Diffeomorphism Invariant Theories Research Articles

Teleparallel gravity and dimensional reductions of noncommutative gauge theory

We study dimensional reductions of noncommutative electrodynamics on flat space which lead to gauge theories of gravitation. For a general class of such reductions, we show that the noncommutative gauge fields naturally yield a Weitzenbock geometry on spacetime and that the induced diffeomorphism invariant field theory can be made equivalent to a teleparallel formulation of gravity which macroscopically describes general relativity. The Planck length is determined in this setting by the Yang-Mills coupling constant and the noncommutativity scale. The effective field theory can also contain higher-curvature and non-local terms which are characteristic of string theory. Some applications to D-brane dynamics and generalizations to include the coupling of ordinary Yang-Mills theory to gravity are also described.

Making classical and quantum canonical general relativity computable through a power series expansion in the inverse cosmological constant.

We consider general relativity with a cosmological constant as a perturbative expansion around a completely solvable diffeomorphism invariant field theory. This theory is the lambda --> infinity limit of general relativity. This allows an explicit perturbative computational setup in which the quantum states of the theory and the classical observables can be explicitly computed. An unexpected relationship arises at a quantum level between the discrete spectrum of the volume operator and the allowed values of the cosmological constant.

Ghost equations and diffeomorphism-invariant theories

Four-dimensional Einstein gravity in the Palatini first-order formalismis shown to possess a vector supersymmetry of the same type as foundin the topological theories for Yang-Mills fields. A peculiar feature of thegravitational theory, characterized by diffeomorphism invariance, is adirect link of vector supersymmetry with the field equation of motion forthe Faddeev-Popov ghost of diffeomorphisms.

Free-field realization of D-dimensional cylindricalgravitational waves

We find two-dimensional free-field variables for D-dimensionalgeneral relativity on spacetimes with D-2 commutingspacelike Killing vector fields and non-compact spatial sections for D>4.We show that there is a canonical transformation which mapsthe corresponding two-dimensional dilaton gravity theory into atwo-dimensional diffeomorphism-invariant theory of the free-fieldvariables.We also show that the spacetime metric components can be expressed asasymptotic series in negative powers of the dilaton, with coefficientswhichcan be determined in terms of the free fields.

Symmetry reduction for quantizeddiffeomorphism-invariant theories of connections

Given a symmetry group acting on a principal fibre bundle, symmetricstates of the quantum theory of a diffeomorphism-invariant theory ofconnections on this fibre bundle are defined. These symmetricstates, equipped with a scalar product derived from theAshtekar-Lewandowski measure for loop quantum gravity, form aHilbert space of their own. Restriction to this Hilbert space yieldsa quantum symmetry reduction procedure within the framework ofspin-network states, the structure of which is analysed in detail. Threeillustrating examples are discussed: reduction of (3+1)- to(2+1)-dimensional quantum gravity, spherically symmetric quantumelectromagnetism and spherically symmetric quantum gravity. In thelatter system the eigenvalues of the area operator applied to thespherically symmetric spin-network states have the form An∝(n(n + 2))1/2, n = 0,1,2,..., giving An∝n for large n.This result clarifies (and reconciles) the relationship between themore complicated spectrum of the general (non-symmetric) areaoperator in loop quantum gravity and the old Bekenstein proposal thatAn∝n.

Statistical entropy of BTZ black hole in higher curvature gravity

For the BTZ black hole in the Einstein gravity, a statistical entropy has been calculated to be equal to the Bekenstein–Hawking entropy. In this paper, the statistical entropy of the BTZ black hole in the higher curvature gravity is calculated and shown to be equal to the one derived by using the Noether charge method. This suggests that the equivalence of the geometrical and statistical entropies of the black hole is retained in the general diffeomorphism invariant theories of gravity. A relation between the cosmic censorship conjecture and the unitarity of the conformal field theory on the boundary of AdS 3 is also discussed.

