Quantum Koszul formula on quantum spacetime

JOURNAL OF GEOMETRY AND PHYSICS

卷:129

Article

Majid, Shahn¹  Williams, Liam¹ 

[1] Sch Math, Mile End Rd, London E1 4NS, England

关键词: Noncommutative geometry;    Quantum Riemannian geometry;    Quantum gravity;    Codifferential;    Central extension;    Differential graded algebra;   

DOI  :  10.1016/j.geomphys.2018.02.012

来源: Elsevier

【 摘 要 】

Noncommutative or quantum Riemannian geometry has been proposed as an effective theory for aspects of quantum gravity. Here the metric is an invertible bimodule map Omega(1)circle times(A)Omega(1)-> A where A is a possibly noncommutative or 'quantum' spacetime coordinate algebra and (Omega(1), d) is a specified bimodule of 1-forms or 'differential calculus' over it. In this paper we explore the proposal of a 'quantum Koszul formula' in Majid [12] with initial data a degree -2 bilinear map perpendicular to on the full exterior algebra Omega obeying the 4-term relations (-l)(|eta|)(omega eta) perpendicular to + (omega perpendicular to eta)? = omega perpendicular to (eta zeta) + (-1)(|omega|+|eta|)omega(eta perpendicular to zeta), for all omega, eta, zeta epsilon Omega and a compatible degree -1 'codifferential' map delta. These provide a quantum metric, interior product and a canonical bimodule connection on all degrees. The theory is also more general than classically in that we do not assume symmetry of the metric nor that delta is obtained from the metric. We solve and interpret the (delta, perpendicular to) data on the bicrossproduct model quantum spacetime [r, t] = lambda r for its two standard choices of Omega. For the alpha-family calculus the construction includes the quantum Levi-Civita connection for a general quantum symmetric metric, while for the more standard beta = 1 calculus we find the quantum Levi-Civita connection for a quantum 'metric' that in the classical limit is antisymmetric. This suggests to consider quantum Riemannian and symplectic geometry on a more equal footing than is currently the case. (c) 2018 Elsevier B.V. All rights reserved.

【 授权许可】

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