Spatial Manifold Research Articles

Quasilocal energy for spin-net gravity

The Hamiltonian of the gravitational field defined in a bounded region is quantized. The classical Hamiltonian, and starting point for the regularization, is a boundary term required by functional differentiability of the Hamiltonian constraint. It is the quasilocal energy of the system and becomes the ADM mass in asymptopia. The quantization is carried out within the framework of canonical quantization using spin networks. The result is a gauge-invariant, well-defined operator on the Hilbert space induced by the state space on the whole spatial manifold. The spectrum is computed. An alternative form of the operator, with the correct naive classical limit, but requiring a restriction on the Hilbert space, is also defined. Comparison with earlier work and several consequences are briefly explored.

Topological Field Theory and Quantum Holonomy Representations of Motion Groups

Canonical quantization of abelian BF-type topological field theory coupled to extended sources on generic d-dimensional manifolds and with curved line bundles is studied. Sheaf cohomology is used to construct the appropriate topological extension of the action and the topological flux quantization conditions, in terms of the Čech cohomology of the underlying spatial manifold, as required for topological invariance of the quantum field theory. The wavefunctions are found in the Hamiltonian formalism and are shown to carry multi-dimensional projective representations of various topological groups of the space. Expressions for generalized linking numbers in any dimension are thereby derived. In particular, new global aspects of motion group presentations are obtained in any dimension. Applications to quantum exchange statistics of objects in various dimensionalities are also discussed.

String holonomy and extrinsic geometry in four-dimensional topological gauge theory

The most general gauge-invariant marginal deformation of four-dimensional abelian BF-type topological field theory is studied. It is shown that the deformed quantum field theory is topological and that its observables compute, in addition to the usual linking numbers, smooth intersection indices of immersed surfaces which are related to the Euler and Chern characteristic classes of their normal bundles in the underlying space-time manifold. Canonical quantization of the theory coupled to non-dynamical particle and string sources is carried out in the Hamiltonian formalism and explicit solutions of the Schr̈odinger equation are obtained. The wavefunctions carry a one-dimensional unitary representation of the particle-string exchange holonomies and of non-topological string-string intersection holonomies given by adiabatic limits of the world-sheet Euler numbers. They also carry a multi-dimensional projective representation of the deRham complex of the underlying spatial manifold and define a generalization of the presentation of its motion group from Euclidean space to an arbitrary 3-manifold. Some potential physical applications of the topological field theory as a dual model for effective vortex strings are discussed.

Global anomalies in canonical gravity

In this note we study the structure of diffeomorphism anomalies in 3 + 1 canonical gravity coupled to a chiral massless fermion. We find that when the spatial manifold Σ is S 3 or a Lens space L( p, q), the first homotopy group of the related diffeomorphism group can be non-trivial and hence the question of global anomalies becomes relevant. Here we show that for gravity coupled to SU(2) chiral fermions, assuming the strong form of the Hatcher conjecture, SU(2)-induced diffeomorphism anomalies do not occur whenever Σ is S 3 or a Lens space.

Quantum gravity with linear action and intrinsic rigidity of spacetime

An earlier proposed theory with linear-gonihedric action A( M 4) for quantum gravity, which requires the existence of a fundamental constant of dimension one is reviewed. One can consider this theory as a “square root” of classical gravity. We also demonstrate that the partition function for the discretized version of the Einstein-Hilbert action found by Regge in 1961 can be represented as a superposition of random surfaces with Euler character as an action, and in the case of linear gravity, as a superposition of three-dimensional manifolds with an action which is proportional to the total solid angle deficit of these manifolds. This representation allows us to construct the transfer matrix which describes the propagation of spatial manifold. We discuss the so called gonihedric principle which allows to define a discrete version of higher derivative terms in quantum gravity and to introduce generalized deficit angles which suppress non-flat vertices, thus introducing an intrinsic rigidity of spacetime. This note is based on a talk delivered at the II meeting on constrained dynamics and quantum gravity at Santa Margherita Ligure.

Reduced Hamiltonians in general

In this first of a pair of papers we resurrect a standard construction of analytical mechanics dating from the last century. The technique allows one to pass from any dynamical system whose first order evolution equations are known, and whose bracket algebra is not degenerate, to a system of canonical variables and a non-zero Hamiltonian that generates their evolution. The construction agrees with the usual results for gauge theories and can be applied as well to gravity, even when the spatial manifold is closed. Although our results are classical they can be formally quantized to give the naive functional formalism. This is not only an effective starting point for calculations, it also seems to provide a formulation of the quantum theory which is non-perturbative, at least in principle.

Vortices in the Bogomol'nyi limit of Einstein-Maxwell-Higgs theory with or without external sources.

