Manifestly-covariant <Emphasis Type="Italic">p</Emphasis><Superscript>3</Superscript> calculation of nucleon Compton scattering

We compute the Compton scattering off the nucleons in the framework of manifestly covariant baryon chiral perturbation theory (BχPT). The results for observables differ substantially from the corresponding calculations in heavy-baryon chiral perturbation theory (HBχPT), most appreciably in the forward kinematics. We verify that the covariant p ³ result fulfills the forward-Compton-scattering sum rules. We also explore the effect of the Δ(1232) resonance at order p ⁴/Δ, with Δ ≈ 300 MeV, the resonance excitation energy. We find that the substantial effect of the Δ-excitation on the nucleon polarizabilities can naturally be accommodated in the manifestly covariant calculation. The article is published in the original.     

On the cosmological constant problem and brane-world geometry

The cosmological constant problem is examined within the context of the covariant brane-world gravity, based on Nash’s embedding theorem for Riemannian geometries. We show that the vacuum structure of the brane-world is more complex than General Relativity’s because it involves extrinsic elements, in specific, the extrinsic curvature. In other words, the shape (or local curvature) of an object becomes a relative concept, instead of the “absolute shape” of General Relativity. We point out that the immediate consequence is that the cosmological constant and the energy density of the vacuum quantum fluctuations have different physical meanings: while the vacuum energy density remains confined to the four-dimensional brane-world, the cosmological constant is a property of the bulk’s gravitational field that leads to the conclusion that these quantities cannot be compared, as it is usually done in General Relativity. Instead, the vacuum energy density contributes to the extrinsic curvature, which in turn generates Nash’s perturbation of the gravitational field. On the other hand, the cosmological constant problem ceases to be in the brane-world geometry, reappearing only in the limit where the extrinsic curvature vanishes.     

Structure theorem for covariant bundles on quantum homogeneous spaces

The natural generalization of the notion of bundle in quantum geometry is that of bimodule. If the base space has quantum group symmetries, one is particularly interested in bimodules covariant (equivariant) under these symmetries. Most attention has so far been focused on the case with maximal symmetry — where the base space is a quantum group and the bimodules are bicovariant. The structure of bicovariant bimodules is well understood through their correspondence with crossed modules. We investigate the “next best” case — where the base space is a quantum homogeneous space and the bimodules are covariant. We present a structure theorem that resembles the one for bicovariant bimodules. Thus, there is a correspondence between covariant bimodules and a new kind of “crossed” modules which we define. The latter are attached to the pair of quantum groups which defines the quantum homogeneous space. We apply our structure theorem to differential calculi on quantum homogeneous spaces and discuss a related notion of induced differential calculus. Presented at the 10th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June 2001. This work was supported by a NATO fellowship grant.     

Non-commutative Lorentz symmetry and the origin of the Seiberg–Witten map

We show that the non-commutative Yang–Mills field forms an irreducible representation of the (undeformed) Lie algebra of rigid translations, rotations and dilatations. The non-commutative Yang–Mills action is invariant under combined conformal transformations of the Yang–Mills field and of the non-commutativity parameter . The Seiberg–Witten differential equation results from a covariant splitting of the combined conformal transformations and can be computed as the missing piece to complete a covariant conformal transformation to an invariance of the action. Received: 6 November 2001 / Published online: 5 April 2002     

Extreme Covariant Observables for Type I Symmetry Groups

The structure of covariant observables—normalized positive operator measures (POMs)—is studied in the case of a type I symmetry group. Such measures are completely determined by kernels which are measurable fields of positive semidefinite sesquilinear forms. We produce the minimal Kolmogorov decompositions for the kernels and determine those which correspond to the extreme covariant observables. Illustrative examples of the extremals in the case of the Abelian symmetry group are given. Dedicated to Pekka J. Lahti in honor of his sixtieth birthday     

Hawking radiation from black holes in de Sitter spaces via covariant anomalies

We apply the covariant anomaly cancellation method to compute the Hawking fluxes from the event and cosmic horizons of the Schwarzschild–de Sitter black hole. The derivation is new from the existing ones as we split the space in three different regions (near to and away from the event and cosmic horizons) and write down the covariant energy–momentum tensor using three step functions which covers the whole region leading elegantly to the conditions required to compute the Hawking fluxes from the event and cosmic horizons.     

Covariant realizations of kappa-deformed space

We study a Lie algebra type κ-deformed space with an undeformed rotation algebra and commutative vector-like Dirac derivatives in a covariant way. The space deformation depends on an arbitrary vector. Infinitely many covariant realizations in terms of commuting coordinates of undeformed space and their derivatives are constructed. The corresponding coproducts and star products are found and related in a new way. All covariant realizations are physically equivalent. Specially, a few simple realizations are found and discussed. The scalar fields, invariants and the notion of invariant integration is discussed in the natural realization.     

