The geometry of a q-deformed phase space

The geometry of the q-deformed line is studied. A real differential calculus is introduced and the associated algebra of forms represented on a Hilbert space. It is found that there is a natural metric with an associated linear connection which is of zero curvature. The metric, which is formally defined in terms of differential forms, is in this simple case identifiable as an observable. Received: 26 November 1998 / Published online: 27 April 1999     

Conditions on a connection to be a metric connection

It is shown that a torsion free linear connection is determined by a metric of given signature if and only if its holonomy group is a subgroup of the orthogonal group corresponding to the signature.     

Lambda polarization measured at HERMES and lambda spin structure

Several examples and models based on noncommutative differential calculi on commutative algebras indicate that a metric should be regarded as an element of the left-linear tensor product of the space of 1-forms with itself. We show how the metric compatibility condition with a linear connection generalizes to this framework.     

Critical remarks on Finsler modifications of gravity and cosmology by Zhe Chang and Xin Li

I do not agree with the authors of papers arXiv:0806.2184 and arXiv:0901.1023v1 (published in [Zhe Chang, Xin Li, Phys. Lett. B 668 (2008) 453] and [Zhe Chang, Xin Li, Phys. Lett. B 676 (2009) 173], respectively). They consider that “In Finsler manifold, there exists a unique linear connection – the Chern connection … It is torsion freeness and metric compatibility …”. There are well-known results (for example, presented in monographs by H. Rund and R. Miron and M. Anastasiei) that in Finsler geometry there exist an infinite number of linear connections defined by the same metric structure and that the Chern and Berwald connections are not metric compatible. For instance, the Chern's one (being with zero torsion and “weak” compatibility on the base manifold of tangent bundle) is not generally compatible with the metric structure on total space. This results in a number of additional difficulties and sophistication in definition of Finsler spinors and Dirac operators and in additional problems with further generalizations for quantum gravity and noncommutative/string/brane/gauge theories. I conclude that standard physics theories can be generalized naturally by gravitational and matter field equations for the Cartan and/or any other Finsler metric compatible connections. This allows us to construct more realistic models of Finsler spacetimes, anisotropic field interactions and cosmology.     

Special Symplectic Connections and Poisson Geometry

By a special symplectic connection we mean a torsion free connection which is either the Levi-Civita connection of a Bochner-Kähler metric of arbitrary signature, a Bochner-bi-Lagrangian connection, a connection of Ricci type or a connection with special symplectic holonomy. A manifold or orbifold with such a connection is called special symplectic. We show that any special symplectic connection can be constructed using symplectic realizations of quadratic deformations of a certain linear Poisson structure. Moreover, we show that these Poisson structures cannot be symplectically integrated by a Hausdorff groupoid. As a consequence, we obtain a canonical principal line bundle over any special symplectic manifold or orbifold, and we deduce numerous global consequences.     

Probability measures on distribution spaces and quantum field theoretical models

It is well known that there is a close connection between (Abelian) fields and probability measures on distribution space, or more generally, on infinite-dimensional vector spaces, and their associated random processes. We establish general criteria for mutual singularity of such measures and apply them to quantum fields. In particular, it is shown that a kind of generalized clustering implies singularity. Then a condition for singularity is given in terms of a natural metric introduced previously. It is used to show that the translate of a measure by a linear functional which is not continuous for this metric is singular to the untranslated measure. The results are applied to processes with independent values at each point, corresponding to the ultra-local model. It is shown that each such measuris singular to any of its translates although its finite-dimensional projections may be equivalent to Lebesgue measure.     

The classification of 4-dimensional non-degenerate affine hypersurfaces with parallel cubic form

In this paper, we complete the classification of 4-dimensional non-degenerate affine hypersurfaces with parallel cubic form with respect to the Levi-Civita connection of the affine Berwald–Blaschke metric.     