The Abelian Higgs model with or without external particles is considered in curved space. Using the dual transformation, we rewrite the model in terms of dual gauge fields and derive the Bogomol'nyi-type bound. We find all possible cylindrically symmetric vortex solutions and vortex-particle composites by examining the Einstein equations and the first-order Bogomol'nyi equations. The underlying spatial manifold of these objects comprises a cylinder asymptotically and a two-sphere in addition to the well-known cone.

CURRENT ALGEBRA AND CONFORMAL FIELD THEORY ON A FIGURE EIGHT

We examine the dynamics of a free massless scalar field on a figure eight network. Upon requiring the scalar field to have a well-defined value at the junction of the network, it is seen that the conserved currents of the theory satisfy Kirchhoff’s law, that is that the current flowing into the junction equals the current flowing out. We obtain the corresponding current algebra and show that, unlike on a circle, the left- and right-moving currents on the figure eight do not in general commute in quantum theory. Since a free scalar field theory on a one-dimensional spatial manifold exhibits conformal symmetry, it is natural to ask whether an analogous symmetry can be defined for the figure eight. We find that, unlike in the case of a manifold, the action plus boundary conditions for the network are not invariant under separate conformal transformations associated with left- and right-movers. Instead, the system is, at best, invariant under only a single set of transformations. Its conserved current is also found to satisfy Kirchhoff’s law at the junction. We obtain the associated conserved charges, and show that they generate a Virasoro algebra. Its conformal anomaly (central charge) is computed for special values of the parameters characterizing the network.

Symplectic structure of 2D dilaton gravity

We analyze the symplectic structure of 2D dilaton gravity by evaluating the symplectic form on the space of classical solutions. The case when the spatial manifold is compact is studied in depth. For the free theory we find that the reduced space is a cotangent bundle and we determine the Hilbert space of the quantum theory. In the non-compact case the symplectic form is not well defined due to an unresolved ambiguity in the choice of the boundary terms.

THE EDGE STATES OF THE BF SYSTEM AND THE LONDON EQUATIONS

It is known that the 3D Chern–Simons interaction describes the scaling limit of a quantum Hall system and predicts edge currents in a sample with boundary, the currents generating a chiral U(1) Kac-Moody algebra. It is no doubt also recognized that, in a somewhat similar way, the 4D BF interaction (with B a two-form, dB the dual *j of the electromagnetic current, and F the electromagnetic field form) describes the scaling limit of a superconductor. We show in this paper that there are edge excitations in this model as well for manifolds with boundaries. They are the modes of a scalar field with invariance under the group of diffeomorphisms (diffeos) of the bounding spatial two-manifold. Not all diffeos of this group seem implementable by operators in quantum theory, the implementable group being a subgroup of volume-preserving diffeos. The BF system in this manner can lead to the w1+∞ algebra and its variants. Lagrangians for fields on the bounding manifold which account for the edge observables on quantization are also presented. They are the analogs of the (1+1)-dimensional massless scalar field Lagrangian describing the edge modes of an Abelian Chern-Simons theory with a disk as the spatial manifold. We argue that the addition of “Maxwell” terms constructed from F∧*F and dB∧*dB does not affect the edge states, and that the augmented Lagrangian has an infinite number of conserved charges—the aforementioned scalar field modes—localized at the edges. This Lagrangian is known to describe London equations and a massive vector field. A (3+1)-dimensional generalization of the Hall effect involving vortices coupled to B is also proposed.

Ashtekar formulation of (2 + 1)-dimensional gravity on a torus

Pure (2 + 1)-dimensional Eisntein gravity is analysed in the Ashtekar formulation, when the spatial manifold is a torus. We have found a set of globally defined observables, forming a closed algebra. This allowed us to solve the quantum constraints, and to show that the reduced phase space of the Ashtekar formulation is larger than the corresponding space of the Witten formulation. Furthermore, we have found a globally defined time variable which satisfies all the requirements of an extrinsic time variable in quantum gravity.

Exactly solvable models of 2D dilaton quantum gravity

We study canonical quantization of a class of 2D dilaton gravity models, which contains the model proposed by Callan, Giddings, Harvey and Strominger. A set of non-canonical phase space variables is found, forming an SL(2, R )×U(1) current algebra, such that the constraints become quadratic in these new variables. In the case when the spatial manifold is compact, the corresponding quantum theory can be solved exactly, since it reduces to a problem of finding the cohomology of a free-field Virasoro algebra. In the non-compact case, which is relevant for 2D black holes, this construction is likely to break down, since the most general field configuration cannot be expanded into Fourier modes. Strategy for circumventing this problem is discussed.

Hamiltonian lattice gravity. Deformations of discrete manifolds.