Covariant canonical quantization

We present a manifestly covariant quantization procedure based on the de Donder–Weyl Hamiltonian formulation of classical field theory. This procedure agrees with conventional canonical quantization only if the parameter space is d=1 dimensional time. In d>1 quantization requires a fundamental length scale, and any bosonic field generates a spinorial wave function, leading to the purely quantum-theoretical emergence of spinors as a byproduct. We provide a probabilistic interpretation of the wave functions for the fields, and we apply the formalism to a number of simple examples. These show that covariant canonical quantization produces both the Klein–Gordon and the Dirac equation, while also predicting the existence of discrete towers of identically charged fermions with different masses. Covariant canonical quantization can thus be understood as a “first” or pre-quantization within the framework of conventional QFT. PACS 04.62.+v; 11.10.Ef; 12.10.Kt     

Covariant derivative on non-linear fiber bundles

A gauge field is usually described as a connection on a principal bundle. It induces a covariant derivative on associated vector bundles, sections of which represent matter fields. In general, however, it is not possible to define a covariant derivative on non-linear fiber bundles, i.e. on those which are not vector bundles. We definelogarithmic covariant derivatives acting on two special non-linear fiber bundles — on the principal bundle and on the local gauge group bundle. The logarithmic derivatives map from sections of these bundles to the sections of the local gauge algebra bundle. Some properties of the logarithmic derivatives are formulated.     

Vacuum fluctuations of the supersymmetric field in curved background

We study a supersymmetric model in curved background spacetime. We calculate the effective action and the vacuum expectation value of the energy momentum tensor using a covariant regularization procedure. A soft supersymmetry breaking induces a nonzero contribution to the vacuum energy density and pressure. Assuming the presence of a cosmic fluid in addition to the vacuum fluctuations of the supersymmetric field an effective equation of state is derived in a self-consistent approach at one loop order. The net effect of the vacuum fluctuations of the supersymmetric fields in the leading adiabatic order is a renormalization of the Newton and cosmological constants.     

Covariant anomaly and Hawking radiation from the modified black hole in the rainbow gravity theory

Recently, Banerjee and Kulkarni (R. Banerjee, S. Kulkarni, arXiv: 0707. 2449 [hep-th]) suggested that it is conceptually clean and economical to use only the covariant anomaly to derive Hawking radiation from a black hole. Based upon this simplified formalism, we apply the covariant anomaly cancellation method to investigate Hawking radiation from a modified Schwarzschild black hole in the theory of rainbow gravity. Hawking temperature of the gravity’s rainbow black hole is derived from the energy-momentum flux by requiring it to cancel the covariant gravitational anomaly at the horizon. We stress that this temperature is exactly the same as that calculated by the method of cancelling the consistent anomaly.     

Simplectic Geometry and the Canonical Variables for Dirac–Nambu–Goto and Gauss–Bonnet System in String Theory

Using a strongly covariant formalism given by Carter for the deformations dynamics of p-branes in a curved background and a covariant and gauge invariant geometric structure constructed on the corresponding Witten's phase space, we identify the canonical variables for Dirac–Nambu–Goto (DNG) and Gauss–Bonnet (GB) system in string theory. Future extensions of the present results are outlined.     

Natural generation and amplification of magnetic seed fields

We show, using a covariant and gauge– invariant approach to cosmological perturbation theory, that velocity and gravitational wave perturbations of the Friedmann– Lemaître– Robertson– Walker (FLRW) model can lead to the generation and amplification of cosmic magnetic fields. It is argued that under certain conditions these fields can reach strengths capable of supporting the galactic dynamo mechanism.     

Local Wick Polynomials and Time Ordered Products¶of Quantum Fields in Curved Spacetime

