Jeanette Nelson | Università degli Studi di Torino (original) (raw)

Papers by Jeanette Nelson

Research paper thumbnail of gr-qc/9311007 Equivalent Quantisations of (2+1)-Dimensional Gravity*

For spacetimes with the topology IR×T 2, the action of (2+1)-dimensional gravity with negative co... more For spacetimes with the topology IR×T 2, the action of (2+1)-dimensional gravity with negative cosmological constant Λ is written uniquely in terms of the time-independent traces of holonomies around two intersecting noncontractible paths on T 2. The holonomy parameters are related to the moduli on slices of constant mean curvature by a time-dependent canonical transformation which introduces an effective Hamiltonian. The quantisation of the two classically equivalent formulations differs by terms of order O(¯h 3), negligible for small |Λ|.

Research paper thumbnail of The canonical Hamiltonian for vierbein general relativity

Classical and Quantum Gravity, 1986

... very similar to that of the Hamiltonian constraint in the ADM formalism (Arnowitt et a1 1962)... more ... very similar to that of the Hamiltonian constraint in the ADM formalism (Arnowitt et a1 1962) who use metric variables g, given by (2.5) and corresponding momenta nr', and indeed also to the Hamiltonians of Deser and Isham (1976), Henneaux (1978) and Nelson and Teitelboim ...

Research paper thumbnail of Equivalent quantizations of (2+1)-dimensional gravity

We compare three approaches to the quantization of (2+1)dimensional gravity with a negative cosmo... more We compare three approaches to the quantization of (2+1)dimensional gravity with a negative cosmological constant: reduced phase space quantization with the York time slicing, quantization of the algebra of holonomies, and quantization of the space of classical solutions. The relationships among these quantum theories allow us to define and interpret time-dependent operators in the “frozen time ” holonomy formulation.

Research paper thumbnail of Quantum geometry from 2 + 1 AdS

Wilson observables for 2 + 1 quantum gravity with negative cosmological constant, when the spatia... more Wilson observables for 2 + 1 quantum gravity with negative cosmological constant, when the spatial manifold is a torus, exhibit several novel features: signed area phases relate the observables assigned to homotopic loops, and their commutators describe loop intersections, with properties that are not yet fully understood. We describe progress in our study of this bracket, which can be interpreted as a q-deformed Goldman bracket, and provide a geometrical interpretation in terms of a quantum version of Pick's formula for the area of a polygon with integer vertices.

Research paper thumbnail of Interest in the construction of a quantum bracket (commutator) for intersect- ing loops on surfaces arose from the study of quantum gravity for spacetimes

Gen Relativ Gravit, 2011

Wilson observables for 2 + 1 quantum gravity with negative cosmological constant, when the spatia... more Wilson observables for 2 + 1 quantum gravity with negative cosmological constant, when the spatial manifold is a torus, exhibit several novel features: signed area phases relate the observables assigned to homotopic loops, and their commutators describe loop intersections, with properties that are not yet fully understood. We describe progress in our study of this bracket, which can be interpreted as a q-deformed Goldman bracket, and provide a geometrical interpretation in terms of a quantum version of Pick's formula for the area of a polygon with integer vertices.

Research paper thumbnail of Quantum holonomies and the Heisenberg group

Modern Physics Letters A, 2019

Quantum holonomies of closed paths on the torus [Formula: see text] are interpreted as elements o... more Quantum holonomies of closed paths on the torus [Formula: see text] are interpreted as elements of the Heisenberg group [Formula: see text]. Group composition in [Formula: see text] corresponds to path concatenation and the group commutator is a deformation of the relator of the fundamental group [Formula: see text] of [Formula: see text], making explicit the signed area phases between quantum holonomies of homotopic paths. Inner automorphisms of [Formula: see text] adjust these signed areas, and the discrete symplectic transformations of [Formula: see text] generate the modular group of [Formula: see text].

