Canonical Formulation Of General Relativity Research Articles

Yang–Mills Theory of Gravity

The canonical formulation of general relativity (GR) is based on decomposition space–time manifold M into R × Σ , where R represents the time, and Ksi is the three-dimensional space-like surface. This decomposition has to preserve the invariance of GR, invariance under general coordinates, and local Lorentz transformations. These symmetries are associated with conserved currents that are coupled to gravity. These symmetries are studied on a three dimensional space-like hypersurface Σ embedded in a four-dimensional space–time manifold. This implies continuous symmetries and conserved currents by Noether’s theorem on that surface. We construct a three-form E i ∧ D A i (D represents covariant exterior derivative) in the phase space ( E i a , A a i ) on the surface Σ , and derive an equation of continuity on that surface, and search for canonical relations and a Lagrangian that correspond to the same equation of continuity according to the canonical field theory. We find that Σ i 0 a is a conjugate momentum of A a i and Σ i a b F a b i is its energy density. We show that there is conserved spin current that couples to A i , and show that we have to include the term F μ ν i F μ ν i in GR. Lagrangian, where F i = D A i , and A i is complex S O ( 3 ) connection. The term F μ ν i F μ ν i includes one variable, A i , similar to Yang–Mills gauge theory. Finally we couple the connection A i to a left-handed spinor field ψ , and find the corresponding beta function.

Regarding the ‘Hole Argument’ and the ‘Problem of Time’

The canonical formalism of general relativity affords a particularly interesting characterization of the infamous hole argument. It also provides a natural formalism in which to relate the hole argument to the problem of time in classical and quantum gravity. Conceptual and formal inadequacies within the representative language of canonical gravity will be shown to be at the heart of both the canonical hole argument and the problem of time. Interesting and fruitful work at the interface of physics and philosophy relates to the challenge of resolving such inadequacies.

Felix Arnold Edward Pirani

Felix Arnold Edward Pirani

Discrete Hamiltonian for general relativity

Beginning from the Ashtekar formulation of canonical general relativity, we derive a physical Hamiltonian written in terms of (classical) loop gravity variables. This is done by gauge-fixing the gravitational fields within a complex of three-dimensional cells such that curvature and torsion vanish within each cell. The resulting theory is holographic, with the bulk dynamics being captured completely by degrees of freedom living on cell boundaries. Quantization is readily obtainable by existing methods.

BV-BFV approach to general relativity: Einstein-Hilbert action

The present paper shows that general relativity in the Arnowitt-Deser-Misner formalism admits a BV-BFV formulation. More precisely, for any d + 1 ≠ 2 (pseudo-) Riemannian manifold M with space-like or time-like boundary components, the BV data on the bulk induces compatible BFV data on the boundary. As a byproduct, the usual canonical formulation of general relativity is recovered in a straightforward way.

A CANONICAL RANK-THREE TENSOR MODEL WITH A SCALING CONSTRAINT

A rank-three tensor model in canonical formalism has recently been proposed. The model describes consistent local-time evolutions of fuzzy spaces through a set of first-class constraints which form an on-shell closed algebra with structure functions. In fact, the algebra provides an algebraically consistent discretization of the Dirac–DeWitt constraint algebra in the canonical formalism of general relativity. However, the configuration space of this model contains obvious degeneracies of representing identical fuzzy spaces. In this paper, to delete the degeneracies, another first-class constraint representing a scaling symmetry is added to propose a new canonical rank-three tensor model. A consequence is that, while classical solutions of the previous model have typically runaway or vanishing behaviors, the new model has a compact configuration space and its classical solutions asymptotically approach either fixed points or cyclic orbits in time evolution. Among others, fixed points contain configurations with group symmetries, and may represent stationary symmetric fuzzy spaces. Another consequence on the uniqueness of the local Hamiltonian constraint is also discussed, and a minimal canonical tensor model, which is unique, is given.

Quantisation, Representation and Reduction; How Should We Interpret the Quantum Hamiltonian Constraints of Canonical Gravity?

