trace over elements of Poincaré algebra and Iwahori–Hecke algebra. This corresponds to a fact that, in a spinor-momenta formalism, gravitational theories are invariant under spacetime translations and diffeomorphism. The former symmetry is embedded in tangent spaces of frame fields while the latter is realized by a braid trace. We make a detailed analysis on the gravitational Chan–Paton factor and show that an S-matrix functional for graviton amplitudes can be expressed in terms of a supersymmetric version of the holonomy operator. This formulation will shed a new light on studies of quantum gravity and cosmology in four dimensions.     

Potential–density pairs for axisymmetric galaxies: the influence of scalar fields

We present a formulation for potential–density pairs to describe axi-symmetric galaxies in the Newtonian limit of scalar–tensor theories of gravity. The scalar field is described by a modified Helmholtz equation with a source that is coupled to the standard Poisson equation of Newtonian gravity. The net gravitational force is given by two contributions: the standard Newtonian potential plus a term stemming from massive scalar fields. General solutions have been found for axisymmetric systems and the multipole expansion of the Yukawa potential is given. In particular, we have computed potential–density pairs of galactic disks for an exponential profile and their rotation curves.     

Stability at a local maximum in higher dimensional anti-deSitter space and applications to supergravity

The stability of anti-deSitter background solutions of gravity/scalar systems with respect to small fluetuations of the scalar fields is analyzed. As in the four dimensional case, one finds stability even about a local maximum provided that the effective scalar field masses are not “too tachyonic,” and for sufficiently low mass there are two possible boundary conditions that allow a well-defined and conserved energy. Unlike the d = 4 case only one boundary condition is applicable to the known gauged supergravity theories in d = 5 and d = 7. For d = 7 the stability criterion is satisfied while for d = 5 stability is marginal.     

Time‐reparametrization invariance and Hamilton‐Jacobi approach to the cosmological σ‐model

The construction of physical models with local time‐reparametrization invariance is reviewed. Negative‐energy contributions to the hamiltonian are shown to be crucial for the realization of this reparametrization symmetry. The covariant formulation of the dynamics is used to develop a time and gauge invariant Hamilton‐Jacobi theory. This formalism is applied to solve for the cosmology of a homogeneous universe of the Friedmann‐Lemaître‐Robertson‐Walker type. After a discussion of empty universes, the FLRW theory is extended with homogeneous scalar fields generically described by a σ‐model on some scalar manifold. An explicit gauge‐invariant solution is constructed for the non‐linear O(N)‐models.     

Braneworld models of scalar fields with generalized dynamics

This work deals with braneworld models driven by real scalar fields with nonstandard dynamics. We develop the first-order formalism for models with standard gravity but with the scalar fields having generalized dynamics. We illustrate the results with examples of current interest, and we find analytical and numerical solutions for warp factors and scalar fields. The results indicate that the generalized braneworld scenario is classically stable, and capable of localizing gravity.     

Non-minimally coupled scalar fields, Holst action and black hole mechanics

The paper deals with the extension of the Weak Isolated Horizon (WIH) formulation of black hole horizons to the non-minimally coupled scalar fields. In the early part of the paper, we introduce an appropriate Holst type action to incorporate scalar fields non-minimally coupled to gravity and construct the covariant phase space of the theory. Using this phase space, we proceed to prove the laws of black hole mechanics. Further, we show that with a gauge fixing, the symplectic structure on the horizon reduces to that of a U(1) Chern–Simons theory. The level of the Chern–Simons theory is shown to depend on the non-minimally coupled scalar field.     

Spinorial Matter in Affine Theory of Gravity and the Space Problem

The purely affine theory of gravity possesses a canonical formulation. For this and other reasons, it could be a promising candidate for quantum gravity. Motivated by these perspectives, we discuss spinorial matter coupled to gravity, where the latter is described by a connection having no a priori relation to a metric. We show that one can establish a truncated spinor formalism which, for special or approximate solutions to the gravitational equations, reduces to the standard formalism. As a consequence, one arrives at "matter-induced" Riemann–Cartan spaces solving the Weyl-Cartan space problem.     