Research paper thumbnail of Surface integrals and the gravitational action

Journal of Physics A: Mathematical and General, 1983

The authors discuss the modifications needed to free the Einstein-Hilbert action of gravitation f... more The authors discuss the modifications needed to free the Einstein-Hilbert action of gravitation from all second derivatives of fields, and give explicitly the resulting action applicable to either metric or vierbein variables. Variation of this action leads to Einstein's equations without boundary conditions. It vanishes for flat space-time and contains one arbitrary real parameter.

Research paper thumbnail of Explicit form of the constraint algebra in tetrad gravity

Classical and Quantum Gravity, 1988

... It is not our purpose to present an exhaustive review of tetrad gravity, but for completeness... more ... It is not our purpose to present an exhaustive review of tetrad gravity, but for completeness we note previous contributions in this field (Castellani et a1 1982, Deser and Isham 1976, Henneaux 1978, 1983, Lerda et a1 1985, Nelson and Regge 1986, Nelson and Teitelboim ...

Research paper thumbnail of Surface Integrals and the Gravitational Action

Research paper thumbnail of Homotopy Groups and 2+1 Dimansional Quantum Gravity

Research paper thumbnail of Classical and quantum geometry of moduli spaces in three-dimensional gravity

Proceedings of Xiii Fall Workshop on Geometry and Physics Murcia Spain September 20 22 2004 2005 Isbn 84 933610 6 2 Pags 104 114, Feb 20, 2005

We describe some results concerning the phase space of 3-dimensional Einstein gravity when space ... more We describe some results concerning the phase space of 3-dimensional Einstein gravity when space is a torus and with negative cosmological constant. The approach uses the holonomy matrices of flat SL(2,R) connections on the torus to parametrise the geometry. After quantization, these matrices acquire non-commuting entries, in such a way that they satisfy q-commutation relations and exhibit interesting geometrical properties. In particular they lead to a quantization of the Goldman bracket.

Research paper thumbnail of Lorentz Invariance in tangent space

Geometrical and Algebraic Aspects of Nonlinear Field Theory, 1989

Research paper thumbnail of Generalized Link-Invariants on 3-Manifolds 2;h x (0, 1) from Chern-Simons Gauge and Gravity Theories

Lett Math Phys, 1991

We show that the recently found generalized Jones and Homfly polynomials for links in ∑ h × [0, 1... more We show that the recently found generalized Jones and Homfly polynomials for links in ∑ h × [0, 1], where ∑ h is a closed oriented Riemann surface, may be also obtained by the canonical quantization of a Chern-Simons non-Abelian gauge theory on ∑ h × [0, 1]. As a particular case, one may consider the 2+1-dimensional Euclidean quantum gravity with a positive cosmological constant.

Research paper thumbnail of Canonical theories from the group manifolds

Research paper thumbnail of Constants of motion and the conformal anti-de Sitter algebra in (2+1)-Dimensional Gravity

International Journal of Modern Physics D

Constants of motion are calculated for 2+1 dimensional gravity with topology IR × T 2 and negativ... more Constants of motion are calculated for 2+1 dimensional gravity with topology IR × T 2 and negative cosmological constant. Certain linear combinations of them satisfy the anti -de Sitter algebra so(2, 2) in either ADM or holonomy variables. Quantisation is straightforward in terms of the holonomy parameters. On inclusion of the Hamiltonian three new global constants are derived and the quantum algebra extends to that of the conformal algebra so(2, 3). The modular group appears as a discrete subgroup of the conformal group. Its quantum action is generated by these conserved quantities.

Research paper thumbnail of The Constraints of 2+1 Quantum Gravity

The study of the gravitational field in 2+1 spacetime dimensions (2 space, 1 time) has blossomed ... more The study of the gravitational field in 2+1 spacetime dimensions (2 space, 1 time) has blossomed in the last few years into a substantial industry, after important contributions by Leutwyler [1], Deser, Jackiw and 'tHooft [2] and Witten .

Research paper thumbnail of Covariant canonical formalism for gravity and supergravity

The authors construct a covariant canonical formalism (CCF) for the group manifold, applied to gr... more The authors construct a covariant canonical formalism (CCF) for the group manifold, applied to gravity and supergravity. This CCF has no preferred time direction, and gives the first class Hamiltonian as a functional of the primary constraints, using a form/superform bracket. The authors establish the correspondence between the CCF and the canonical vierbein formalism (CVF).

