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Gray categories with duals and their diagrams (2024)
Journal Article
Barrett, J. W., Meusburger, C., & Schaumann, G. (2024). Gray categories with duals and their diagrams. Advances in Mathematics, 450, Article 109740. https://doi.org/10.1016/j.aim.2024.109740

The geometric and algebraic properties of Gray categories with duals are investigated by means of a diagrammatic calculus. The diagrams are three-dimensional stratifications of a cube, with regions, surfaces, lines and vertices labelled by Gray categ... Read More about Gray categories with duals and their diagrams.

Spectral estimators for finite non-commutative geometries (2019)
Journal Article
Barrett, J. W., Druce, P., & Glaser, L. (2019). Spectral estimators for finite non-commutative geometries. Journal of Physics A: Mathematical and Theoretical, 52(27), https://doi.org/10.1088/1751-8121/ab22f8

A finite non-commutative geometry consists of a fuzzy space together with a Dirac operator satisfying the axioms of a real spectral triple. This paper addreses the question of how to extract information about these geometries from the spectrum of the... Read More about Spectral estimators for finite non-commutative geometries.

Dichromatic state sum models for four-manifolds from pivotal functors (2017)
Journal Article
Bärenz, M., & Barrett, J. W. (2018). Dichromatic state sum models for four-manifolds from pivotal functors. Communications in Mathematical Physics, 360(2), https://doi.org/10.1007/s00220-017-3012-9

A family of invariants of smooth, oriented four-dimensional manifolds is defined via handle decompositions and the Kirby calculus of framed link diagrams. The invariants are parametrised by a pivotal functor from a spherical fusion category into a ri... Read More about Dichromatic state sum models for four-manifolds from pivotal functors.

Monte Carlo simulations of random non-commutative geometries (2016)
Journal Article
Barrett, J. W., & Glaser, L. (2016). Monte Carlo simulations of random non-commutative geometries. Journal of Physics A: Mathematical and Theoretical, 49(24), Article 245001. https://doi.org/10.1088/1751-8113/49/24/245001

© 2016 IOP Publishing Ltd. Random non-commutative geometries are introduced by integrating over the space of Dirac operators that form a spectral triple with a fixed algebra and Hilbert space. The cases with the simplest types of Clifford algebra are... Read More about Monte Carlo simulations of random non-commutative geometries.

Two-dimensional state sum models and spin structures (2014)
Journal Article
Barrett, J. W., & Tavares, S. O. G. (2014). Two-dimensional state sum models and spin structures. Communications in Mathematical Physics, 336(1), 63-100. https://doi.org/10.1007/s00220-014-2246-z

The state sum models in two dimensions introduced by Fukuma, Hosono and Kawai are generalised by allowing algebraic data from a non-symmetric Frobenius algebra. Without any further data, this leads to a state sum model on the sphere. When the data is... Read More about Two-dimensional state sum models and spin structures.

Integrability for Relativistic Spin Networks (2001)
Journal Article
Barrett, J. W., & Baez, J. C. (2001). Integrability for Relativistic Spin Networks. Classical and Quantum Gravity, 18(4683-4),

The evaluation of relativistic spin networks plays a fundamental role in the Barrett-Crane state sum model of Lorentzian quantum gravity in 4 dimensions. A relativistic spin network is a graph labelled by unitary irreducible representations of the Lo... Read More about Integrability for Relativistic Spin Networks.

A Lorentzian Signature Model for Quantum General Relativity (2000)
Journal Article
Barrett, J. W., & Crane, L. (2000). A Lorentzian Signature Model for Quantum General Relativity

We give a relativistic spin network model for quantum gravity based on the Lorentz group and its q-deformation, the Quantum Lorentz Algebra.
We propose a combinatorial model for the path integral given by an integral over suitable representations of... Read More about A Lorentzian Signature Model for Quantum General Relativity.

The asymptotics of an amplitude for the 4-simplex (1999)
Journal Article
Barrett, J. W., & Williams, R. M. (1999). The asymptotics of an amplitude for the 4-simplex. https://doi.org/10.4310/atmp.1999.v3.n2.a1

An expression for the oscillatory part of an asymptotic formula for the relativistic spin network amplitude for a 4-simplex is given. The amplitude depends on specified areas for each two-dimensional face in the 4-simplex. The asymptotic formula has... Read More about The asymptotics of an amplitude for the 4-simplex.

The classical evaluation of relativistic spin networks (1998)
Journal Article
Barrett, J. W. (1998). The classical evaluation of relativistic spin networks. https://doi.org/10.4310/atmp.1998.v2.n3.a7

The evaluation of a relativistic spin network for the classical case of the Lie group SU(2) is given by an integral formula over copies of SU(2). For the graph determined by a 4-simplex this gives the evaluation as an integral over a space of geometr... Read More about The classical evaluation of relativistic spin networks.

State sum models for quantum gravity
Journal Article
Barrett, J. W. State sum models for quantum gravity

This paper reviews the construction of quantum field theory on a 4-dimensional spacetime by combinatorial methods, and discusses the recent developments in the direction of a combinatorial construction of quantum gravity.

Unlinked Embedded Graphs
Journal Article
Barrett, J. W. Unlinked Embedded Graphs

This paper is a self-contained development of an invariant of graphs embedded in three-dimensional Euclidean space using the Jones polynomial and skein theory. Some examples of the invariant are computed. An unlinked embedded graph is one that contai... Read More about Unlinked Embedded Graphs.