HAL (Le Centre pour la Communication Scientifique Directe), Apr 12, 2001
We study a family of physical observable quantities in quantum gravity. We denote them W function... more We study a family of physical observable quantities in quantum gravity. We denote them W functions, or n-net functions. They represent transition amplitudes between quantum states of the geometry, are analogous to the n-point functions in quantum field theory, but depend on spin networks with n connected components. In particular, they include the three-geometry to three-geometry transition amplitude. The W functions are scalar under four-dimensional diffeomorphism, and fully gauge invariant. They capture the physical content of the quantum gravitational theory. We show that W functions are the natural n-point functions of the field theoretical formulation of the gravitational spin foam models. They can be computed from a perturbation expansion, which can be interpreted as a sum-over-four-geometries. Therefore the W functions bridge between the canonical (loop) and the covariant (spinfoam) formulations of quantum gravity. Following Wightman, the physical Hilbert space of the theory can be reconstructed from the W functions, if a suitable positivity condition is satisfied. We compute explicitly the W functions in a "free" model in which the interaction giving the gravitational vertex is shut off, and we show that, in this simple case, we have positivity, the physical Hilbert space of the theory can be constructed explicitly and the theory admits a well defined interpretation in terms of diffeomorphism invariant transition amplitudes between quantized geometries.
HAL (Le Centre pour la Communication Scientifique Directe), Apr 12, 2001
We study a family of physical observable quantities in quantum gravity. We denote them W function... more We study a family of physical observable quantities in quantum gravity. We denote them W functions, or n-net functions. They represent transition amplitudes between quantum states of the geometry, are analogous to the n-point functions in quantum field theory, but depend on spin networks with n connected components. In particular, they include the three-geometry to three-geometry transition amplitude. The W functions are scalar under four-dimensional diffeomorphism, and fully gauge invariant. They capture the physical content of the quantum gravitational theory. We show that W functions are the natural n-point functions of the field theoretical formulation of the gravitational spin foam models. They can be computed from a perturbation expansion, which can be interpreted as a sum-over-four-geometries. Therefore the W functions bridge between the canonical (loop) and the covariant (spinfoam) formulations of quantum gravity. Following Wightman, the physical Hilbert space of the theory can be reconstructed from the W functions, if a suitable positivity condition is satisfied. We compute explicitly the W functions in a "free" model in which the interaction giving the gravitational vertex is shut off, and we show that, in this simple case, we have positivity, the physical Hilbert space of the theory can be constructed explicitly and the theory admits a well defined interpretation in terms of diffeomorphism invariant transition amplitudes between quantized geometries.
Weaves are eigenstates of geometrical operators in nonperturbative quan- tum gravity, which appro... more Weaves are eigenstates of geometrical operators in nonperturbative quan- tum gravity, which approximate ¯ at space (or other smooth geometries) at large scales. We describe two such states, which diagonalize the area as well as the volume operators. The existence of such ...
We extend the definition of the "flipped" loop-quantum-gravity vertex to the case of a finite Imm... more We extend the definition of the "flipped" loop-quantum-gravity vertex to the case of a finite Immirzi parameter γ. We cover both the euclidean and lorentzian cases. We show that the resulting dynamics is defined on a Hilbert space isomorphic to the one of loop quantum gravity, and that the area operator has the same discrete spectrum as in loop quantum gravity. This includes the correct dependence on γ, and, remarkably, holds in the lorentzian case as well. The ad hoc flip of the symplectic structure that was required to derive the flipped vertex is not anymore required for finite γ. These results establish a bridge between canonical loop quantum gravity and the spinfoam formalism in four dimensions.
We study the quantum fermions + gravity system, that is, the gravitational counterpart of QED. We... more We study the quantum fermions + gravity system, that is, the gravitational counterpart of QED. We start from the standard Einstein-Weyl theory, reformulated in terms of Ashtekar variables; and we construct its non-perturbative quantum theory by extending the loop representation of general relativity. To this aim, we construct the fermion equivalent of the loop variables, and we define the quantum theory as a representation of their Poisson algebra. Not surprisingly, fermions can be accounted for in the loop representation simply by including open curves into loop space, as expected from lattice Yang-Mills theory. We explicitly construct the diffeomorphism and Hamiltonian operators. The first can be fully solved as in pure gravity. The second is constructed by using a background independent regularization technique. The theory retains the clean geometrical features of pure quantum gravity. In particular, the Hamiltonian constraint admits the same simple geometrical interpretation as its pure gravity counterpart: it is the operator that shifts curves along themselves (shift operator). Quite surprisingly, we believe, this simple action codes the full dynamics of the interacting fermion-gravity theory. To unravel the dynamics of the theory we study the evolution of the fermion-gravity system in the physical time defined by an additional coupled (clock-) scalar field. We explicitly construct the Hamiltonian operator that evolves the system in this physical time. We show that this Hamiltonian is finite, diffeomorphism invariant, and has a simple geometrical action confined to the intersections and the end points of the "loops". The quantum theory of fermions + gravity evolving in the clock time is finally given by the combinatorial and geometrical action of this Hamiltonian on a set of graphs with a finite number of end points. This geometrical action defines the topological Feynman rules of the theory.
