Generally covariant quantum field theory and scaling limits

The formulation of a generally covariant quantum field theory is described. It demands the elimination of global features and a characterization of the theory in terms of the allowed germs of families of states. A simple application is the computation of counting rates of accelerated idealized detectors. As a first orientation we discuss here the consequences of the assumption that the states have a short distance scaling limit. The scaling limit at a point gives a reduction of the theory to tangent space. It contains kinematical information but not the full dynamical laws. The reduced theory will, under rather general conditions, be invariant under translations and under a proper subgroup of the linear transformations in tangent space. One interesting possibility is that it is invariant under SLR(4). Then the macroscopic metric must evolve as a cooperative effect in finite size regions. The other natural possibility is that each family (coherent folium) of states defines a microscopic metric by the scaling limit and the tangent space theory reduces to a theory of free massless fields in a Minkowski space. Irrespective of the assumption of a scaling limit we show that the theory can be constructed from strictly local information.     

Dimensionality Reduction of SPD Data Based on Riemannian Manifold Tangent Spaces and Isometry

Symmetric positive definite (SPD) data have become a hot topic in machine learning. Instead of a linear Euclidean space, SPD data generally lie on a nonlinear Riemannian manifold. To get over the problems caused by the high data dimensionality, dimensionality reduction (DR) is a key subject for SPD data, where bilinear transformation plays a vital role. Because linear operations are not supported in nonlinear spaces such as Riemannian manifolds, directly performing Euclidean DR methods on SPD matrices is inadequate and difficult in complex models and optimization. An SPD data DR method based on Riemannian manifold tangent spaces and global isometry (RMTSISOM-SPDDR) is proposed in this research. The main contributions are listed: (1) Any Riemannian manifold tangent space is a Hilbert space isomorphic to a Euclidean space. Particularly for SPD manifolds, tangent spaces consist of symmetric matrices, which can greatly preserve the form and attributes of original SPD data. For this reason, RMTSISOM-SPDDR transfers the bilinear transformation from manifolds to tangent spaces. (2) By log transformation, original SPD data are mapped to the tangent space at the identity matrix under the affine invariant Riemannian metric (AIRM). In this way, the geodesic distance between original data and the identity matrix is equal to the Euclidean distance between corresponding tangent vector and the origin. (3) The bilinear transformation is further determined by the isometric criterion guaranteeing the geodesic distance on high-dimensional SPD manifold as close as possible to the Euclidean distance in the tangent space of low-dimensional SPD manifold. Then, we use it for the DR of original SPD data. Experiments on five commonly used datasets show that RMTSISOM-SPDDR is superior to five advanced SPD data DR algorithms.     

The geodesic approximation for the Yang-Mills-Higgs equations

In this paper we consider the dynamics of the monopole solutions of Yang-Mills-Higgs theory on Minkowski space. The monopoles are solutions of the Yang-Mills-Higgs equations on three dimensional Euclidean space. It is of interest to understand how they evolve in time when considered as solutions of the Yang-Mills-Higgs equations on Minkowski space-i.e. the time dependent equations. It was suggested by Manton that in certain situations the monopole dynamics could be understood in terms of geodesics with respect to a certain, metric on the space of guage equivalence classes of monopoles-the moduli space. The metric is defined by taking theL ² inner product of tangent vectors to this space. In this paper we will prove that Manton's approximation is indeed valid in the right circumstances, which correspond to the slow motion of monopoles. The metric on the moduli space of monopoles was analysed in a book by Atiyah and Hitchin, so together with the results of this paper a detailed and rigorous understanding of the low energy dynamics of monopoles in Yang-Mills-Higgs theory is obtained. The strategy of the proof is to develop asymptotic expansions using appropriate gauge conditions, and then to use energy estimates to prove their validity. For the case of monopoles to be considered here there is a technical obstacle to be overcome-when the equations are linearised about the monopole the continuous spectrum extends all the way to the origin. This is overcome by using a norm introduced by Taubes in a discussion of index, theory for the Yang-Mills-Higgs functional.Supported by grant DMS-9214067 from the National Science Foundation.     

Weak gravitational field in Finsler–Randers space and Raychaudhuri equation

The linearized form of the metric of a Finsler–Randers space is studied in relation to the equations of motion, the deviation of geodesics and the generalized Raychaudhuri equation are given for a weak gravitational field. This equation is also derived in the framework of a tangent bundle. By using Cartan or Berwald-like connections we get some types “gravito-electromagnetic” curvature. In addition we investigate the conditions under which a definite Lagrangian in a Randers space leads to Einstein field equations under the presence of electromagnetic field. Finally, some applications of the weak field in a generalized Finsler spacetime for gravitational waves are given.     

