Abstract
We investigate a formulation of continuum 4D gravity in terms of a constrained BF theory, in the spirit of the Plebanski formulation, but involving only linear constraints, of the type used recently in the spin foam approach to quantum gravity. We identify both the continuum version of the linear simplicity constraints used in the quantum discrete context and a linear version of the quadratic volume constraints that are necessary to complete the reduction from the topological theory to gravity. We also illustrate and discuss the discrete counterpart of the same continuum linear constraints. Moreover, we show under which additional conditions the discrete volume constraints follow from the simplicity constraints, thus playing the role of secondary constraints.
Highlights
The equations of general relativity can be derived from several different action principles [1], leading to equivalent classical theories
Among them we can mention, in addition to the Einstein-Hilbert action [2], the Palatini first order formulation and its modification proposed by Holst [3]. This last one is of special interest because it is the classical, covariant starting point for the canonical quantisation leading to Loop Quantum Gravity [4, 5]
The most recent developments in the spin foam and group field theory approach to quantum gravity are based on a linear set of discrete constraints, which can be shown to be slightly stronger, in the restrictions they impose on the original BF configurations, than some of the original discrete quadratic constraints
Summary
The equations of general relativity can be derived from several different action principles [1], leading to equivalent classical theories (in the case of pure gravity, at least). Among them we can mention, in addition to the Einstein-Hilbert action [2], the Palatini first order formulation and its modification proposed by Holst [3]. This last one is of special interest because it is the classical, covariant starting point for the canonical quantisation leading to Loop Quantum Gravity [4, 5]. The most recent developments in the spin foam and group field theory approach to quantum gravity are based on a linear set of discrete constraints, which can be shown to be slightly stronger, in the restrictions they impose on the original BF configurations, than some of the original discrete quadratic constraints.
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