R. Picken - Academia.edu (original) (raw)

Papers by R. Picken

Letters in Mathematical Physics, 2002

The moduli space of flat SL(2, R)-connections modulo gauge transformations on the torus may be de... more The moduli space of flat SL(2, R)-connections modulo gauge transformations on the torus may be described by ordered pairs of commuting SL(2, R) matrices modulo simultaneous conjugation by SL(2, R) matrices. Their spectral properties allow a classification of the equivalence classes, and a unique canonical form is given for each of these. In this way the moduli space becomes explicitly parametrized, and has a simple structure, resembling that of a cell complex, allowing it to be depicted. Finally, a Hausdorff topology based on this classification and parametrization is proposed.

Journal of Physics A: Mathematical and Theoretical, 2008

In the context of quantum gravity for spacetimes of dimension 2 + 1, we describe progress in the ... more In the context of quantum gravity for spacetimes of dimension 2 + 1, we describe progress in the construction of a quantum Goldman bracket for intersecting loops on surfaces. Using piecewise linear paths in R 2 (representing loops on the spatial manifold, i.e. the torus) and a quantum connection with noncommuting components, we review how holonomies and Wilson loops for two homotopic paths are related by phases in terms of the signed area between them. Paths rerouted at intersection points with other paths occur on the r.h.s. of the Goldman bracket. To better understand their nature we introduce the concept of integer points inside the parallelogram spanned by two intersecting paths, and show that the rerouted paths must necessarily pass through these integer points.

International Journal of Modern Physics A, 2009

In the context of (2+1)-dimensional gravity, we use holonomies of constant connections which gene... more In the context of (2+1)-dimensional gravity, we use holonomies of constant connections which generate a q-deformed representation of the fundamental group to derive signed area phases which relate the quantum matrices assigned to homotopic loops. We use these features to determine a quantum Goldman bracket (commutator) for intersecting loops on surfaces, and discuss the resulting quantum geometry.

The notion of quantum matrix pairs is defined. These are pairs of matrices with non-commuting ent... more The notion of quantum matrix pairs is defined. These are pairs of matrices with non-commuting entries, which have the same pattern of internal relations, q-commute with each other under matrix multiplication, and are such that products of powers of the matrices obey the same pattern of internal relations as the original pair. Such matrices appear in an approach by the authors to quantizing gravity in 2 space and 1 time dimensions with negative cosmological constant on the torus. Explicit examples and transformations which generate new pairs from a given pair are presented.

We describe an approach to the quantization of (2+1)-dimensional gravity with topology R × T2 and... more We describe an approach to the quantization of (2+1)-dimensional gravity with topology R × T2 and negative cosmological constant, which uses two quantum holonomy matrices satisfying a q-commutation relation. Solutions of diagonal and upper-triangular form are constructed, which in the latter case exhibit additional, non-trivial internal relations for each holonomy matrix. This leads to the notion of quantum matrix pairs.

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.

General Relativity and Gravitation, 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 quantum brackets (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.

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.

Arxiv preprint math-ph/0501051, 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 noncommuting 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.

Advances in Theoretical and Mathematical Physics, 2005

In the context of 2 + 1-dimensional quantum gravity with negative cosmological constant and topol... more In the context of 2 + 1-dimensional quantum gravity with negative cosmological constant and topology R×T 2 , constant matrix-valued connections generate a q-deformed representation of the fundamental group, and signed area phases relate the quantum matrices assigned to homotopic loops. Some features of the resulting quantum geometry are explored, and as a consequence a quantum version of the Goldman bracket is obtained. e-print archive:

Letters in Mathematical Physics, 2002

The moduli space of flat SL(2, R)-connections modulo gauge transformations on the torus may be de... more The moduli space of flat SL(2, R)-connections modulo gauge transformations on the torus may be described by ordered pairs of commuting SL(2, R) matrices modulo simultaneous conjugation by SL(2, R) matrices. Their spectral properties allow a classification of the equivalence classes, and a unique canonical form is given for each of these. In this way the moduli space becomes explicitly parametrized, and has a simple structure, resembling that of a cell complex, allowing it to be depicted. Finally, a Hausdorff topology based on this classification and parametrization is proposed.

Journal of Physics A: Mathematical and Theoretical, 2008

In the context of quantum gravity for spacetimes of dimension 2 + 1, we describe progress in the ... more In the context of quantum gravity for spacetimes of dimension 2 + 1, we describe progress in the construction of a quantum Goldman bracket for intersecting loops on surfaces. Using piecewise linear paths in R 2 (representing loops on the spatial manifold, i.e. the torus) and a quantum connection with noncommuting components, we review how holonomies and Wilson loops for two homotopic paths are related by phases in terms of the signed area between them. Paths rerouted at intersection points with other paths occur on the r.h.s. of the Goldman bracket. To better understand their nature we introduce the concept of integer points inside the parallelogram spanned by two intersecting paths, and show that the rerouted paths must necessarily pass through these integer points.

International Journal of Modern Physics A, 2009

In the context of (2+1)-dimensional gravity, we use holonomies of constant connections which gene... more In the context of (2+1)-dimensional gravity, we use holonomies of constant connections which generate a q-deformed representation of the fundamental group to derive signed area phases which relate the quantum matrices assigned to homotopic loops. We use these features to determine a quantum Goldman bracket (commutator) for intersecting loops on surfaces, and discuss the resulting quantum geometry.

The notion of quantum matrix pairs is defined. These are pairs of matrices with non-commuting ent... more The notion of quantum matrix pairs is defined. These are pairs of matrices with non-commuting entries, which have the same pattern of internal relations, q-commute with each other under matrix multiplication, and are such that products of powers of the matrices obey the same pattern of internal relations as the original pair. Such matrices appear in an approach by the authors to quantizing gravity in 2 space and 1 time dimensions with negative cosmological constant on the torus. Explicit examples and transformations which generate new pairs from a given pair are presented.

We describe an approach to the quantization of (2+1)-dimensional gravity with topology R × T2 and... more We describe an approach to the quantization of (2+1)-dimensional gravity with topology R × T2 and negative cosmological constant, which uses two quantum holonomy matrices satisfying a q-commutation relation. Solutions of diagonal and upper-triangular form are constructed, which in the latter case exhibit additional, non-trivial internal relations for each holonomy matrix. This leads to the notion of quantum matrix pairs.

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.

General Relativity and Gravitation, 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 quantum brackets (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.

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.

Arxiv preprint math-ph/0501051, 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 noncommuting 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.

Advances in Theoretical and Mathematical Physics, 2005

In the context of 2 + 1-dimensional quantum gravity with negative cosmological constant and topol... more In the context of 2 + 1-dimensional quantum gravity with negative cosmological constant and topology R×T 2 , constant matrix-valued connections generate a q-deformed representation of the fundamental group, and signed area phases relate the quantum matrices assigned to homotopic loops. Some features of the resulting quantum geometry are explored, and as a consequence a quantum version of the Goldman bracket is obtained. e-print archive: