A discretization of Holst’s action for general relativity. (English) Zbl 1528.83021
Summary: We present a simplicial model for gravity written in terms of a discretized Lorentz connection and a discretized tetrad field. The continuum limit of its action is Holst’s action for general relativity. With the intention of using it to construct spin foam modes for quantum gravity, we write two other equivalent models written in terms of a discretized and constrained \(B\) field. The differences between our model and existing models are most likely inessential in the sense that a quantization would lead to equivalent quantum theories in the Wilsonian continuum limit. Nevertheless, we mention two features leading to possible advantages: Curvature degrees of freedom are described at the level of each 4-simplex. Our model offers a picture of bulk geometry leading to actions for matter couplings that split as a sum over 4-simplices.
MSC:
| 83C40 | Gravitational energy and conservation laws; groups of motions |
| 05E45 | Combinatorial aspects of simplicial complexes |
| 83C45 | Quantization of the gravitational field |
| 83C60 | Spinor and twistor methods in general relativity and gravitational theory; Newman-Penrose formalism |
| 65M22 | Numerical solution of discretized equations for initial value and initial-boundary value problems involving PDEs |
| 81T27 | Continuum limits in quantum field theory |
| 53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |
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