On General Solutions for Field Equations in Einstein and?Higher Dimension Gravity

We prove that the Einstein equations can be solved in a very general form for arbitrary spacetime dimensions and various types of vacuum and non-vacuum cases following a geometric method of anholonomic frame deformations for constructing exact solutions in gravity. The main idea of this method is to introduce on (pseudo) Riemannian manifolds an alternative (to the Levi-Civita connection) metric compatible linear connection which is also completely defined by the same metric structure. Such a canonically distinguished connection is with nontrivial torsion which is induced by some nonholonomy frame coefficients and generic off-diagonal terms of metrics. It is possible to define certain classes of adapted frames of reference when the Einstein equations for such an alternative connection transform into a system of partial differential equations which can be integrated in very general forms. Imposing nonholonomic constraints on generalized metrics and connections and adapted frames (selecting Levi-Civita configurations), we generate exact solutions in Einstein gravity and extra dimension generalizations.     

Einstein Gravity in Almost Kähler Variables and Stability of Gravity with Nonholonomic Distributions and Nonsymmetric Metrics

We argue that the Einstein gravity theory can be reformulated in almost Kähler (nonsymmetric) variables with effective symplectic form and compatible linear connection uniquely defined by a (pseudo) Riemannian metric. A class of nonsymmetric theories of gravitation on manifolds enabled with nonholonomic distributions is considered. We prove that, for certain types of nonholonomic constraints, there are modelled effective Lagrangians which do not develop instabilities. It is also elaborated a linearization formalism for anholonomic noncommutative gravity theories models and analyzed the stability of stationary ellipsoidal solutions defining some nonholonomic and/or nonsymmetric deformations of the Schwarzschild metric. We show how to construct nonholonomic distributions which remove instabilities in nonsymmetric gravity theories. It is concluded that instabilities do not consist a general feature of theories of gravity with nonsymmetric metrics but a particular property of some models and/or unconstrained solutions.     

On parallel displacement within the light cone and its application in the electrodynamics of charges moving with the velocity of light

It is shown that there exists on the light cone an affine connection which is metric, semisymmetric and locally integrable. There exists a correspondence between this connection and a system of charges moving with the velocity of light. The correspondence reveals in the case of commensurable charges a symmetry which disappears for non-commensurable charges.     

High-energy behavior and second class constraints in quantum gravity

It is proposed that sensible high-energy behavior in a quantum theory of gravity may be achieved in a class of theories in which the connection and metric are independent and unconstrained and where the action is chosen so that no derivatives of the metric appear. This is because in these theories all ten of the metric field equations are realized as second class constraints. These can in principle be solved, expressing the operators g_μν as functions of the operators for the components of the connection and their canonical momenta. Thus, the metric has no independent quantum fluctuations, and the instabilities resulting from the usual curvature squared terms are eliminated. Furthermore, there is no need to assume metric compatibility, as it is automatically restored in the low-energy limit by the dominance of dimension-two terms.In order to explore these ideas a toy model with two degrees of freedom, corresponding to a metric and a connection variable, is quantized and shown to have a sensible high energy limit, while a related model, in which a constraint analogous to metric compatibility is imposed, is found to be unstable at high energies.     

Einstein gravity in almost Kähler and Lagrange–Finsler variables and deformation quantization

A geometric procedure is elaborated for transforming (pseudo) Riemannian metrics and connections into canonical geometric objects (metric and nonlinear and linear connections) for effective Lagrange, or Finsler, geometries which, in turn, can be equivalently represented as almost Kähler spaces. This allows us to formulate an approach to quantum gravity following standard methods of deformation quantization. Such constructions are performed not on tangent bundles, as in usual Finsler geometry, but on spacetimes enabled with nonholonomic distributions defining 2+2$2+2$ splitting with associate nonlinear connection structure. We also show how the Einstein equations can be written in terms of Lagrange–Finsler variables and corresponding almost symplectic structures and encoded into the zero-degree cohomology coefficient for a quantum model of Einstein manifolds.     

