Diffeomorphism Invariant Theories Research Articles

Gravity as a diffeomorphism-invariant gauge theory

A general diffeomorphism invariant SU(2) gauge theory is a gravity theory with two propagating polarizations of the graviton. We develop this description of gravity, in particular for future applications to the perturbative quantization. Thus, the linearized theory, gauge symmetries, gauge fixing are discussed in detail, and the propagator is obtained. The propagator takes a simple form of that of Yang-Mills theory with an additional projector on diffeomorphism equivalence classes of connections inserted. In our approach the gravitational perturbation theory takes a rather unusual form in that the Planck length is no longer fundamental.

Parent formulation at the Lagrangian level

The recently proposed first-order parent formalism at the level of equations of motion is specialized to the case of Lagrangian systems. It is shown that for diffeomorphism-invariant theories the parent formulation takes the form of an AKSZ-type sigma model. The proposed formulation can be also seen as a Lagrangian version of the BV-BRST extension of the Vasiliev unfolded approach. We also discuss its possible interpretation as a multidimensional generalization of the Hamiltonian BFV--BRST formalism. The general construction is illustrated by examples of (parametrized) mechanics, relativistic particle, Yang--Mills theory, and gravity.

Can Hamiltonians be boundary observables in parametrized field theories?

It has been argued that holography in gravitational theories is related to the existence of a particularly useful Gauss Law that allows energy to be measured at the boundary. This work investigates the extent to which such Gauss Laws follow from diffeomorphism invariance. We study parametrized field theories, which are a class of diffeomorphism-invariant theories without gravity. We find that the Hamiltonian for non-gravitational parametrized field theories vanishes on shell even in the presence of a boundary and under a variety of boundary conditions. We conclude that such theories have no useful Gauss Law, consistent with the absence of holography.

EQUIPARTITION OF MICROSCOPIC DEGREES OF FREEDOM, SPACE–TIME ENTROPY AND HOLOGRAPHY

One can identify the number density of the microscopic space–time degrees of freedom in any diffeomorphism-invariant theory of gravity by using the principle of equipartition, applied to the area elements of a surface [Formula: see text] which are at the local Unruh temperature. The entropy associated with these degrees of freedom, which matches with the Wald entropy for the theory, can be used to obtain the field equations of the theory through an extremization priciple. When the microscopic degrees of freedom are in local thermal equilibrium, the entropy of a bulk region of space–time resides on its boundary. These facts support an emergent perspective of gravity.

Unification of gravity, gauge fields and Higgs bosons

We consider a diffeomorphism invariant theory of a gauge field valued in a Lie algebra that breaks spontaneously to the direct sum of the spacetime Lorentz algebra, a Yang–Mills algebra and their complement. Beginning with a fully gauge invariant action—an extension of the Plebanski action for general relativity—we recover the action for gravity, Yang–Mills and Higgs fields. The low-energy coupling constants, obtained after symmetry breaking, are all functions of the single parameter present in the initial action and the vacuum expectation value of the Higgs.

Surface density of spacetime degrees of freedom from equipartition law in theories of gravity

I show that the principle of equipartition, applied to area elements of a surface which are in equilibrium at the local Davies-Unruh temperature, allows one to determine the surface number density of the microscopic spacetime degrees of freedom in any diffeomorphism invariant theory of gravity. The entropy associated with these degrees of freedom matches with the Wald entropy for the theory. This result also allows one to attribute an entropy density to the spacetime in a natural manner. The field equations of the theory can then be obtained by extremising this entropy. Moreover, when the microscopic degrees of freedom are in local thermal equilibrium, the spacetime entropy of a bulk region resides on its boundary.

ENTROPY DENSITY OF SPACE–TIME AND GRAVITY: A CONCEPTUAL SYNTHESIS

I show that combining the principle of equivalence and the principle of general covariance with the known properties of local Rindler horizons, perceived by accelerated observers, leads to the following inescapable conclusion: The field equations describing gravity in any diffeomorphism-invariant theory must have a thermodynamic interpretation. This synthesis of quantum theory, thermodynamics and gravity shows that the gravitational dynamics can be interpreted completely in terms of entropy balance between matter and space–time. This idea has far-reaching implications for the microstructure of space–time and quantum gravity.

A comment on Kerr–CFT and Wald entropy

We point out that the entropies of black holes in general diffeomorphism invariant theories, computed using the Kerr–CFT correspondence and the Wald formula (as implemented in the entropy function formalism), need not always agree. A simple way to illustrate this is to consider Einstein–Gauss–Bonnet gravity in four dimensions, where the Gauss–Bonnet term is topological. This means that the central charge of Kerr–CFT computed in the Barnich–Brandt–Compere formalism remains the same as in Einstein gravity, while the entropy computed using the entropy function gives a universal correction proportional to the Gauss–Bonnet coupling. We argue that at least in this example, the Kerr–CFT result is the physically reasonable one. The resolution to this discrepancy might lie in a better understanding of boundary terms.

Quantization of Diffeomorphism Invariant Theories of Connections with a Non-Compact Structure Group—an Example

A simple diffeomorphism invariant theory of connections with the non-compact structure group R of real numbers is quantized. The theory is defined on a four-dimensional 'space-time' by an action resembling closely the self-dual Plebanski action for general relativity. The space of quantum states is constructed by means of projective techniques by Kijowski. Except this point the applied quantization procedure is based on Loop Quantum Gravity methods.

Emergent diffeomorphism invariance in a discrete loop quantum gravity model

Several approaches to the dynamics of loop quantum gravity involve discretizing the equations of motion. The resulting discrete theories are known to be problematic since the first-class algebra of constraints of the continuum theory becomes second class upon discretization. If one treats the second-class constraints properly, the resulting theories have very different dynamics and number of degrees of freedom than those of the continuum theory. It is therefore questionable how these theories could be considered a starting point for the quantization and the definition of a continuum theory through a continuum limit. We show explicitly in a model that the uniform discretizations approach to the quantization of constrained systems overcomes these difficulties. We consider here a simple diffeomorphism invariant one-dimensional model and complete the quantization using uniform discretizations. The model can be viewed as a spherically symmetric reduction of the well-known Husain–Kuchař model of diffeomorphism invariant theory. We show that the correct quantum continuum limit can be satisfactorily constructed for this model. This opens the possibility of treating (1 + 1)-dimensional dynamical situations of great interest in quantum gravity taking into account the full dynamics of the theory and preserving the spacetime covariance at a quantum level.

Quasilocal energy flux of spacetime perturbation

A general expression for quasilocal energy flux for spacetime perturbation is derived from covariant Hamiltonian formulation using functional differentiability and symplectic structure invariance, which is independent of the choice of the canonical variables and the possible boundary terms one initially puts into the Lagrangian in the diffeomorphism invariant theories. The energy flux expression depends on a displacement vector field and the 2-surface under consideration. We apply and test the expression in Vaidya spacetime. At null infinity the expression leads to the Bondi type energy flux obtained by Lindquist, Schwartz, and Misner. On dynamical horizons with a particular choice of the displacement vector, it gives the area balance law obtained by Ashtekar and Krishnan.

Topological field theories inn-dimensional spacetimes and Cartan’s equations

Action principles of the BF type for diffeomorphism invariant topological field theories living in n-dimensional spacetime manifolds are presented. Their construction is inspired by Cuesta and Montesinos' recent paper where Cartan's first and second structure equations together with first and second Bianchi identities are treated as the equations of motion for a field theory. In opposition to that paper, the current approach involves also auxiliary fields and holds for arbitrary n-dimensional spacetimes. Dirac's canonical analysis for the actions is detailedly carried out in the generic case and it is shown that these action principles define topological field theories, as mentioned. The current formalism is a generic framework to construct geometric theories with local degrees of freedom by introducing additional constraints on the various fields involved that destroy the topological character of the original theory. The latter idea is implemented in two-dimensional spacetimes where gravity coupled to matter fields is constructed out, which has indeed local excitations.