Research paper thumbnail of Normal coordinates in canonical form

The Dirac (1958) formalism for constrained system is used to fix the coordinate gage as Riemann n... more The Dirac (1958) formalism for constrained system is used to fix the coordinate gage as Riemann normal coordinates. The transformations within the restricted set of metric in Riemann normal form are generated by the Dirac brackets. The canonical structure produces a detailed description of the gravitational field in the neighborhood of the origin of the metric tensor (O), but does not give a global account of the field on the whole mainfold Sigma.

Research paper thumbnail of Constants of the Motion and the Quantum Modular Group in (2+1) - Dimensional Gravity

Constants of motion are calculated for 2+1 dimensional gravity with topology IR × T 2 and negativ... more Constants of motion are calculated for 2+1 dimensional gravity with topology IR × T 2 and negative cosmological constant. Certain linear combinations of them satisfy the anti -de Sitter algebra so(2, 2) in either ADM or holonomy variables. Quantisation is straightforward in terms of the holonomy parameters, and the modular group is generated by these conserved quantities. On inclusion of the Hamiltonian three new global constants are derived and the quantum algebra extends to the conformal algebra so(2, 3).

Research paper thumbnail of Theory of intersecting loops on a torus

Advances in Theoretical and Mathematical Physics

We continue our investigation into intersections of closed paths on a torus, to further our under... more We continue our investigation into intersections of closed paths on a torus, to further our understanding of the commutator algebra of Wilson loop observables in 2+1 quantum gravity, when the cosmological constant is negative. We give a concise review of previous results, e.g. that signed area phases relate observables assigned to homotopic loops, and present new developments in this theory of intersecting loops on a torus. We state precise rules to be applied at intersections of both straight and crooked/rerouted paths in the covering space mathbbR2\mathbb{R}^2mathbbR2. Two concrete examples of combinations of different rules are presented.

Research paper thumbnail of gr-qc/9311007 Equivalent Quantisations of (2+1)-Dimensional Gravity*

For spacetimes with the topology IR×T 2, the action of (2+1)-dimensional gravity with negative co... more For spacetimes with the topology IR×T 2, the action of (2+1)-dimensional gravity with negative cosmological constant Λ is written uniquely in terms of the time-independent traces of holonomies around two intersecting noncontractible paths on T 2. The holonomy parameters are related to the moduli on slices of constant mean curvature by a time-dependent canonical transformation which introduces an effective Hamiltonian. The quantisation of the two classically equivalent formulations differs by terms of order O(¯h 3), negligible for small |Λ|.

Research paper thumbnail of The canonical Hamiltonian for vierbein general relativity

Classical and Quantum Gravity, 1986

... very similar to that of the Hamiltonian constraint in the ADM formalism (Arnowitt et a1 1962)... more ... very similar to that of the Hamiltonian constraint in the ADM formalism (Arnowitt et a1 1962) who use metric variables g, given by (2.5) and corresponding momenta nr', and indeed also to the Hamiltonians of Deser and Isham (1976), Henneaux (1978) and Nelson and Teitelboim ...

Research paper thumbnail of Equivalent quantizations of (2+1)-dimensional gravity

We compare three approaches to the quantization of (2+1)dimensional gravity with a negative cosmo... more We compare three approaches to the quantization of (2+1)dimensional gravity with a negative cosmological constant: reduced phase space quantization with the York time slicing, quantization of the algebra of holonomies, and quantization of the space of classical solutions. The relationships among these quantum theories allow us to define and interpret time-dependent operators in the “frozen time ” holonomy formulation.