Hamiltonian constraints feature in the canonical formulation of general relativity. Unlike typical constraints they cannot be associated with a reduction procedure leading to a non-trivial reduced phase space and this means the physical interpretation of their quantum analogues is ambiguous. In particular, can we assume that “quantisation commutes with reduction” and treat the promotion of these constraints to operators annihilating the wave function, according to a Dirac type procedure, as leading to a Hilbert space equivalent to that reached by quantisation of the problematic reduced space? If not, how should we interpret Hamiltonian constraints quantum mechanically? And on what basis do we assert that quantisation and reduction commute anyway? These questions will be refined and explored in the context of modern approaches to the quantisation of canonical general relativity.

Double-Slit Interference Pattern from Single-Slit Screen and Its Gravitational Analogues

The double slit experiment (DSE) is known as an important cornerstone in the foundations of physical theories such as Quantum Mechanics and Special Relativity. A large number of different variants of it were designed and performed over the years. We perform and discuss here a new version with the somewhat unexpected results of obtaining interference pattern from single-slit screen. We show using either the Brill’s version of the canonical formulation of general relativity or the linearized version of it that one may find corresponding and analogous situations in the framework of general relativity.

Brane-world quantum gravity

The Arnowitt–Deser–Misner canonical formulation of general relativity is extended to the covariant brane-world theory in arbitrary dimensions. The exclusive probing of the extra dimensions makes a substantial difference, allowing for the construction of a non-constrained canonical theory. The quantum states of the brane-world geometry are defined by the Tomonaga–Schwinger equation, whose integrability conditions are determined by the classical perturbations of submanifolds contained in the Nash's differentiable embedding theorem. In principle, quantum brane-world theory can be tested by current experiments in astrophysics and by near future laboratory experiments at Tev energy. The implications to the black-hole information loss problem, to the accelerating cosmology, and to a quantum mathematical theory of four-sub manifolds are briefly commented.

A perturbative approach to Dirac observables and their spacetime algebra

We introduce a general approximation scheme in order to calculate gauge invariant observables in the canonical formulation of general relativity. Using this scheme we will show how the observables and the dynamics of field theories on a fixed background or equivalently the observables of the linearized theory can be understood as an approximation to the observables in full general relativity. Gauge invariant corrections can be calculated up to an arbitrary high order and we will explicitly calculate the first non-trivial correction. Furthermore, we will make a first investigation into the Poisson algebra between observables corresponding to fields at different spacetime points and consider the locality properties of the observables.

Towards conformal loop quantum gravity

A discussion is given of recent developments in canonical gravity that assimilates the conformal analysis of gravitational degrees of freedom. The work is motivated by the problem of time in quantum gravity and is carried out at the metric and the triad levels. At the metric level, it is shown that by extending the Arnowitt-Deser-Misner (ADM) phase space of general relativity (GR), a conformal form of geometrodynamics can be constructed. In addition to the Hamiltonian and Diffeomorphism constraints, an extra first class constraint is introduced to generate conformal transformations. This phase space consists of York's mean extrinsic curvature time, conformal three-metric and their momenta. At the triad level, the phase space of GR is further enlarged by incorporating spin-gauge as well as conformal symmetries. This leads to a canonical formulation of GR using a new set of real spin connection variables. The resulting gravitational constraints are first class, consisting of the Hamiltonian constraint and the canonical generators for spin-gauge and conformorphism transformations. The formulation has a remarkable feature of being parameter-free. Indeed, it is shown that a conformal parameter of the Barbero-Immirzi type can be absorbed by the conformal symmetry of the extended phase space. This gives rise to an alternative approach to loop quantum gravity that addresses both the conceptual problem of time and the technical problem of functional calculus in quantum gravity.

Unambiguous spin-gauge formulation of canonical general relativity with conformorphism invariance

We present a parameter-free gauge formulation of general relativity in terms of a new set of real spin connection variables. The theory is constructed by extending the phase space of the recently formulated conformal geometrodynamics to accommodate a spin-gauge description. This leads to a further enlarged set of first class gravitational constraints consisting of a reduced Hamiltonian constraint and the canonical generators for spin-gauge and conformorphism transformations. Owing to the incorporated conformal symmetry, the new theory is shown to be free from an ambiguity of the Barbero-Immirzi type.