Wess–Zumino Currents on S3 × R Spacetime

Based on the group theory powerful techniques, as a rigorous tool for treating fields on S ³ × R spacetime, which is the manifold of SU(2), we put the supersymmetric Wess–Zumino model on the S ³ × R background. After deriving the system of Klein–Gordon–Dirac-type equations, for the scalar and Majorana fields, we get in the corresponding current, besides the supercurrent, an additional term due to the coupling of spin to gravity. Finally, considerations on the solutions of the fields equations are made, pointing out significant differences from the Minkowskian case.     

Gravitational Waves and Black Holes

In these lectures general relativity is outlined as the classical field theory of gravity, emphasizing physical phenomena rather than formalism. Dynamical solutions representing traveling waves as well as stationary fields like those of black holes are discussed. Their properties are investigated by studying the geodesic structure of the corresponding space-times, as representing the motion of point-like test particles. The interaction between gravitational, electro-magnetic and scalar fields is also considered.     

Weyl geometric gravity and electroweak symmetry “breaking”

A Weyl geometric scale covariant approach to gravity due to Omote, Dirac, and Utiyama (1971ff) is reconsidered. It can be extended to the electroweak sector of elementary particle fields, taking into account their basic scaling freedom. Already Cheng (1988) indicated that electroweak symmetry breaking, usually attributed to the Higgs field with a boson expected at 0.1–0.3 TeV, may be due to a coupling between Weyl geometric gravity and electroweak interactions. Weyl geometry seems to be well suited for treating questions of elementary particle physics, which relate to scale invariance and its “breaking”. This setting suggests the existence of a scalar field boson at the surprisingly low energy of ～ 1 eV. That may appear unlikely; but, as a payoff, the acquirement of mass arises as a result of coupling to gravity in agreement with the understanding of mass as the gravitational charge of fields.     

Effects Due to a Scalar Coupling on the Particle-Antiparticle Production in the Duffin-Kemmer-Petiau Theory

The Duffin-Kemmer-Petiau formalism with vector and scalar potentials is used to point out a few misconceptions diffused in the literature. It is explicitly shown that the scalar coupling makes the DKP formalism not equivalent to the Klein-Gordon formalism or to the Proca formalism, and that the spin-1 sector of the DKP theory looks formally like the spin-0 sector. With proper boundary conditions, scattering of massive bosons in an arbitrary mixed vector-scalar square step potential is explored in a simple way and effects due to the scalar coupling on the particle-antiparticle production and localization of bosons are analyzed in some detail.     

Matter Loops Corrected Modified Gravity in Palatini Formulation

Recently, corrections to the standard Einstein-Hilbert action were proposed to explain the current cosmic acceleration in stead of introducing dark energy. In the Palatini formulation of those modified gravity models, there is an important observation due to Arkani-Hamed: matter loops will give rise to a correction to the modified gravity action proportional to the Ricci scalar of the metric. In the presence of such a term, we show that the current forms of modified gravity models in Palatini formulation, specifically, the 1/R gravity and ln R gravity, will have phantoms. Then we study the possible instabilities due to the presence of phantom fields. We show that the strong instability in the metric formulation of 1/R gravity indicated by Dolgov and Kawasaki will not appear and the decay timescales for the phantom fields may be long enough for the theories to make sense as effective field theory . On the other hand, if we change the sign of the modification terms to eliminate the phantoms, some other inconsistencies will arise for the various versions of the modified gravity models. Finally, we comment on the universal property of the Palatini formulation of the matter loops corrected modified gravity models and its implications.     

Time-dependent scalar Beltrami-Hertz potentials in free space

We have solved the Beltrami-Maxwell equations for free space in terms of time-dependent scalar functions, the so-called scalar Beltrami-Hertz potentials. The two Beltrami fields have been represented in terms of scalar Beltrami-Hertz potentials. While the method is formulated for general sources, it is at its most powerful when the impressed source current densities are unidirectional: each Beltrami field, a complex-valued vector, can then be derived from a single scalar Beltrami-Hertz potential. We have calculated the corresponding scalar Green function explicity and given closed-form solutions for dipolar sources. Finally, the connection between the Beltrami-Maxwell formalism and conventional electromagnetic theory has been re-affirmed.     