Twisted gauge and gravity theories on the Groenewold-Moyal plane

Recent work indicates an approach to the formulation of diffeomorphism invariant quantum field theories (qft's) on the Groenewold-Moyal plane. In this approach to the qft's, statistics gets twisted and the $S$ matrix in the nongauge qft's becomes independent of the noncommutativity parameter ${\ensuremath{\theta}}^{\ensuremath{\mu}\ensuremath{\nu}}$. Here we show that the noncommutative algebra has a commutative spacetime algebra as a substructure: the Poincar\'e, diffeomorphism and gauge groups are based on this algebra in the twisted approach as is known already from the earlier work. It is natural to base covariant derivatives for gauge and gravity fields as well on this algebra. Such an approach will, in particular, introduce no additional gauge fields as compared to the commutative case and also enable us to treat any gauge group [and not just $U(N)$]. Then classical gravity and gauge sectors are the same as those for ${\ensuremath{\theta}}^{\ensuremath{\mu}\ensuremath{\nu}}=0$, but their interactions with matter fields are sensitive to ${\ensuremath{\theta}}^{\ensuremath{\mu}\ensuremath{\nu}}$. We construct quantum noncommutative gauge theories (for arbitrary gauge groups) by requiring consistency of twisted statistics and gauge invariance. In a subsequent paper (whose results are summarized here), the locality and Lorentz invariance properties of the $S$ matrices of these theories will be analyzed, and new nontrivial effects coming from noncommutativity will be elaborated. This paper contains further developments of an earlier paper of ours and a new formulation based on its approach.

Dimensional reduction, truncations, constraints and the issue of consistency

A brief overview of dimensional reductions for diffeomorphism invariant theories is given. The distinction between the physical idea of compactification and the mathematical problem of a consistent truncation is discussed, and the typical ingredients of the latter-reduction of spacetime dimensions and the introduction of constraints-are examined. The consistency in the case of of group manifold reductions, when the structure constants satisfy the unimodularity condition, is shown together with the associated reduction of the gauge group. The problem of consistent truncations on coset spaces is also discussed and we comment on examples of some remarkable consistent truncations that have been found in this context.

Asymptotic generators of fermionic charges and boundary conditions preserving supersymmetry

We use a covariant phase space formalism to give a general prescription for defining Hamiltonian generators of bosonic and fermionic symmetries in diffeomorphism invariant theories, such as supergravities. A simple and general criterion is derived for a choice of boundary condition to lead to conserved generators of the symmetries on the phase space. In particular, this provides a criterion for the preservation of supersymmetries. For bosonic symmetries corresponding to diffeomorphisms, our prescription coincides with the method of Wald et al. We then illustrate these methods in the case of certain supergravity theories in d = 4. In minimal AdS supergravity, boundary conditions such that the supercharges exist as Hamiltonian generators of supersymmetry transformations are unique within the usual framework in which the boundary metric is fixed. In extended AdS supergravity, or more generally in the presence of chiral matter superfields, we find that there exist many boundary conditions preserving supersymmetry for which corresponding generators exist. These choices are shown to correspond to a choice of certain arbitrary boundary ‘superpotentials’, for suitably defined ‘boundary superfields’. We also derive corresponding formulae for the conserved bosonic charges, such as energy, in those theories, and we argue that energy is always positive, for any supersymmetry-preserving boundary conditions. We finally comment on the relevance and interpretation of our results within the AdS/CFT correspondence.

