arXiv subjects
Publications and source records attributed to W. Kummer.
The trace anomaly for nonminimally coupled scalars in spherically reduced gravity obtained by Bousso and Hawking (hep-th/9705236) is incorrect. We explain the reasons for the deviations from our correct (published) result which is supported by several other recent papers.
We demonstrate that in the absence of `matter' fields to all orders of perturbation theory and for all 2D dilaton theories the quantum effective action coincides with the classical one. This resolves the apparent contradiction between the well established results of Dirac quantization and perturbative (path-integral) approaches which seemed to yield non-trivial quantum corrections. For the Jackiw--Teitelboim (JT) model, our result is even extended to the situation when a matter field is present.
All 1+1 dimensional dipheomorphism-invariant models can be viewed in a unified manner. This includes also general dilaton theories and especially spherically symmetric gravity (SSG) and Witten's dilatonic black hole (DBH). A common feature --- also in the presence of matter fields of any type --- is the appearance of an absolutely conserved quantity C which is determined by the influx of matter. Only for a subclass of generalized dilaton theories the singularity structure vanishes together with C. Such `physical' theories include, of course, SSG and DBH. It seems to have been overlooked until recently that the (classical) 'black hole' singularity of the DBH deviates from SSG in a physically nontrivial manner. At the quantum level for all generalized dilaton theories --- in the absence of matter --- the local quantum effects are shown to disappear. This enables us to compute e.g. the second loop order correction to the Polyakov term. For non-minimal scalar coupling we also believe to have settled the controversial issue of Hawking radiation to infinity with a somewhat puzzling result for the case of SSG.
Performing an nonperturbative path integral for the geometric part of a large class of 2d theories without kinetic term for the dilaton field, the quantum effects from scalar matter fields are treated as a perturbation. When integrated out to two-loops they yield a correction to the Polyakov term which is still exact in the geometric part. Interestingly enough the effective action only experiences a renormalization of the dilaton potential.
It is well known that spherically symmetric reduction of General Relativity (SSG) leads to non-minimally coupled scalar matter. We generalize (and correct) recent results to Hawking radiation for a class of dilaton models which share with the Schwarzschild black hole non-minimal coupling of scalar fields and the basic global structure. An inherent ambiguity of such models (if they differ from SSG) is discussed. However, for SSG we obtain the rather disquieting result of a negative Hawking flux at infinity, if the usual recipe for such calculations is applied.
Within a first-order framework, we comprehensively examine the role played by boundary conditions in the canonical formulation of a completely general two-dimensional gravity model. Our analysis particularly elucidates the perennial themes of mass and energy. The gravity models for which our arguments are valid include theories with dynamical torsion and so-called generalized dilaton theories (GDTs). Our analysis of the canonical action principle (i) provides a rigorous correspondence between the most general first-order two-dimensional Einstein-Cartan model (ECM) and GDT and (ii) allows us to extract in a virtually simultaneous manner the ``true degrees of freedom'' for both ECMs and GDTs. For all such models, the existence of an absolutely conserved (in vacuo) quantity C is a generic feature, with (minus) C corresponding to the black-hole mass parameter in the important special cases of spherically symmetric four-dimensional general relativity and standard two-dimensional dilaton gravity. The mass C also includes (minimally coupled) matter into a ``universal mass function.'' We place particular emphasis on the (quite general) class of models within GDT possessing a Minkowski-like groundstate solution (allowing comparison between $C$ and the Arnowitt-Deser-Misner mass for such models).
General matterless models of gravity include dilaton gravity, arbitrary powers in curvature, but also dynamical torsion. They are a special class of "Poisson-sigma-models" whose solutions are known completely, together with their general global structure. Beside the ordinary black hole, arbitrary singularity structures can be studied. It is also possible to derive an action "backwards", starting from a given manifold. The role of conservation laws, Noether charge and the quantization have been investigated. Scalar and fermionic matter fields may be included as well.
Local path integral quantization of generic 2D dilaton gravity is considered. Locality means that we assume asymptotic fall off conditions for all fields. We demonstrate that in the absence of `matter' fields to all orders of perturbation theory and for all 2D dilaton theories the quantum effective action coincides with the classical one. This resolves the apparent contradiction between the well established results of Dirac quantization and perturbative (path-integral) approaches which seemed to yield non-trivial quantum corrections. For a particular case, the Jackiw--Teitelboim model, our result is even extended to the situation when a matter field is present.
The black hole of the widely used ordinary 2d--dilaton model (DBH) deviates from the Schwarzschild black hole (SBH) of General Relativity in one important feature: Whereas non-null extremals or geodesics show the expected incompleteness this turns out {\it not to be the case for the null extremals}. After a simple analysis in Kruskal coordinates for singularities with power behavior of this -- apparently till now overlooked -- property we discuss the global structure of a large family of generalized dilaton theories which does not only contain the DBH and SBH but also other proposed dilaton theories as special cases. For large ranges of the parameters such theories are found to be free from this defect and exhibit global SBH behavior.
The $t\bar{t}$ system allows a truly perturbative treatment of the potential. Completing previous computations, we calculate the contributions of QCD box graph corrections, which make the ``relativistic'' $O(\a^4)$ corrections in the non-Abelian case differ from the well known corrections of the same order in QED.