Energy, Hamiltonian, Noether charge, and black holes

It is shown that in general the energy ${\cal E}$ and the Hamiltonian ${\cal H}$ of matter fields on the black hole exterior play different roles. ${\cal H}$ is a generator of the time evolution along the Killing time while ${\cal E}$ enters the first law of black hole thermodynamics. For non-minimally coupled fields the difference ${\cal H}-{\cal E}$ is not zero and is a Noether charge $Q$ analogous to that introduced by Wald to define the black hole entropy. If fields vanish at the spatial boundary, $Q$ is reduced to an integral over the horizon. The analysis is carried out and an explicit expression for $Q$ is found for general diffeomorphism invariant theories. As an extension of the results by Wald et al, the first law of black hole thermodynamics is derived for arbitrary weak matter fields.

Geometric holography, the renormalization group and the c-theorem

In this paper the whole geometrical set-up giving a conformally invariant holographic projection of a diffeomorphism invariant bulk theory is clarified. By studying the renormalization group flow along null geodesic congruences a holographic version of Zamolodchikov's c-theorem is proven.

Combinatorial Space from Loop Quantum Gravity

The canonical quantization of diffeomorphism invariant theories of connections in terms of loop variables is revisited. Such theories include general relativity described in terms of Ashtekar-Barbero variables and extension to Yang-Mills fields (with or without fermions) coupled to gravity. It is argued that the operators induced by classical diffeomorphism invariant or covariant functions are respectably invariant or covariant under a suitable completion of the diffeomorphism group. The canonical quantization in terms of loop variables described here, yields a representation of the algebra of observables in a separable Hilbert space. Furthermore, the resulting quantum theory is equivalent to a model for diffeomorphism invariant gauge theories which replaces space with a manifestly combinatorial object.

Kinematical Hilbert spaces for fermionic and Higgs quantum field theories

We extend the recently developed kinematical framework for diffeomorphism-invariant theories of connections for compact gauge groups to the case of a diffeomorphism-invariant quantum field theory which includes, besides connections, also fermions and Higgs fields. This framework is appropriate for coupling matter to quantum gravity. The presence of diffeomorphism invariance forces us to choose a representation which is a rather non-Fock-like one: the elementary excitations of the connection are along open or closed strings, while those of the fermions or Higgs fields are at the end points of the string. Nevertheless we are able to promote the classical reality conditions to quantum adjointness relations which, in turn, uniquely fixes the gauge- and diffeomorphism-invariant probability measure that underlies the Hilbert space. Most of the fermionic part of this work is independent of the recent preprint by Baez and Krasnov and earlier work by Rovelli and Morales-Tecótl because we use new canonical fermionic variables, so-called Grassman-valued half-densities, which enable us to solve the difficult fermionic adjointness relations.

Quantum spin dynamics (QSD): III. Quantum constraint algebra and physical scalar product in quantum general relativity

This paper deals with several technical issues of non-perturbative four-dimensional Lorentzian canonical quantum gravity in the continuum that arose in connection with the recently constructed Wheeler-DeWitt quantum constraint operator. The Wheeler-DeWitt constraint of quantum general relativity mixes the diffeomorphism superselection sectors for diffeomorphism-invariant theories of connections that were previously discussed in the literature. From it one can construct diffeomorphism-invariant operators which do not necessarily commute with the Hamiltonian constraint but which still mix those sectors and which, at the diffeomorphism-invariant level, encode physical information. Thus, if one adopts, as before in the literature, the strategy to solve the diffeomorphism constraint before the Hamiltonian constraint then those sectors become spurious. The inner product for diffeomorphism-invariant states can be fixed by requiring that diffeomorphism group averaging is a partial isometry. The established non-anomalous constraint algebra is clarified by computing commutators of duals of constraint operators. The full classical constraint algebra is faithfully implemented on the diffeomorphism-invariant Hilbert space in an appropriate sense. The Hilbert space of diffeomorphism-invariant states can be made separable if a natural new superselection principle is satisfied. We propose a natural physical scalar product for quantum general relativity by extending the group-average approach to the case of non-self-adjoint constraint operators like the Wheeler-DeWitt constraint. Equipped with this inner product, the construction of physical observables is straightforward.