Uniqueness of Diffeomorphism Invariant States on Holonomy–Flux Algebras

Loop quantum gravity is an approach to quantum gravity that starts from the Hamiltonian formulation in terms of a connection and its canonical conjugate. Quantization proceeds in the spirit of Dirac: First one defines an algebra of basic kinematical observables and represents it through operators on a suitable Hilbert space. In a second step, one implements the constraints. The main result of the paper concerns the representation theory of the kinematical algebra: We show that there is only one cyclic representation invariant under spatial diffeomorphisms. While this result is particularly important for loop quantum gravity, we are rather general: The precise definition of the abstract *-algebra of the basic kinematical observables we give could be used for any theory in which the configuration variable is a connection with a compact structure group. The variables are constructed from the holonomy map and from the fluxes of the momentum conjugate to the connection. The uniqueness result is relevant for any such theory invariant under spatial diffeomorphisms or being a part of a diffeomorphism invariant theory.

Noether charges and black hole mechanics in Einstein-aether theory

The Noether charge method for defining the Hamiltonian of a diffeomorphism-invariant field theory is applied to ``Einstein-aether'' theory, in which gravity couples to a dynamical, timelike, unit-norm vector field. Using the method, expressions are obtained for the total energy, momentum, and angular momentum of an Einstein-aether space-time. The method is also used to discuss the mechanics of Einstein-aether black holes. The derivation of Wald, and Iyer and Wald, of the first law of black hole thermodynamics fails for this theory because the unit-vector is necessarily singular at the bifurcation surface of the Killing horizon. A general identity relating variations of energy and angular momentum to a surface integral at the horizon is obtained, but a thermodynamic interpretation, including a definitive expression for the black hole entropy, is not found.

Diffeomorphism invariance and black hole entropy

The Noether-charge realization and the Hamiltonian realization for the $\diff({\cal M})$ algebra in diffeomorphism invariant gravitational theories are studied in a covariant formalism. For the Killing vector fields, the Nother-charge realization leads to the mass formula as an entire vanishing Noether charge for the vacuum black hole spacetimes in general relativity and the corresponding first law of the black hole mechanics. It is analyzed in which sense the Hamiltonian functionals form the $\diff({\cal M})$ algebra under the Poisson bracket and shown how the Noether charges with respect to the diffeomorphism generated by vector fields and their variations in general relativity form this algebra. The asymptotic behaviors of vector fields generating diffeomorphism of the manifold with boundaries are discussed. In order to get more precise estimation for the "central extension" of the algebra, it is analyzed in the Newman-Penrose formalism and shown that the "central extension" for a large class of vector fields is always zero on the Killing horizon. It is also checked whether the Virasoro algebra may be picked up by choosing the vector fields near the horizon. The conclusion is unfortunately negative.

Non-existence of a dilaton gravity action for the exact string black hole

We prove that no local diffeomorphism invariant two-dimensional theory of the metric and the dilaton without higher derivatives can describe the exact string black hole solution found a decade ago by Dijkgraaf, Verlinde and Verlinde. One of the key points in this proof is the concept of dilaton-shift invariance. We present and solve (classically) all dilaton-shift invariant theories of two-dimensional dilaton gravity. Two such models, resembling the exact string black hole and generalizing the CGHS model, are discussed explicitly.

GENERALISED PSEUDO-RIEMANNIAN GEOMETRY FOR GENERAL RELATIVITY

The study of singular spacetimes by distributional methods faces the fundamental obstacle of the inherent nonlinearity of the field equations. Staying strictly within the distributional (in particular: linear) regime, as determined by Geroch and Traschen2 excludes a number of physically interesting examples (e.g., cosmic strings). In recent years, several authors have therefore employed nonlinear theories of generalized functions (Colombeau algebras, in particular) to tackle general relativistic problems1,5,8. Under the influence of these applications in general relativity the nonlinear theory of generalized functions itself has undergone a rapid development lately, resulting in a diffeomorphism invariant global theory of nonlinear generalized functions on manifolds3,4,6. In particular, a generalized pseudo-Riemannian geometry allowing for a rigorous treatment of generalized (distributional) spacetime metrics has been developed7. It is the purpose of this talk to present these new mathematical methods themselves as well as a number of applications in mathematical relativity.