A C*-Algebra for Quantized Principal U(1)-Connections on Globally Hyperbolic Lorentzian Manifolds

The aim of this work is to complete our program on the quantization of connections on arbitrary principal U(1)-bundles over globally hyperbolic Lorentzian manifolds. In particular, we show that one can assign via a covariant functor to any such bundle an algebra of observables which separates gauge equivalence classes of connections. The C*-algebra we construct generalizes the usual CCR-algebras, since, contrary to the standard field-theoretic models, it is based on a presymplectic Abelian group instead of a symplectic vector space. We prove a no-go theorem according to which neither this functor, nor any of its quotients, satisfies the strict axioms of general local covariance. As a byproduct, we prove that a morphism violates the locality axiom if and only if a certain induced morphism of cohomology groups is non-injective. We show then that, fixing any principal U(1)-bundle, there exists a suitable category of subbundles for which a quotient of our functor yields a quantum field theory in the sense of Haag and Kastler. We shall provide a physical interpretation of this feature and we obtain some new insights concerning electric charges in locally covariant quantum field theory.     

Geodesic Curves on Quantized Manifolds

A general definition of curves and geodesics associated with a given connection on a quantized manifold is given. In the particular case of functional quantization, we define geodesics in the same way as in the classical case and we will show that the two definitions are compatible. As an example, we examine our results for the quantum Manin plane.     

Sheaves and D-Modules on Lorentzian Manifolds

We introduce a class of causal manifolds which contains the globally hyperbolic spacetimes and prove global propagation theorems for sheaves on such manifolds. As an application, we solve globally the Cauchy problem for hyperfunction solutions of hyperbolic systems.     

Injectivity Radius of Lorentzian Manifolds

Motivated by the application to general relativity we study the geometry and regularity of Lorentzian manifolds under natural curvature and volume bounds, and we establish several injectivity radius estimates at a point or on the past null cone of a point. Our estimates are entirely local and geometric, and are formulated via a reference Riemannian metric that we canonically associate with a given observer (p, T) –where p is a point of the manifold and T is a future-oriented time-like unit vector prescribed at p only. The proofs are based on a generalization of arguments from Riemannian geometry. We first establish estimates on the reference Riemannian metric, and then express them in terms of the Lorentzian metric. In the context of general relativity, our estimate on the injectivity radius of an observer should be useful to investigate the regularity of spacetimes satisfying Einstein field equations.     

Quantized Flag Manifolds and Irreducible *-Representations

In this paper we fill some gaps in the arguments of our previous papers [1,2]. In particular, we give a proof that the L operators of Conformal Field Theory indeed satisfy the defining relations of the Yang–Baxter algebra. Among other results we present a derivation of the functional relations satisfied by T and Q operators and a proof of the basic analyticity assumptions for these operators used in [1,2]. Received: 20 May 1998 / Accepted: 7 July 1998     

Scarring on Invariant Manifolds for Perturbed Quantized Hyperbolic Toral Automorphisms

We exhibit scarring for the quantization of certain nonlinear ergodic maps on the torus. We consider perturbations of hyperbolic toral automorphisms preserving certain co-isotropic submanifolds. The classical dynamics is ergodic, hence, in the semiclassical limit almost all quantum eigenstates converge to the volume measure of the torus. Nevertheless, we show that for each of the invariant submanifolds, there are also eigenstates which localize and converge to the volume measure of the corresponding submanifold.     

Curvature Properties of Lorentzian Manifolds with Large Isometry Groups

The curvature of Lorentzian manifolds (M ⁿ ,g), admitting a group of isometries of dimension at least 1/2n(n − 1) + 1, is completely described. Interesting behaviours are found, in particular as concerns local symmetry, local homogeneity and conformal flatness. Second and third authors supported by funds of MURST and the University of Salento.     