The Immirzi parameter is a constant appearing in the general relativity action used as a starting... more The Immirzi parameter is a constant appearing in the general relativity action used as a starting point for the loop quantization of gravity. The parameter is commonly believed not to show up in the equations of motion, because it appears in front of a term in the action that vanishes on shell. We show that in the presence of fermions, instead, the Immirzi term in the action does not vanish on shell, and the Immirzi parameter does appear in the equations of motion. It determines the coupling constant of a four-fermion interaction. Therefore the Immirzi parameter leads to effects that are observable in principle, even independently from nonperturbative quantum gravity.
We consider the elementary radiative-correction terms in loop quantum gravity. These are a twover... more We consider the elementary radiative-correction terms in loop quantum gravity. These are a twovertex "elementary bubble" and a five-vertex "ball"; they correspond to the one-loop self-energy and the one-loop vertex correction of ordinary quantum field theory. We compute their naive degree of (infrared) divergence.
A new representation for quantum general relativity is described, which is defined in terms of fu... more A new representation for quantum general relativity is described, which is defined in terms of functionals of sets of loops in three-space. In this representation exact solutions of the quantum constraints may be obtained. This result is related to the simplification of the constraints in Ashtekar s new formalism. We give in closed form the general solution of the diff'eomorphism constraints and a large class of solutions of the full set of constraints. These are classified by the knot and link classes of the spatial three-manifold.
The Lorentzian "normalized balanced state sum model" of quantum general relativity is finite on a... more The Lorentzian "normalized balanced state sum model" of quantum general relativity is finite on any nondegenerate triangulation. It provides a candidate for a background independent, perturbatively finite, quantum theory of general relativity in four dimensions and with Lorentzian signature.
We study the graviton propagator in euclidean loop quantum gravity, using the spinfoam formalism.... more We study the graviton propagator in euclidean loop quantum gravity, using the spinfoam formalism. We use boundary-amplitude and group-field-theory techniques, and compute one component of the propagator to first order, under a number of approximations, obtaining the correct spacetime dependence. In the large distance limit, the only term of the vertex amplitude that contributes is the exponential of the Regge action: the other terms, that have raised doubts on the physical viability of the model, are suppressed by the phase of the vacuum state, which is determined by the extrinsic geometry of the boundary.
We argue that the statistical entropy relevant for the thermal interactions of a black hole with ... more We argue that the statistical entropy relevant for the thermal interactions of a black hole with its surroundings is (the logarithm of) the number of quantum microstates of the hole which are distinguishable from the hole's exterior, and which correspond to a given hole's macroscopic configuration. We compute this number explicitly from first principles, for a Schwarzschild black hole, using nonperturbative quantum gravity in the loop representation. We obtain a black hole entropy proportional to the area, as in the Bekenstein-Hawking formula.
We compute the transition amplitude between coherent quantum-states of geometry peaked on homogen... more We compute the transition amplitude between coherent quantum-states of geometry peaked on homogeneous isotropic metrics. We use the holomorphic representations of loop quantum gravity and the Kaminski-Kisielowski-Lewandowski generalization of the new vertex, and work at first order in the vertex expansion, second order in the graph (multipole) expansion, and first order in volume −1. We show that the resulting amplitude is in the kernel of a differential operator whose classical limit is the canonical hamiltonian of a Friedmann-Robertson-Walker cosmology. This result is an indication that the dynamics of loop quantum gravity defined by the new vertex yields the Friedmann equation in the appropriate limit.
We present a spinfoam formulation of Lorentzian quantum General Relativity. The theory is based o... more We present a spinfoam formulation of Lorentzian quantum General Relativity. The theory is based on a simple generalization of an Euclidean model defined in terms of a field theory over a group. The model is an extension of a recently introduced Lorentzian model, in which both timelike and spacelike components are included. The spinfoams in the model, corresponding to quantized 4-geometries, carry a natural non-perturbative local causal structure induced by the geometry of the algebra of the internal gauge (sl(2, C)). Amplitudes can be expressed as integrals over the spacelike unit-vectors hyperboloid in Minkowski space, or the imaginary Lobachevskian space.