The structure constants appearing in the Poisson-brackets relations among constraints in the Hamiltonian theory of gravity depend on the properties of deformations of a d-dimensional spatial manifold embedded in a (d+1)-dimensional continuum. The study of such deformations is extended to the case where the d-dimensional manifold is a piecewise flat Regge lattice. Although we cannot say whether the algebra of deformations does or does not close for the general lattice, should it close we provide a prescription for obtaining the structure constants. We discuss the case of small lattices where closure can be demonstrated explicitly.

Places as Possibilities of Location

A common sense picture of space portrays it as a collection of entities (places) which stand in unchanging spatial relations to one another and which may or may not have objects located at them. To impose manageability on the project to be pursued here, I will assume additionally that on the common sense or prescientific view, which is as much as I will be concerned with, each place is at a definite distance from every other, distinct places are distinguished by their distance relations to other places, and places do not stand in causal relations either to themselves or to material objects.1 This common sense picture is congenial to two kinds of realist about places, the absolutist (or substantivalist) and the reductionist. Both kinds of realist are happy to say that are places, and mean nothing unusual by there are, but the reductive realist offers an identification of places with other, purportedly less problematic, entities; or, to look from the bottom up, he or she offers a logical construction of places from the less problematic entities. Reductive realism is midway between absolutism and eliminativism, the latter position offering a translation of talk about places into a language free of any ontological commitment to them. This paper investigates the prospects for a traditional version of reductive realism, known as relationism.2 Relationists hold that places are possibilities of location. The fundamental idea is Leibniz's; he held that we arrive at our conception of the spatial manifold by some kind of abstraction from the conception of objects standing in a system of spatial relation-

General relativity as the surface action of a five-dimensional gauge theory

We show that the action of euclidean general relativity is the surface term of a five-dimensional O(5) gauge theory, thus realizing Hawking's conjecture that the functional integral be restricted to compact manifolds without boundary. Our treatment of general relativity uncovers and exploits analogies with monopoles and instantons in the O(5) gauge theory. We review the metric-independent treatment of general relativity, its implications for the renormalizability of pure gravity and reexamine the positivity problem for the euclidean action. The new phenomenon of spontaneous dimensional reduction, by which a spatial manifold loses a dimension when a scalar field acquires a vacuum value, is described - an approach to higher dimensions distinct from the usual Kaluza-Klein approach. We propose a five-dimensional gauge field model with positive action - for which the surface action is that of general relativity - and explore some of its consequences.

Longitudinal non-euclidean networks: Applying Galileo

This article discusses the theoretical utility of using a non-Euclidean spatial manifold when describing social networks. It proposes that a variant of metric MDS, the Galileo System, can be particularly useful in analyzing social networks and their changes over time, partially because it does not impose Euclidean assumptions on the data. Two sets of longitudinal network data are examined with Galileo. One is the American air traffic network from 1968–1981. The other is ten groups engaged in a computer conference over a 24 month period. In both cases, the results indicate that a Riemannian spatial manifold is required to describe the network structure. Consistent theoretically valid results based upon the non-Euclidean components of spatial manifold are obtained. Further, they could be readily explained by exogenous factors. The implications of these results for network analysis are discussed.

Representations of spacetime diffeomorphisms. II. Canonical geometrodynamics

Our method of constructing representations of the spacetime diffeomorphism group DiffMin parametrized field theories is generalized to canonical geometrodynamics. The gravitational configuration space Riem Σ is extended by the space of embeddings of the spatial manifold Σ in the spacetimeM. Spacetime metrics are limited by Gaussian conditions with respect to an auxiliary foliation structure. As a result of these conditions, the super-Hamiltonian and supermomentum constraints are temporarily suspended. There are, however, new constraints in the theory associated with the canonical pair of the embedding variables and their conjugate momenta. By smearing the new constraint functions by vector fields V ∈ LDiffMrestricted to the embeddings, we construct a homomorphism from the Lie algebra of the spacetime diffeomorphism group into the Poisson bracket algebra of the dynamical variables on the extended geometrodynamical phase space. The dynamical evolution generated by such dynamical variables automatically preserves both the new and the old constraints and builds a Ricci-flat spacetime. The implications of the scheme for the canonical quantization of gravity are discussed.

Is Anisotropic Kaluza-Klein Model of Universe Chaotic ?

We investigate the dynamical evolution of a 7-dimensional Kaluza-Klein space-time which oscillates anisotropically. Our model has the spatial manifold of a direct product of a fiat three space and a group manifold of 50(3). Using the Hamiltonian formalism and numerical computations for this dynamical system, we show that the chaotic behavior of the scale factors cannot occur unless the fiat three space is static.