Quantization of diffeomorphism-invariant theories with fermions

We extend ideas developed for the loop representation of quantum gravity to diffeomorphism-invariant gauge theories coupled to fermions. Let P→Σ be a principal G-bundle over space and let F be a vector bundle associated to P whose fiber is a sum of continuous unitary irreducible representations of the compact connected gauge group G, each representation appearing together with its dual. We consider theories whose classical configuration space is A×F, where A is the space of connections on P and F is the space of sections of F, regarded as a collection of Grassmann-valued fermionic fields. We construct the “quantum configuration space” Ā×F̄ as a completion of A×F. Using this, we construct a Hilbert space L2(Ā×F̄) for the quantum theory on which all automorphisms of P act as unitary operators, and determine an explicit “spin network basis” of the subspace L2((Ā×F̄)/Ḡ) consisting of gauge-invariant states. We represent observables constructed from holonomies of the connection along paths together with fermionic fields and their conjugate momenta as operators on L2((Ā×F̄)/Ḡ). We also construct a Hilbert space Hdiff of diffeomorphism-invariant states using the group averaging procedure of Ashtekar, Lewandowski, Marolf, Mourão and Thiemann.

Diffeomorphism invariant integrable field theories and hypersurface motions in Riemannian manifolds

We discuss hypersurface motions in Riemannian manifolds whose normal velocity is a function of the induced hypersurface volume element and derive a second-order partial differential equation for the corresponding time function τ(x) at which the hypersurface passes the point x. Equivalently, these motions may be described in a Hamiltonian formulation as the singlet sector of certain diffeomorphism-invariant field theories. At least in some (infinite class of) cases, which could be viewed as a large-volume limit of Euclidean M-branes moving in an arbitrary M+1-dimensional Riemannian manifold, the models are integrable: In the time-function formulation the equation becomes linear [with τ(x) a harmonic function on the embedding Riemannian manifold]. We explicitly compute solutions to the large volume limit of Euclidean membrane dynamics in R3 by methods used in electrostatics and point out an additional gradient flow structure in Rn. In the Hamiltonian formulation we discover infinitely many hierarchies of integrable, multidimensional, N-component theories possessing infinitely many diffeomorphism invariant, Poisson commuting, conserved charges.

A combinatorial approach to diffeomorphism invariant quantum gauge theories

Quantum gauge theory in the connection representation uses functions of holonomies as configuration observables. Physical observables (gauge and diffeomorphism invariant) are represented in the Hilbert space of physical states; physical states are gauge and diffeomorphism invariant distributions on the space of functions of the holonomies of the edges of a certain family of graphs. Then a family of graphs embedded in the space manifold (satisfying certain properties) induces a representation of the algebra of physical observables. We construct a quantum model from the set of piecewise linear graphs on a piecewise linear manifold, and another manifestly combinatorial model from graphs defined on a sequence of increasingly refined simplicial complexes. Even though the two models are different at the kinematical level, they provide unitarily equivalent representations of the algebra of physical observables in separable Hilbert spaces of physical states (their s-knot basis is countable). Hence, the combinatorial framework is compatible with the usual interpretation of quantum field theory.

Determination and reduction of large diffeomorphisms

For diffeomorphism invariant theories we consider the problem of how to determine and reduce diffeomorphisms which are not in the component of the identity.

On diffeomorphism invariance for lattice theories

We consider the role of the diffeomorphism constraint in the quantization of lattice formulations of diffeomorphism invariant theories of connections. It has been argued that in working with abstract lattices one automatically takes care of the diffeomorphism constraint in the quantum theory. We use two systems in order to show that imposing the diffeomorphism constraint is imperative to obtain a physically acceptable quantum theory. First, we consider 2 + 1 gravity where an exact lattice formulation is available. Next, general theories of connections for compact gauge groups are treated, where the quantum theories are known - for both the continuum and the lattice - and can be compared.