HAL (Le Centre pour la Communication Scientifique Directe), Apr 12, 2001
We study a family of physical observable quantities in quantum gravity. We denote them W function... more We study a family of physical observable quantities in quantum gravity. We denote them W functions, or n-net functions. They represent transition amplitudes between quantum states of the geometry, are analogous to the n-point functions in quantum field theory, but depend on spin networks with n connected components. In particular, they include the three-geometry to three-geometry transition amplitude. The W functions are scalar under four-dimensional diffeomorphism, and fully gauge invariant. They capture the physical content of the quantum gravitational theory. We show that W functions are the natural n-point functions of the field theoretical formulation of the gravitational spin foam models. They can be computed from a perturbation expansion, which can be interpreted as a sum-over-four-geometries. Therefore the W functions bridge between the canonical (loop) and the covariant (spinfoam) formulations of quantum gravity. Following Wightman, the physical Hilbert space of the theory can be reconstructed from the W functions, if a suitable positivity condition is satisfied. We compute explicitly the W functions in a "free" model in which the interaction giving the gravitational vertex is shut off, and we show that, in this simple case, we have positivity, the physical Hilbert space of the theory can be constructed explicitly and the theory admits a well defined interpretation in terms of diffeomorphism invariant transition amplitudes between quantized geometries.
HAL (Le Centre pour la Communication Scientifique Directe), Apr 12, 2001
We study a family of physical observable quantities in quantum gravity. We denote them W function... more We study a family of physical observable quantities in quantum gravity. We denote them W functions, or n-net functions. They represent transition amplitudes between quantum states of the geometry, are analogous to the n-point functions in quantum field theory, but depend on spin networks with n connected components. In particular, they include the three-geometry to three-geometry transition amplitude. The W functions are scalar under four-dimensional diffeomorphism, and fully gauge invariant. They capture the physical content of the quantum gravitational theory. We show that W functions are the natural n-point functions of the field theoretical formulation of the gravitational spin foam models. They can be computed from a perturbation expansion, which can be interpreted as a sum-over-four-geometries. Therefore the W functions bridge between the canonical (loop) and the covariant (spinfoam) formulations of quantum gravity. Following Wightman, the physical Hilbert space of the theory can be reconstructed from the W functions, if a suitable positivity condition is satisfied. We compute explicitly the W functions in a "free" model in which the interaction giving the gravitational vertex is shut off, and we show that, in this simple case, we have positivity, the physical Hilbert space of the theory can be constructed explicitly and the theory admits a well defined interpretation in terms of diffeomorphism invariant transition amplitudes between quantized geometries.
Weaves are eigenstates of geometrical operators in nonperturbative quan- tum gravity, which appro... more Weaves are eigenstates of geometrical operators in nonperturbative quan- tum gravity, which approximate ¯ at space (or other smooth geometries) at large scales. We describe two such states, which diagonalize the area as well as the volume operators. The existence of such ...
We extend the definition of the "flipped" loop-quantum-gravity vertex to the case of a finite Imm... more We extend the definition of the "flipped" loop-quantum-gravity vertex to the case of a finite Immirzi parameter γ. We cover both the euclidean and lorentzian cases. We show that the resulting dynamics is defined on a Hilbert space isomorphic to the one of loop quantum gravity, and that the area operator has the same discrete spectrum as in loop quantum gravity. This includes the correct dependence on γ, and, remarkably, holds in the lorentzian case as well. The ad hoc flip of the symplectic structure that was required to derive the flipped vertex is not anymore required for finite γ. These results establish a bridge between canonical loop quantum gravity and the spinfoam formalism in four dimensions.
We study the quantum fermions + gravity system, that is, the gravitational counterpart of QED. We... more We study the quantum fermions + gravity system, that is, the gravitational counterpart of QED. We start from the standard Einstein-Weyl theory, reformulated in terms of Ashtekar variables; and we construct its non-perturbative quantum theory by extending the loop representation of general relativity. To this aim, we construct the fermion equivalent of the loop variables, and we define the quantum theory as a representation of their Poisson algebra. Not surprisingly, fermions can be accounted for in the loop representation simply by including open curves into loop space, as expected from lattice Yang-Mills theory. We explicitly construct the diffeomorphism and Hamiltonian operators. The first can be fully solved as in pure gravity. The second is constructed by using a background independent regularization technique. The theory retains the clean geometrical features of pure quantum gravity. In particular, the Hamiltonian constraint admits the same simple geometrical interpretation as its pure gravity counterpart: it is the operator that shifts curves along themselves (shift operator). Quite surprisingly, we believe, this simple action codes the full dynamics of the interacting fermion-gravity theory. To unravel the dynamics of the theory we study the evolution of the fermion-gravity system in the physical time defined by an additional coupled (clock-) scalar field. We explicitly construct the Hamiltonian operator that evolves the system in this physical time. We show that this Hamiltonian is finite, diffeomorphism invariant, and has a simple geometrical action confined to the intersections and the end points of the "loops". The quantum theory of fermions + gravity evolving in the clock time is finally given by the combinatorial and geometrical action of this Hamiltonian on a set of graphs with a finite number of end points. This geometrical action defines the topological Feynman rules of the theory.