New measures for nonrenormalizable quantum field theory

Generic interactions characteristic of so-called nonrenormalizable scalar and spinor quantum field theories are interpreted as discontinuous perturbations in the sense that the theory does not return to the unperturbed theory as the interaction coupling vanishes. To proceed beyond this interpretation specific alternatives to conventional quantization schemes are developed. Solution of a highly idealized (independent-value), nonrenormalizable scalar field theory automatically entails a formally scale-invariant measure (rather than the conventional translation-invariant measure) in a functional integral formulation, and the success of this measure suggests its use more generally. Such a measure can be motivated (by augmented field theory) on heuristic grounds as taking into account the partial hardcore nature of the interaction responsible for its behavior as a discontinuous perturbation. This modification leads generally to what we call scale-covariant quantization, which can be formulated in terms of unconventional functional differential equations, coupled Green's function equations and operator field equations. Use of affine fields establishes equivalence of these various approaches and enables analogous coupled Green's function equations for models with fermions to be most easily obtained. The basic concepts of this program are illustrated with elementary wave-mechanical examples.     

Destabilizing the gaugino condensate in modular invariant supergravity

We investigate the stability of the hidden sector gaugino condensate in a SL(2,Z)-invariant supergravity model inspired by E₈E₈ heterotic string, using the chiral superfield formalism. We calculate Planck-suppressed corrections to the “truncated approximation” for the condensate value and scalar potential. A transition to a phase with zero condensate occurs near special points in moduli space and at large compactification radius. We discuss the implications for the T-modulus dependence of supersymmetry-breaking.     

On quantum electrodynamics of two-particle bound states containing spinless particles

We develop here the general treatment arising from the Bethe-Salpeter equation for a two-particle bound system in which at least one of the particles is spinless. It is shown that a natural two-component formalism can be formulated for describing the propagators of scalar particles. This leads to a formulation of the Bethe-Salpeter equation in a form very reminiscent of the fermion-fermion case. It is also shown, that using this two-component formulation for spinless particles, the perturbation theory can be systematically developed in a manner similar to that of fermions. Quantum electrodynamics for scalar particles is then developed in the two component formalism, and the problem of bound states, in which one of the constituent particles is spinless, is examined by means of the means of the Bethe-Salpeter equation. For this case, the Bethe-Salpeter equation is cast into a form which is convenient to perform a Foldy-Woutyhuysen transformation which we carry out, keeping the lowest-order relativistic corrections to the nonrelativistic equation. The results are compared with the corresponding fermion-fermion case. It is shown, as might have been expected, that the only spin-independent terms that occur for the fermion-fermion system which do not occur for bound scalar particle cases, is the zitterbewegung contribution. The relevance of the above considerations for systems that are essentially bound by electromagnetic interactions, such as kaonic hydrogen, is discussed.     

A local approach to dimensional reduction: I. General formalism

We present a formalism for dimensional reduction based on the local properties of invariant cross-sections (“fields”) and differential operators. This formalism does not need an ansatz for the invariant fields and is convenient when the reducing group is non-compact.

In the approach presented here, splittings of some exact sequences of vector bundles play a key role. In the case of invariant fields and differential operators, the invariance property leads to an explicit splitting of the corresponding sequences, i.e. to the reduced field/operator. There are also situations when the splittings do not come from invariance with respect to a group action but from some other conditions, which leads to a “non-canonical” reduction.

In a special case, studied in detail in the second part of this article, this method provides an algorithm for construction of conformally invariant fields and differential operators in Minkowski space.     

The nearly Newtonian regime in non-linear theories of gravity

The present paper reconsiders the Newtonian limit of models of modified gravity including higher order terms in the scalar curvature in the gravitational action. This was studied using the Palatini variational principle in Meng and Wang (Gen. Rel. Grav. 36, 1947 (2004)) and Domínguez and Barraco (Phys. Rev. D 70, 043505 (2004)) with contradicting results. Here a different approach is used, and problems in the previous attempts are pointed out. It is shown that models with negative powers of the scalar curvature, like the ones used to explain the present accelerated expansion, as well as their generalization which include positive powers, can give the correct Newtonian limit, as long as the coefficients of these powers are reasonably small. Some consequences of the performed analysis seem to raise doubts for the way the Newtonian limit was derived in the purely metric approach of fourth order gravity [Dick in Gen. Rel. Grav. 36, 217 (2004)]. Finally, we comment on a recent paper [Olmo in Phys. Rev. D 72, 083505 (2005)] in which the problem of the Newtonian limit of both the purely metric and the Palatini formalism is discussed, using the equivalent Brans–Dicke theory, and with which our results partly disagree.