Two dimensional gravity with torsion is proved to be equivalent to special types of generalized 2d dilaton gravity. E.g. in one version, the dilaton field is shown to be expressible by the extra scalar curvature, constructed for an independent Lorentz connection corresponding to a nontrivial torsion. Elimination of that dilaton field yields an equivalent torsionless theory, nonpolynomial in curvature. These theories, although locally equivalent exhibit quite different global properties of the general solution. We discuss the example of a (torsionless) dilaton theory equivalent to the $R^2 + T^2$--model. Each global solution of this model is shown to split into a set of global solutions of generalized dilaton gravity. In contrast to the theory with torsion the equivalent dilaton one exhibits solutions which are asymptotically flat in special ranges of the parameters. In the simplest case of ordinary dilaton gravity we clarify the well known problem of removing the Schwarzschild singularity by a field redefinition.
General matterless--theories in 1+1 dimensions include dilaton gravity, Yang--Mills theory as well as non--Einsteinian gravity with dynamical torsion and higher power gravity, and even models of spherically symmetric d = 4 General Relativity. Their recent identification as special cases of 'Poisson--sigma--models' with simple general solution in an arbitrary gauge, allows a comprehensive discussion of the relation between the known absolutely conserved quantities in all those cases and Noether charges, resp. notions of quasilocal 'energy--momentum'. In contrast to Noether like quantities, quasilocal energy definitions require some sort of 'asymptotics' to allow an interpretation as a (gauge--independent) observable. Dilaton gravitation, although a little different in detail, shares this property with the other cases. We also present a simple generalization of the absolute conservation law for the case of interactions with matter of any type.
The bound state problem for a fermion-antifermion system is considered taking into account a finite decay width of the constituents. We propose an exactly solvable relativistic zero order equation similar to that of Barbieri and Remiddi, but including a constant width. We focus especially on the $t\bar{t}$ system for which we reconsider our recent calculation of the bound state corrections to the toponium width, which was performed in the narrow width approximation and needed the use of second order Bethe-Salpeter perturbation theory. We show that one obtains the same result already in first order BS perturbation theory if one uses our present approach. Furthermore the large cancellations of gauge dependent terms is demonstrated to be a consequence of a Ward identity. This cancellation mechanism is shown to be valid for general fermion-antifermion systems.
Recent evidence for the top mass in the region of 160 $GeV$ for the first time provides an opportunity to use the full power of relativistic quantum field theoretical methods, available also for weakly bound systems. Because of the large decay width $\G$ of the top quark individual energy-levels in "toponium" will be unobservable. However, the potential for the $t\bar{t}$ system, based on a systematic expansion in powers of the strong coupling constant $\a_s$ can be rigorously derived from QCD and plays a central role in the threshold region. It is essential that the neglect of nonperturbative (confining) effects is fully justified here for the first time to a large accuracy, also just {\it because} of the large $\G$. The different contributions to that potential are computed from real level corrections near the bound state poles of the $t\bar{t}$-system which for $\G \ne 0$ move into the unphysical sheet of the complex energy plane. Thus, in order to obtain the different contributions to that potential we may use the level corrections at that (complex) pole. Within the relevant level shifts we especially emphasize the corrections of order $O(\a_s^4 m_t)$ and numerically comparable ones to that order also from electroweak interactions which may become important as well.
Although individual levels of toponium will be unobservable, the top--anti--top system near threshold fulfills all requirements of a rigorous perturbation theory in QCD for weakly bound systems. Corresponding techniques from positronium may thus be transferred successfully to this case. After clarifying the effect of a non-zero width we calculate the $t\bar{t}$ potential to be used for the calculation of e.g. the cross-sections near threshold.
Off-shell and relativistic bound-state corrections for the decay $t \to b+W$ are calculated to $O(\a_s^2)$ making full use of the Bethe-Salpeter formalism for weakly bound systems. Thus we are able to take into account all terms to that order in a systematic and straightforward manner. One of the previously not considered contributions cancels precisely gauge dependent terms which appeared in an earlier off-shell calculation. Important cancellations also determine the gauge-independent part.
The expected large mass $m_t$ of the top quark provides for the first time a chance to discuss the bound-states of the corresponding quantum system as the nonabelian generalization of positronium, using the full power of relativistic quantum field theoretic methods which are available for weakly bound systems. Thus our approach differs in principle from the one used in the vast phenomenological literature on quarkonium potentials. We emphasize especially the corrections of energy levels which are of order $\a^4 m_t$ or numerically comparable to that order, and which have no counterpart in the 'relativistic' corrections of QED. In contrast to previous computations we give analytic expressions for all contributions considered in our present work, hopefully preparing the ground for further similar calculations.
The integrability of $R^2$-gravity with torsion in two dimensions is traced to an ultralocal dynamical symmetry of constraints and momenta in Hamiltonian phase space. It may be interpreted as a quadratically deformed $iso(2,1)$-algebra with the deformation consisting of the Casimir operators of the undeformed algebra. The locally conserved quantity encountered in the explicit solution is identified as an element of the centre of this algebra. Specific contractions of the algebra are related to specific limits of the explicit solutions of this model.