The Immirzi parameter is a constant appearing in the general relativity action used as a starting... more The Immirzi parameter is a constant appearing in the general relativity action used as a starting point for the loop quantization of gravity. The parameter is commonly believed not to show up in the equations of motion, because it appears in front of a term in the action that vanishes on shell. We show that in the presence of fermions, instead, the Immirzi term in the action does not vanish on shell, and the Immirzi parameter does appear in the equations of motion. It determines the coupling constant of a four-fermion interaction. Therefore the Immirzi parameter leads to effects that are observable in principle, even independently from nonperturbative quantum gravity.
We consider the elementary radiative-correction terms in loop quantum gravity. These are a twover... more We consider the elementary radiative-correction terms in loop quantum gravity. These are a twovertex "elementary bubble" and a five-vertex "ball"; they correspond to the one-loop self-energy and the one-loop vertex correction of ordinary quantum field theory. We compute their naive degree of (infrared) divergence.
A new representation for quantum general relativity is described, which is defined in terms of fu... more A new representation for quantum general relativity is described, which is defined in terms of functionals of sets of loops in three-space. In this representation exact solutions of the quantum constraints may be obtained. This result is related to the simplification of the constraints in Ashtekar s new formalism. We give in closed form the general solution of the diff'eomorphism constraints and a large class of solutions of the full set of constraints. These are classified by the knot and link classes of the spatial three-manifold.
The Lorentzian "normalized balanced state sum model" of quantum general relativity is finite on a... more The Lorentzian "normalized balanced state sum model" of quantum general relativity is finite on any nondegenerate triangulation. It provides a candidate for a background independent, perturbatively finite, quantum theory of general relativity in four dimensions and with Lorentzian signature.
We study the graviton propagator in euclidean loop quantum gravity, using the spinfoam formalism.... more We study the graviton propagator in euclidean loop quantum gravity, using the spinfoam formalism. We use boundary-amplitude and group-field-theory techniques, and compute one component of the propagator to first order, under a number of approximations, obtaining the correct spacetime dependence. In the large distance limit, the only term of the vertex amplitude that contributes is the exponential of the Regge action: the other terms, that have raised doubts on the physical viability of the model, are suppressed by the phase of the vacuum state, which is determined by the extrinsic geometry of the boundary.
We argue that the statistical entropy relevant for the thermal interactions of a black hole with ... more We argue that the statistical entropy relevant for the thermal interactions of a black hole with its surroundings is (the logarithm of) the number of quantum microstates of the hole which are distinguishable from the hole's exterior, and which correspond to a given hole's macroscopic configuration. We compute this number explicitly from first principles, for a Schwarzschild black hole, using nonperturbative quantum gravity in the loop representation. We obtain a black hole entropy proportional to the area, as in the Bekenstein-Hawking formula.
We compute the transition amplitude between coherent quantum-states of geometry peaked on homogen... more We compute the transition amplitude between coherent quantum-states of geometry peaked on homogeneous isotropic metrics. We use the holomorphic representations of loop quantum gravity and the Kaminski-Kisielowski-Lewandowski generalization of the new vertex, and work at first order in the vertex expansion, second order in the graph (multipole) expansion, and first order in volume −1. We show that the resulting amplitude is in the kernel of a differential operator whose classical limit is the canonical hamiltonian of a Friedmann-Robertson-Walker cosmology. This result is an indication that the dynamics of loop quantum gravity defined by the new vertex yields the Friedmann equation in the appropriate limit.
We present a spinfoam formulation of Lorentzian quantum General Relativity. The theory is based o... more We present a spinfoam formulation of Lorentzian quantum General Relativity. The theory is based on a simple generalization of an Euclidean model defined in terms of a field theory over a group. The model is an extension of a recently introduced Lorentzian model, in which both timelike and spacelike components are included. The spinfoams in the model, corresponding to quantized 4-geometries, carry a natural non-perturbative local causal structure induced by the geometry of the algebra of the internal gauge (sl(2, C)). Amplitudes can be expressed as integrals over the spacelike unit-vectors hyperboloid in Minkowski space, or the imaginary Lobachevskian space.
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Papers by Carlo Rovelli