Research paper thumbnail of Quantum geometry from 2 + 1 AdS

Wilson observables for 2 + 1 quantum gravity with negative cosmological constant, when the spatia... more Wilson observables for 2 + 1 quantum gravity with negative cosmological constant, when the spatial manifold is a torus, exhibit several novel features: signed area phases relate the observables assigned to homotopic loops, and their commutators describe loop intersections, with properties that are not yet fully understood. We describe progress in our study of this bracket, which can be interpreted as a q-deformed Goldman bracket, and provide a geometrical interpretation in terms of a quantum version of Pick's formula for the area of a polygon with integer vertices.

Research paper thumbnail of Interest in the construction of a quantum bracket (commutator) for intersect- ing loops on surfaces arose from the study of quantum gravity for spacetimes

Gen Relativ Gravit, 2011

Wilson observables for 2 + 1 quantum gravity with negative cosmological constant, when the spatia... more Wilson observables for 2 + 1 quantum gravity with negative cosmological constant, when the spatial manifold is a torus, exhibit several novel features: signed area phases relate the observables assigned to homotopic loops, and their commutators describe loop intersections, with properties that are not yet fully understood. We describe progress in our study of this bracket, which can be interpreted as a q-deformed Goldman bracket, and provide a geometrical interpretation in terms of a quantum version of Pick's formula for the area of a polygon with integer vertices.

Research paper thumbnail of Quantum holonomies and the Heisenberg group

Modern Physics Letters A, 2019

Quantum holonomies of closed paths on the torus [Formula: see text] are interpreted as elements o... more Quantum holonomies of closed paths on the torus [Formula: see text] are interpreted as elements of the Heisenberg group [Formula: see text]. Group composition in [Formula: see text] corresponds to path concatenation and the group commutator is a deformation of the relator of the fundamental group [Formula: see text] of [Formula: see text], making explicit the signed area phases between quantum holonomies of homotopic paths. Inner automorphisms of [Formula: see text] adjust these signed areas, and the discrete symplectic transformations of [Formula: see text] generate the modular group of [Formula: see text].

Research paper thumbnail of Surface integrals and the gravitational action

Journal of Physics A: Mathematical and General, 1983

The authors discuss the modifications needed to free the Einstein-Hilbert action of gravitation f... more The authors discuss the modifications needed to free the Einstein-Hilbert action of gravitation from all second derivatives of fields, and give explicitly the resulting action applicable to either metric or vierbein variables. Variation of this action leads to Einstein's equations without boundary conditions. It vanishes for flat space-time and contains one arbitrary real parameter.

Research paper thumbnail of Explicit form of the constraint algebra in tetrad gravity

Classical and Quantum Gravity, 1988

... It is not our purpose to present an exhaustive review of tetrad gravity, but for completeness... more ... It is not our purpose to present an exhaustive review of tetrad gravity, but for completeness we note previous contributions in this field (Castellani et a1 1982, Deser and Isham 1976, Henneaux 1978, 1983, Lerda et a1 1985, Nelson and Regge 1986, Nelson and Teitelboim ...

Research paper thumbnail of Surface Integrals and the Gravitational Action

Research paper thumbnail of Homotopy Groups and 2+1 Dimansional Quantum Gravity

Research paper thumbnail of Classical and quantum geometry of moduli spaces in three-dimensional gravity

Proceedings of Xiii Fall Workshop on Geometry and Physics Murcia Spain September 20 22 2004 2005 Isbn 84 933610 6 2 Pags 104 114, Feb 20, 2005

We describe some results concerning the phase space of 3-dimensional Einstein gravity when space ... more We describe some results concerning the phase space of 3-dimensional Einstein gravity when space is a torus and with negative cosmological constant. The approach uses the holonomy matrices of flat SL(2,R) connections on the torus to parametrise the geometry. After quantization, these matrices acquire non-commuting entries, in such a way that they satisfy q-commutation relations and exhibit interesting geometrical properties. In particular they lead to a quantization of the Goldman bracket.