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.     

Canonical connections of Finsler metrics and Finslerian connections on Riemannian metrics

A concept of canonical connection of a Finsler metric is developed. Connections that are compatible with Finsler metrics are compared with the canonical connection itself. They are also compared with the corresponding Cartan connection. A necessary and sufficient condition on metric Finsler connections is given for the metric to be Riemannian. This study unearths different ways in which Finsler geometry could be used to generalize the theory of general relativity.     

Energy-momentum complex for nonlinear gravitational Lagrangians in the first-order formalism

It has been recently shown that there is universality of Einstein equations, in the first-order (Palatini) formalism, in the sense that for a generic nonlinear Lagrangian depending only on the scalar curvature of a metric and a torsionless connection one always gets Einstein equations. In this paper the energy-density flow for nonlinear gravitational Lagrangians is investigated in this formalism. It is shown that in the generic case the energy-momentum complex does not depend on the Lagrangian and is in fact equal to the Komar complex, known in the purely metric formalism for the standard linear Hilbert Lagrangian.On leave from the Institute of Theoretical Physics, University of Wrocaw, pl. Maxa Borna 9, 50-204 Wrocaw, Poland     

Conformal transformations in metric‐affine gravity and ghosts

Conformal transformations play a widespread role in gravity theories in regard to their cosmological and other implications. In the pure metric theory of gravity, conformal transformations change the frame to a new one wherein one obtains a conformal‐invariant scalar–tensor theory such that the scalar field, deriving from the conformal factor, is a ghost. In this work, conformal transformations and ghosts will be analyzed in the framework of the metric‐affine theory of gravity. Within this framework, metric and connection are independent variables, and, hence, transform independently under conformal transformations. It will be shown that, if affine connection is invariant under conformal transformations, then the scalar field of concern is a non‐ghost, non‐dynamical field. It is an auxiliary field at the classical level, and might develop a kinetic term at the quantum level. Alternatively, if connection transforms additively with a structure similar to yet more general than that of the Levi‐Civita connection, the resulting action describes the gravitational dynamics correctly, and, more importantly, the scalar field becomes a dynamical non‐ghost field. The equations of motion, for generic geometrical and matter‐sector variables, do not reduce connection to the Levi‐Civita connection, and, hence, independence of connection from metric is maintained. Therefore, metric‐affine gravity provides an arena in which ghosts arising from the conformal factor are avoided thanks to the independence of connection from the metric.     

Projective Invariance and Einstein's Equations

It is shown that a projectively invariant Lagrangian field theory based on linear non-symmetric connections in space-time and arbitrary source fields is equivalent to Einstein's standard theory of gravitation coupled to a source Lagrangian depending solely on the original source fields. A key point is that, as in the case of Lagrangian field theories based on symmetric connections in space-time, the Euler-Lagrange field equations uniquely determine the projective invariant part of the linear connection in terms of the metric, their first-order derivatives, the source fields, and their conjugate momenta.     

Projective versus metric structures

We present a number of conditions which are necessary for an n-dimensional projective structure (M,[∇]) to include the Levi-Civita connection ∇ of some metric on M. We provide an algorithm, which effectively checks whether a Levi-Civita connection is in the projective class and, which finds this connection and the metric, when it is possible. The article also provides basic information on invariants of projective structures, including the treatment via the Cartan normal projective connection. In particular we show that there are a number of Fefferman-like conformal structures, defined on a subbundle of the Cartan bundle of the projective structure, which encode the projectively invariant information about (M,[∇]).     

On the Rainich geometrization of a vector meson field in the Kibble theory

The variables of a vector meson field are determined within the framework of the Kibble theory as the functions of the metric tensor, affine connection and their derivatives and a system of differential equations is found for the metric tensor and affine connection which is equivalent to the equations of motion of gravitational and vector meson fields.