A Variational Theory for Light Rays in Stably Causal Lorentzian Manifolds: Regularity and Multiplicity Results

This paper is dedicated to the study of light rays joining an event p with a timelike curve γ in a light–convex subset &\Lambda; of a stably causal Lorentzian manifold . We set up a functional framework, defined intrinsically, consisting of a family of manifolds and a positive functional Q defined on them. The critical points of Q on approach, as , the lightlike, future pointing geodesics joining p and γ. We prove some regularity results, including the C ¹–regularity of , the C ²–regularity of Q on and the C ²–regularity of its critical points. Using them, we develop a Ljusternik–Schnirelman theory for light rays, obtaining some multiplicity results, depending on the topology of the space of all lightlike curves joining p and γ. Received: 9 April 1996 / Accepted: 27 December 1996     

Comment: Towards a Complete Classification of Spherically Symmetric Lorentzian Manifolds According to Their Ricci Collineations

General expressions for the components of the Ricci collineation vector are derived and the related constraints are obtained. These constraints are then solved to obtain Ricci collineations and the related constraints on the Ricci tensor components for all spacetime manifolds (degenerate or non-degenerate, diagonal or non-diagonal) admitting symmetries larger than so(3) and already known results are recovered. A complete solution is achieved for the spacetime manifolds admitting so(3) as the maximal symmetry group with non-degenerate and non diagonal Ricci tensor components. It is interesting to point out that there appear cases with finite number of Ricci collineations although the Ricci tensor is degenerate and also the cases with infinitely many Ricci collineations even in the case of non-degenerate Ricci tensor. Interestingly, it is found that the spacetime manifolds with so(3) as maximal symmetry group may admit two extra proper Ricci collineations, although they do not admit a G ₅ as the maximal symmetry group. Examples are provided which show and clarify some comments made by Camci et al. [Camci, U., and Branes, A. (2002). Class. Quantum Grav. 19, 393–404]. Theorems are proved which correct the earlier claims made in [Carot, J., Nunez, L. A., and Percoco, U. (1997). Gen. Relativ. Gravit. 29, 1223–1237; Contreras, G., Núñez, L. A., and Percolo, U. (2000). Gen. Relativ. Gravit. 32, 285–294].     

Killing spinors on Lorentzian manifolds

The aim of this paper is to describe some results concerning the geometry of Lorentzian manifolds admitting Killing spinors. We prove that there are imaginary Killing spinors on simply connected Lorentzian Einstein–Sasaki manifolds. In the Riemannian case, an odd-dimensional complete simply connected manifold (of dimension n≠7) is Einstein–Sasaki if and only if it admits a non-trivial Killing spinor to . The analogous result does not hold in the Lorentzian case. We give an example of a non-Einstein Lorentzian manifold admitting an imaginary Killing spinor. A Lorentzian manifold admitting a real Killing spinor is at least locally a codimension one warped product with a special warping function. The fiber of the warped product is either a Riemannian manifold with a real or imaginary Killing spinor or with a parallel spinor, or it again is a Lorentzian manifold with a real Killing spinor. Conversely, all warped products of that form admit real Killing spinors.     

Homogeneous structures on three-dimensional Lorentzian manifolds

We prove that any non-symmetric three-dimensional homogeneous Lorentzian manifold is isometric to a Lie group equipped with a left-invariant Lorentzian metric. We then classify all three-dimensional homogeneous Lorentzian manifolds.     

Twistor spinors on Lorentzian symmetric spaces

An indecomposable Riemannian symmetric space which admits non-trivial twistor spinors has constant sectional curvature. Furthermore, each homogeneous Riemannian manifold with parallel spinors is flat. In the present paper we solve the twistor equation on all indecomposable Lorentzian symmetric spaces explicitly. In particular, we show that there are — in contrast to the Riemannian case — indecomposable Lorentzian symmetric spaces with twistor spinors, which have non-constant sectional curvature and non-flat and non-Ricci flat homogeneous Lorentzian manifolds with parallel spinors.     