Timelike surfaces in Lorentz covariant loop gravity and spin foam models

We apply a recently developed canonical formulation of general relativity, which possesses explicit covariance with respect to Lorentz transformations in the tangent space, to describe the case of a timelike foliation of spacetime. Assuming that such a theory based on a timelike foliation makes sense, we derive the spectrum of the area operator of a two-dimensional surface in the framework of the loop quantization. Its different branches are naturally associated with spacelike and timelike surfaces. The results are compared with the predictions of Lorentzian (Barrett–Crane) spin foam models. A restriction of the representations labelling spin networks leads to perfect agreement between the states as well as the area spectra in the two approaches.

Histories approach to general relativity: II. invariance groups

In this paper we show in detail how the histories description of general relativity carries representations of both the spacetime diffeomorphism group and the Dirac algebra of constraints. We show that the introduction of metric-dependent equivariant foliations leads to the crucial result that the canonical constraints are invariant under the action of spacetime diffeomorphisms. Furthermore, there exists a representation of the group of generalized spacetime mappings that are functionals of the 4-metric: this is a spacetime analogue of the group originally defined by Bergmann and Komar in the context of the canonical formulation of general relativity. Finally, we discuss the possible directions for the quantization of gravity in histories theory.

A NOVEL SPIN-STATISTICS THEOREM IN (2 + 1)DCHERN–SIMONS GRAVITY

It has been known for some time that topological geons in quantum gravity may lead to a complete violation of the canonical spin-statistics relation: There may be no connection between spin and statistics for a pair of geons. We present an algebraic description of quantum gravity in a (2 + 1) D manifold of the form Σ × ℝ, based on the first-order canonical formalism of general relativity. We identify a certain algebra describing the system, and obtain its irreducible representations. We then show that although the usual spin-statistics theorem is not valid, statistics is completely determined by spin for each of these irreducible representations, provided one of the labels of these representations, which we call flux, is superselected. We argue that this is indeed the case. Hence, a new spin-statistics theorem can be formulated.

ADM worldvolume geometry

We describe the dynamics of a relativistic extended object in terms of the geometry of a configuration of constant time. This involves an adaptation of the ADM formulation of canonical general relativity. We apply the formalism to the hamiltonian formulation of a Dirac-Nambu-Goto [DNG] relativistic extended object in an arbitrary background spacetime.

Quantum gravity and Regge calculus

This is an informal review of the formulation of canonical general relativity and of its implications for quantum gravity; the various versions are compared, both in the continuum and in a discretized approximation suggested by Regge calculus. I also show that the weakness of the link with the geometric content of the theory gives rise to what I think is a serious flaw in the claimed derivation of a discrete structure for space at the quantum level.

QUANTUM GRAVITY BY THE COMPLEX CANONICAL FORMULATION

The basic features of the complex canonical formulation of general relativity and the recent developments in the quantum gravity program based on it are reviewed. The exposition is intended to be complementary to the review articles already available and some original arguments are included. In particular the conventional treatment of the Hamiltonian constraint and quantum states in the canonical approach to quantum gravity is criticized and a new formulation is proposed.

General covariance, new variables, and dynamics without dynamics.

We give an example of a generally covariant geometric field theory which leads to the standard Gaussian and supermomentum constraints but which lacks the super-Hamiltonian constraint. This theory is closely linked to Ashtekar’s canonical formulation of general relativity. We discuss the Dirac constraint quantization of our model and comment on the issues which it raises regarding the quantization of generally covariant systems.

Intersecting-loop solutions of the hamiltonian constraint of quantum general relativity

Recently, using Ashtekar's new canonical formulation of general relativity, a large class of solutions of the quantum hamiltonian constraint were found by Jacobson and Smolin. These solutions are the traces of the holonomy for Ashtekar's connection that are based on differentiable loops. In this paper, additional solutions of the quantum hamiltonian constraint are described. These new solutions also involve the holonomy, but are based on multiple loops that intersect at a point. It is shown that no solutions exist for three intersecting loops that have more than two linearly independent tangent vectors at the intersection point.