A Spherical Symmetrical Spacetime Solution in Finsler Gravity

Several physical principles of Finsler gravity are proposed in this paper, and I apply the principles to construct a Finsler gravity action, which satisfy the condition that the action can be reduced to the General Relativity action once the metric is independent from the tangent vector. I also get a spacetime solution in Finsler spacetime with the tangent vector y _φ , moreover the solution indicates that the metric relies on the property of test particle in Finsler spacetime.     

Non-Completeness of Zograf–Takhtajan's Kähler Metric for Teichmüller Space of Punctured Riemann Surfaces

Zograf and Takhtajan introduced a new K?hler metric on the Teichmüller space T _g , _n (n>0), in calculating the first Chern form of the Quillen metric for families of -operators. The metric is described in terms of the Eisenstein–Maass series. We prove that it is incomplete. And we also give an alternative proof of non-completeness of the Weil–Petersson metric. For that, we use the pinching family, constructed by Wolpert, whose tangent vectors are always represented by using the relative Poincaré series associated with the pinched geodesic. Received: 18 September 1995 / Accepted: 7 March 1999     

Dimensional reduction of fermions in generalized gravity

We discuss possibilities of obtaining chiral four-dimensional fermions from dimensional reduction of pure higher dimensional gravity. We explore a modification of riemannian geometry where the Lorentz rotations are treated in close analogy to usual gauge theories. The metric is not the product of two vielbeins and the vielbein may not be invertible everywhere. The bundle structure of Lorentz transformations is distinguished from the bundle structure of tangent space rotations and the gravitational index theorems have to be modified for this case. We also investigate noncompact internal spaces with finite volume in the context of riemannian geometry. Chiral fermions are obtained in the latter case.As a byproduct of this work, we find that for the usual torsion theories the Dirac operator is not the relevant mass operator for dimensional reduction of fermions.     

String Without Strings

Scale invariance provides a principled reason for the physical importance of Hilbert space, the Virasoro algebra, the string mode expansion, canonical commutators and Schrödinger evolution of states, independent of the assumptions of string theory and quantum theory. The usual properties of dimensionful fields imply an infinite, projective tower of conformal weights associated with the tangent space to scale-invariant spacetimes. Convergence and measurability on this tangent tower are guaranteed using a scale-invariant norm, restricted to conformally self-dual vectors. Maps on the resulting Hilbert space are correspondingly restricted to semi-definite conformal weight. We find the maximally- and minimally-commuting, complete Lie algebras of definite-weight operators. The projective symmetry of the tower gives these algebras central charges, giving the canonical commutator and quantum Virasoro algebras, respectively. Using a continuous, m-parameter representation for rank-m tower tensors, we show that the parallel transport equation for the momentum-vector of a particle is the Schrödinger equation, while the associated definite-weight operators obey canonical commutation relations. Generalizing to the set of integral curves of general timelike, self-dual vector-valued weight maps gives a lifting such that the action of the curves parallel transports arbitrary tower vectors. We prove that the full set of Schrödinger-lifted integral curves of a general self-dual map gives an immersion of its 2-dim parameter space into spacetime, inducing a Lorentzian metric on the parameter space. This immersion is shown to satisfy the full variational equations of open string.     

Electroweak Symmetry on the Tangent Bundle

Various symmetries of elementary particles can be represented by gauge transformations acting on a fiber of the tangent bundle. These are diffeomorphisms of linear groups which act on vertical vector fields. It is shown how the electroweak vector boson potentials and a corresponding Kaluza-Klein-like metric can be obtained by application of SU(2) × U(1) to a tangent fiber. This geometry gives a more unified approach to gravitation and gauge symmetries.     

Spacelike metric foliations

In this paper, we investigate spacelike metric foliations in lightlike complete spacetimes. When such a foliation satisfies the strong energy condition Ric^V (e) ≥ 0 for timelike vectors e, it must be totally geodesic, and the metric is of higher rank, in the sense that each point of the spacetime is contained inside a flat, totally geodesic, timelike rectangle. If in addition Ric^V(e) = 0, then the metric is (at least locally) a product metric, with the leaves of the foliation tangent to one of the factors.     

Cohomogeneity One Einstein-Sasaki 5-Manifolds

We consider hypersurfaces in Einstein-Sasaki 5-manifolds which are tangent to the characteristic vector field. We introduce evolution equations that can be used to reconstruct the 5-dimensional metric from such a hypersurface, analogous to the (nearly) hypo and half-flat evolution equations in higher dimensions. We use these equations to classify Einstein-Sasaki 5-manifolds of cohomogeneity one.     

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.     