Research paper thumbnail of Lorentz Invariance in tangent space

Geometrical and Algebraic Aspects of Nonlinear Field Theory, 1989

Research paper thumbnail of Generalized Link-Invariants on 3-Manifolds 2;h x (0, 1) from Chern-Simons Gauge and Gravity Theories

Lett Math Phys, 1991

We show that the recently found generalized Jones and Homfly polynomials for links in ∑ h × [0, 1... more We show that the recently found generalized Jones and Homfly polynomials for links in ∑ h × [0, 1], where ∑ h is a closed oriented Riemann surface, may be also obtained by the canonical quantization of a Chern-Simons non-Abelian gauge theory on ∑ h × [0, 1]. As a particular case, one may consider the 2+1-dimensional Euclidean quantum gravity with a positive cosmological constant.

Research paper thumbnail of Canonical theories from the group manifolds

Research paper thumbnail of Constants of motion and the conformal anti-de Sitter algebra in (2+1)-Dimensional Gravity

International Journal of Modern Physics D

Constants of motion are calculated for 2+1 dimensional gravity with topology IR × T 2 and negativ... more Constants of motion are calculated for 2+1 dimensional gravity with topology IR × T 2 and negative cosmological constant. Certain linear combinations of them satisfy the anti -de Sitter algebra so(2, 2) in either ADM or holonomy variables. Quantisation is straightforward in terms of the holonomy parameters. On inclusion of the Hamiltonian three new global constants are derived and the quantum algebra extends to that of the conformal algebra so(2, 3). The modular group appears as a discrete subgroup of the conformal group. Its quantum action is generated by these conserved quantities.

Research paper thumbnail of The Constraints of 2+1 Quantum Gravity

The study of the gravitational field in 2+1 spacetime dimensions (2 space, 1 time) has blossomed ... more The study of the gravitational field in 2+1 spacetime dimensions (2 space, 1 time) has blossomed in the last few years into a substantial industry, after important contributions by Leutwyler [1], Deser, Jackiw and 'tHooft [2] and Witten .

Research paper thumbnail of Covariant canonical formalism for gravity and supergravity

The authors construct a covariant canonical formalism (CCF) for the group manifold, applied to gr... more The authors construct a covariant canonical formalism (CCF) for the group manifold, applied to gravity and supergravity. This CCF has no preferred time direction, and gives the first class Hamiltonian as a functional of the primary constraints, using a form/superform bracket. The authors establish the correspondence between the CCF and the canonical vierbein formalism (CVF).

Research paper thumbnail of Normal coordinates in canonical form

The Dirac (1958) formalism for constrained system is used to fix the coordinate gage as Riemann n... more The Dirac (1958) formalism for constrained system is used to fix the coordinate gage as Riemann normal coordinates. The transformations within the restricted set of metric in Riemann normal form are generated by the Dirac brackets. The canonical structure produces a detailed description of the gravitational field in the neighborhood of the origin of the metric tensor (O), but does not give a global account of the field on the whole mainfold Sigma.

Research paper thumbnail of Constants of the Motion and the Quantum Modular Group in (2+1) - Dimensional Gravity

Constants of motion are calculated for 2+1 dimensional gravity with topology IR × T 2 and negativ... more Constants of motion are calculated for 2+1 dimensional gravity with topology IR × T 2 and negative cosmological constant. Certain linear combinations of them satisfy the anti -de Sitter algebra so(2, 2) in either ADM or holonomy variables. Quantisation is straightforward in terms of the holonomy parameters, and the modular group is generated by these conserved quantities. On inclusion of the Hamiltonian three new global constants are derived and the quantum algebra extends to the conformal algebra so(2, 3).

Research paper thumbnail of Theory of intersecting loops on a torus

Advances in Theoretical and Mathematical Physics

We continue our investigation into intersections of closed paths on a torus, to further our under... more We continue our investigation into intersections of closed paths on a torus, to further our understanding of the commutator algebra of Wilson loop observables in 2+1 quantum gravity, when the cosmological constant is negative. We give a concise review of previous results, e.g. that signed area phases relate observables assigned to homotopic loops, and present new developments in this theory of intersecting loops on a torus. We state precise rules to be applied at intersections of both straight and crooked/rerouted paths in the covering space mathbbR2\mathbb{R}^2mathbbR2. Two concrete examples of combinations of different rules are presented.