Holst action


In the field of theoretical physics, the Holst action is an equivalent formulation of the Palatini action for General Relativity in terms of vierbeins by adding a part of a topological term which does not alter the classical equations of motion as long as there is no torsion,
where is the tetrad, its determinant, the curvature considered as a function of the connection :
a parameter, and where we recover the Palatini action when. It only works in 4D. To be torsion free means the covariant derivative defined by the connection when acting on the Minkowski metric vanishes, implying the connection is anti-symmetric in its internal indices.
As with the first order tetradic Palatini action where and are taken to be independent variables, variation of the action with respect to the connection implies the curvature be replaced by the usual curvature tensor . Variation of the first term of the action with respect to the tetrad gives the Einstein tensor and variation of the second term with respect to the tetrad gives a quantity that vanishes by symmetries of the Riemann tensor, together these imply Einstein's vacuum field equations hold.

Applications

The canonical 3+1 Hamiltonian formulation of the Holst action with happens to correspond to Ashtekar variables which formulates GR as a special type of Yang-Mills gauge theory. The action was seen simply to be the Palatini action with the curvature tensor replaced by its self-dual part only.
The canonical 3+1 Hamiltonian formulation of the Holst action for real was shown to have a configuration variable which is still a connection, and the theory still a special kind of Yang-Mills gauge theory, but has the advantage that it is real, as is then the corresponding gauge theory. This Hamiltonian formulation is the classical starting point of loop quantum gravity which imports non-perturbative techniques from lattice gauge theory. The parameter defined by is usually referred to as the Barbero-Immirzi parameter The Holst action finds application in most recent versions of spin foam models, which can be considered path integral versions of LQG.