Lagrangian perfect fluids and black hole mechanics

The first law of black hole mechanics (in the form derived by Wald) is expressed in terms of integrals over surfaces, at the horizon and spatial infinity, of a stationary, axisymmetric black hole, in a diffeomorphism-invariant Lagrangian theory of gravity. The original statement of the first law given by Bardeen, Carter, and Hawking for an Einstein-perfect fluid system contained, in addition, volume integrals of the fluid fields, over a spacelike slice stretching between these two surfaces. One would expect that Wald's methods, applied to a Lagrangian Einstein-perfect fluid formulation, would convert these terms to surface integrals. However, because the fields appearing in the Lagrangian of a gravitating perfect fluid are typically nonstationary (even in a stationary black-hole--perfect-fluid spacetime) a direct application of these methods generally yields restricted results. We therefore first approach the problem of incorporating general nonstationary matter fields into Wald's analysis, and derive a first-law-like relation for an arbitrary Lagrangian metric theory of gravity coupled to arbitrary Lagrangian matter fields, requiring only that the metric field be stationary. This relation includes a volume integral of matter fields over a spacelike slice between the black hole horizon and spatial infinity, and reduces to the first law originally derived by Bardeen, Carter, and Hawking when the theory is general relativity coupled to a perfect fluid. We then turn to consider a specific Lagrangian formulation for an isentropic perfect fluid given by Carter, and directly apply Wald's analysis, assuming that both the metric and fluid fields are stationary and axisymmetric in the black hole spacetime. The first law we derive contains only surface integrals at the black hole horizon and spatial infinity, but the assumptions of stationarity and axisymmetry of the fluid fields make this relation much more restrictive in its allowed fluid configurations and perturbations than that given by Bardeen, Carter, and Hawking. In the Appendix, we use the symplectic structure of the Einstein-perfect fluid system to derive a conserved current for perturbations of this system: this current reduces to one derived ab initio for this system by Chandrasekhar and Ferrari.

Asymptotic structure of symmetry-reduced general relativity

Gravitational waves with a space-translation Killing field are considered. In this case, the 4-dimensional Einstein vacuum equations are equivalent to the 3-dimensional Einstein equations with certain matter sources. This interplay between 4- and 3- dimensional general relativity can be exploited effectively to analyze issues pertaining to 4 dimensions in terms of the 3-dimensional structures. An example is provided by the asymptotic structure at null infinity: While these space-times fail to be asymptotically flat in 4 dimensions, they can admit a regular completion at null infinity in 3 dimensions. This completion is used to analyze the asymptotic symmetries, introduce the analog of the 4-dimensional Bondi energy-momentum and write down a flux formula. The analysis is also of interest from a purely 3-dimensional perspective because it pertains to a diffeomorphism invariant 3-dimensional field theory with {\it local} degrees of freedom, i.e., to a midi-superspace. Furthermore, due to certain peculiarities of 3 dimensions, the description of null infinity does have a number of features that are quite surprising because they do not arise in the Bondi-Penrose description in 4 dimensions.

Black hole entropy and the Hamiltonian formulation of diffeomorphism invariant theories.

Path integral methods are used to derive a general expression for the entropy of a black hole in a diffeomorphism invariant theory. The result, which depends on the variational derivative of the Lagrangian with respect to the Riemann tensor, agrees with the result obtained from Noether charge methods by Iyer and Wald. The method used here is based on the direct expression of the density of states as a path integral (the microcanonical functional integral). The analysis makes crucial use of the Hamiltonian form of the action. An algorithm for placing the action of a diffeomorphism invariant theory in Hamiltonian form is presented. Other path integral approaches to the derivation of black hole entropy include the Hilbert action surface term method and the conical deficit angle method. The relationships between these path integral methods are presented.

Differential geometry on the space of connections via graphs and projective limits

In a quantum mechanical treatment of gauge theories (including general relativity), one is led to consider a certain completion A G of the space A G of guage equivalent connections. This space serves as the quantum configuration space, or, as the space of all Euclidean histories over which one must integrate in the quantum theory A G is a very large is a very large space and serves as a “universal home” for measures in theories in which the Wilson loop observables are well defined. In this paper, A G is considered as the projective limit of a projective family of compact Hausdorff manifolds, labelled by graphs (which can be regarded as “floating lattices” in the physics terminology). Using this characterization, differential geometry is developed through algebraic methods. In particular, we are able to introduce the following notions on A G : differential forms, exterio derivatives, volume forms, vector fields and Lie brackets between them, divergence of a vector field with respect to a volume form, Laplacians and associated heat kernels and heat kernel measures. Thus, although A G is very large, it is small enough to be mathematically interesting and physically useful. A key feature of this approach is that it does not require a background metric. The geometrical framework is therefore well suited for diffeomorphism invariant theories such as quantum general relativity.