In order to have well defined rules for the perturbative calculation of quantities of interest in an interacting quantum field theory in curved spacetime, it is necessary to construct Wick polynomials and their time ordered products for the noninteracting theory. A construction of these quantities has recently been given by Brunetti, Fredenhagen, and K?hler, and by Brunetti and Fredenhagen, but they did not impose any “locality” or “covariance” condition in their constructions. As a consequence, their construction of time ordered products contained ambiguities involving arbitrary functions of spacetime point rather than arbitrary parameters. In this paper, we construct an “extended Wick polynomial algebra”– large enough to contain the Wick polynomials and their time ordered products – by generalizing a construction of Dütsch and Fredenhagen to curved spacetime. We then define the notion of a local, covariant quantum field, and seek a definition of local Wick polynomials and their time ordered products as local, covariant quantum fields. We introduce a new notion of the scaling behavior of a local, covariant quantum field, and impose scaling requirements on our local Wick polynomials and their time ordered products as well as certain additional requirements – such as commutation relations with the free field and appropriate continuity properties under variations of the spacetime metric. For a given polynomial order in powers of the field, we prove that these conditions uniquely determine the local Wick polynomials and their time ordered products up to a finite number of parameters. (These parameters correspond to the usual renormalization ambiguities occurring in Minkowski spacetime together with additional parameters corresponding to the coupling of the field to curvature.) We also prove existence of local Wick polynomials. However, the issue of existence of local time ordered products is deferred to a future investigation. Received: 27 March 2001 / Accepted: 6 June 2001     

Gauge-invariant localization of infinitely many gravitational energies from all possible auxiliary structures

The problem of finding a covariant expression for the distribution and conservation of gravitational energy–momentum dates to the 1910s. A suitably covariant infinite-component localization is displayed, reflecting Bergmann’s realization that there are infinitely many conserved gravitational energy–momenta. Initially use is made of a flat background metric (or rather, all of them) or connection, because the desired gauge invariance properties are obvious. Partial gauge-fixing then yields an appropriate covariant quantity without any background metric or connection; one version is the collection of pseudotensors of a given type, such as the Einstein pseudotensor, in every coordinate system. This solution to the gauge covariance problem is easily adapted to any pseudotensorial expression (Landau–Lifshitz, Goldberg, Papapetrou or the like) or to any tensorial expression built with a background metric or connection. Thus the specific functional form can be chosen on technical grounds such as relating to Noether’s theorem and yielding expected values of conserved quantities in certain contexts and then rendered covariant using the procedure described here. The application to angular momentum localization is straightforward. Traditional objections to pseudotensors are based largely on the false assumption that there is only one gravitational energy rather than infinitely many.     

Scalar Tachyons in the de Sitter Universe

We provide a construction of a class of local and de Sitter covariant tachyonic quantum fields which exist for discrete negative values of the squared mass parameter and which have no Minkowskian counterpart. These quantum fields satisfy an anomalous non-homogeneous Klein–Gordon equation. The anomaly is a covariant field which can be used to select the physical subspace (of finite co-dimension) where the homogeneous tachyonic field equation holds in the usual form. We show that the model is local and de Sitter invariant on the physical space. Our construction also sheds new light on the massless minimally coupled field, which is a special instance of it.     

The Cauchy Problem for Chern–Simons–Proca–Higgs Equations

We study initial value problems for Chern–Simons–Proca–Higgs equations. We prove that the Cauchy problem is locally well posed under the Lorentz gauge condition. In the case of repulsive potential, the global existence of the solution is proved using the covariant version of the Brezis–Gallouet inequality. In the case of attractive potential, we show that for initial data having negative energy, the solution has a finite time singularity.     

Covariant Sectors with Infinite Dimension¶and Positivity of the Energy

Let ? be a local conformal net of von Neumann algebras on S ¹ and ρ a M?bius covariant representation of ?, possibly with infinite dimension. If ρ has finite index, ρ has automatically positive energy. If ρ has infinite index, we show the spectrum of the energy always to contain the positive real line, but, as seen by an example, it may contain negative values. We then consider nets with Haag duality on ℝ, or equivalently sectors with non-solitonic extension to the dual net; we give a criterion for irreducible sectors to have positive energy, namely this is the case iff there exists an unbounded M?bius covariant left inverse. As a consequence the class of sectors with positive energy is stable under composition, conjugation and direct integral decomposition. Received: 21 April 1997 / Accepted: 23 September 1997     

Super <Emphasis Type="Italic">D</Emphasis>-Differentiation for <Emphasis Type="Italic">R</Emphasis><Superscript> ∞ </Superscript>-Supermanifolds

The concept of ‘D-Differentiation’, which, in the context of smooth manifolds, generalises Lie and covariant differentiation, is extended to R ^∞-supermanifolds under the name of ‘Super D-Differentiation’. This is done by defining new (non-linear) mappings, called ‘μ-mappings’ and by relating their non-linearity to the Leibniz rule that a derivation must satisfy when it acts on a tensor product. The resulting axiomatics, which is basis-independent and coordinate-free, is then expressed in a general basis (not necessarily holonomic). Super Lie and Super covariant differentiation are, amongst others, special cases of Super D-Differentiation. In particular, the transformation rules for the connection coefficients and the commutation coefficients of non-holonomic bases are obtained. These special cases are found to be in agreement with the DeWitt Super covariant and Super Lie derivatives.