Electrodynamics of the Maxwell-Lorentz type in the ten-dimensional space of the testing of special relativity: A case for Finsler type connections

It has recently been shown by Vargas, ⁽⁴⁾ that the passive coordinate transformations that enter the Robertson test theory of special relativity have to be considered as coordinate transformations in a seven-dimensional space with degenerate metric. It has also been shown by Vargas that the corresponding active coordinate transformations are not equal in general to the passive ones and that the composite active-passive transformations act on a space whose number of dimensions is ten (one-particle case) or larger (more than one particle).In this paper, two different (families of) electrodynamics are constructed in ten-dimensional space upon the coordinate free form of the Maxwell and Lorentz equations. The two possibilities arise from the two different assumptions that one can naturally make with respect to the acceleration fields of charges, when these fields are related to their relativistic counterparts. Both theories present unattractive features, which indicates that the Maxwell-Lorentz framework is unsuitable for the construction of an electrodynamics for the Robertson test theory of the Lorentz transformations. It is argued that this construction would first require the formulation of Maxwell-Lorentz electrodynamics in the form of a connection in Finsler space. If such formulation is possible, the sought generalization would consist in simply changing bases in the tangent spaces of the manifold that supports the connection. In addition, the number of dimensions of the space of the Robertson transformations would be ten, but not greater than ten.     

Riemannian Geometry of Quantum Groups¶and Finite Groups with Nonuniversal Differentials

We construct noncommutative “Riemannian manifold” structures on dual quasitriangular Hopf algebras such as ℂ _q [SU ₂] with its standard bicovariant differential calculus, using the quantum frame bundle approach introduced previously. The metric is provided by the braided-Killing form on the braided-Lie algebra on the tangent space and the n-bein by the Maurer–Cartan form. We also apply the theory to finite sets and in particular to finite group function algebras ℂ[G] with differential calculi and Killing forms determined by a conjugacy class. The case of the permutation group ℂ[S ₃] is worked out in full detail and a unique torsion free and cotorsion free or “Levi–Civita” connection is obtained with noncommutative Ricci curvature essentially proportional to the metric (an Einstein space). We also construct Dirac operators in the metric background, including on finite groups such as S ₃. In the process we clarify the construction of connections from gauge fields with nonuniversal calculi on quantum principal bundles of tensor product form. Received: 22 June 2000 / Accepted: 26 August 2001     

Vacuum Rank 1 Space-Time Perturbations

We study Kerr-Schild type perturbations with a non-null perturbation vector in the vacuum case. The perturbation equations are derived and it is shown that they lead to constraints on the background space-time which can be interpreted in terms of the curvature of 3-spaces. The first order perturbation equations are used to construct new Petrov type D solutions tangent to the Schwarzschild metric.     

Calculation of Metric for Gaussian Weave State

Using the recoupling theorem and graph calculation in loop quantum gravity, it is demonstrated that the action of metric matrix operator on Gaussian weave state is an eigenaction, the representation matrix elements of the metric operator and their expectation values are calculated. The values of length of tangent vectors of edges adjacent to the vertex of Gaussian weave state, as well as the angles between them are also obtained in the cases of k=0 and k=2.     

Renormalisation group flow and geodesics in the O(N) model for large N

A metric is introduced on the space of parameters (couplings) describing the large N limit of the O(N) model in Euclidean space. The geometry associated with this metric is analysed in the particular case of the infinite volume limit in three dimensions and it is shown that the Ricci curvature diverges at the ultra-violet (Gaussian) fixed point but is finite and tends to constant negative curvature at the infra-red (Wilson-Fisher) fixed point. The renormalisation group flow is examined in terms of geodesics of the metric. The critical line of cross-over from the Wilson-Fisher fixed point to the Gaussian fixed point is shown to be a geodesic but all other renormalisation group trajectories, which are repulsed from the Gaussian fixed point in the ultraviolet, are not geodesics. The geodesic flow is interpreted in terms of a maximisation principle for the relative entropy.     

Projective geometry and special relativity

Some concepts of real and complex projective geometry are applied to the fundamental physical notions that relate to Minkowski space and the Lorentz group. In particular, it is shown that the transition from an infinite speed of propagation for light waves to a finite one entails the replacement of a hyperplane at infinity with a light cone and the replacement of an affine hyperplane – or rest space – with a proper time hyperboloid. The transition from the metric theory of electromagnetism to the pre‐metric theory is discussed in the context of complex projective geometry, and ultimately, it is proposed that the geometrical issues are more general than electromagnetism, namely, they pertain to the transition from point mechanics to wave mechanics.     

Spacelike energy of timelike unit vector fields on a Lorentzian manifold

On a Lorentzian manifold, we define a new functional on the space of unit timelike vector fields given by the L₂ norm of the restriction of the covariant derivative of the vector field to its orthogonal complement. This spacelike energy is related with the energy of the vector field as a map on the tangent bundle endowed with the Kaluza–Klein metric, but it is more adapted to the situation. We compute the first and second variation of the functional and we exhibit several examples of critical points on cosmological models as generalized Robertson–Walker spaces and Gödel universe, on Einstein and contact manifolds and on Lorentzian Berger’s spheres. For these critical points we have also studied to what extent they are stable or even absolute minimizers.