Twistor Spinors with Zero on Lorentzian 5-Space

We present in this paper a C ¹-metric of Lorentzian signature (1,4) on an open neighbourhood of the origin in \(\mathbb{R}^{5}\), which admits a solution to the twistor equation for spinors with a unique isolated zero at the origin. The metric is not conformally flat in any neighbourhood of the origin and the associated conformal Killing vector to the twistor generates a one-parameter group of essential conformal transformations. The construction is based on the Eguchi-Hanson metric in dimension 4.     

Witten Spinors on Nonspin Manifolds

Motivated by Witten’s spinor proof of the positive mass theorem, we analyze asymptotically constant harmonic spinors on complete asymptotically flat nonspin manifolds with nonnegative scalar curvature.     

Remarks on Symplectic Connections

This note contains a short survey on some recent work on symplectic connections: properties and models for symplectic connections whose curvature is determined by the Ricci tensor, and a procedure to build examples of Ricci-flat connections. For a more extensive survey, see Bieliavsky et al. [Int. J. Geom. Methods Mod. Phys. 3, 375–420 2006]. This note also includes a moment map for the action of the group of symplectomorphisms on the space of symplectic connections, an algebraic construction of a large class of Ricci-flat symmetric symplectic spaces, and an example of global reduction in a non-symmetric case.     

Abelian gauge theories on homogeneous spaces

An algebraic technique of separation of gauge modes in Abelian gauge theories on homogeneous spaces is proposed. An effective potential for the Maxwell-Chern-Simons theory on S ³ is calculated. A generalization of the Chern-Simons action is suggested and analyzed with the example of SU(3)/U(1) X U(1).     

Symmetries on Partially Ordered Abelian Groups

Recently, Foulis (Foulis, D. J. (2003). Compressible groups, Mathematica Slovaca 53, 433–455.) characterized compressions on effect rings, which were introduced as a generalization of unital C*-algebras in the context of ordered abelian groups with order units. In the present paper, we characterize a class of symmetries on effect rings and show their relations to compressions. This characterization leads to a generalization of the notion of orthosymmetric orthoposets (Mayet, R., Pulmannová, S. (1994). Nearly orthosymmetric ortholattices and Hilbert spaces, Foundations of Physics 24, 1425–1437.) to symmetric ordered abelian groups with order units.     

Entropic Dynamics on Gibbs Statistical Manifolds

Entropic dynamics is a framework in which the laws of dynamics are derived as an application of entropic methods of inference. Its successes include the derivation of quantum mechanics and quantum field theory from probabilistic principles. Here, we develop the entropic dynamics of a system, the state of which is described by a probability distribution. Thus, the dynamics unfolds on a statistical manifold that is automatically endowed by a metric structure provided by information geometry. The curvature of the manifold has a significant influence. We focus our dynamics on the statistical manifold of Gibbs distributions (also known as canonical distributions or the exponential family). The model includes an “entropic” notion of time that is tailored to the system under study; the system is its own clock. As one might expect that entropic time is intrinsically directional; there is a natural arrow of time that is led by entropic considerations. As illustrative examples, we discuss dynamics on a space of Gaussians and the discrete three-state system.     

Quantum Projector Method on Curved Manifolds

A generalized stochastic method for projecting out the ground state of the quantum many-body Schrödinger equation on curved manifolds is introduced. This random-walk method is of wide applicability to any second order differential equation (first order in time), in any spatial dimension. The technique reduces to determining the proper quantum corrections for the Euclidean short-time propagator that is used to build up their path-integral Monte Carlo solutions. For particles with Fermi statistics the Fixed-Phase constraint (which amounts to fixing the phase of the many-body state) allows one to obtain stable, albeit approximate, solutions with a variational property. We illustrate the method by applying it to the problem of an electron moving on the surface of a sphere in the presence of a Dirac magnetic monopole.