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| {{main|tetradic Palatini action|frame field}}
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| The '''Holst action'''<ref name="Holst">
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| {{cite journal | last = Holst | first = Sören | authorlink = Sören Holst | date = 15 May 1996 | title = Barbero's Hamiltonian derived from a generalized Hilbert-Palatini action | journal = Physical Review | volume = 53 | issue = 10 | series = D | pages = 5966–5969
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| | doi = 10.1103/PhysRevD.53.5966 | url = http://link.aps.org/doi/10.1103/PhysRevD.53.5966 |archiveurl = http://arxiv.org/abs/gr-qc/9511026 | archivedate = 7 Nov 1995|arxiv = gr-qc/9511026 |bibcode = 1996PhRvD..53.5966H }}
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| </ref> is an equivalent formulation of the [[Palatini action]] for [[General Relativity]] (GR) in terms of [[vierbein]]s (4D space-time frame field) by adding a part of a topological term (Nieh-Yan) which does not alter the classical equations of motion as long as there is no [[torsion tensor|torsion]],
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| :<math>S = \frac{1}{2} \int e e^{\alpha}_{\ I} e^{\beta}_{\ J} | |
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| (F_{\alpha\beta}^{\ \ \ IJ} - \alpha \ast F_{\alpha\beta}^{\ \ \ IJ})
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| \equiv \frac{1}{2} \int e e^{\alpha}_{\ I} e^{\beta}_{\ J}
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| (F_{\alpha\beta}^{\ \ \ IJ} - \frac{\alpha}{2} \epsilon^{IJ}_{\;\;\;KL}
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| F_{\alpha\beta}^{\ \ \ KL})</math>
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| where <math>e^{\alpha}_{\ I}</math> is the tetrad, <math>e</math> its determinant, <math>F_{\alpha\beta}^{\ \ \ IJ}</math> the curvature considered as a function of the connection <math>A_{\alpha\beta}^{\ \ \ IJ}</math>, <math>\alpha</math> a (complex) parameter, and where we recover the Palatini action when <math>\alpha = 0</math>. It only works in 4D. To be torsion free means the [[covariant derivative]] defined by the connection <math>A_{\alpha\beta}^{\ \ \ IJ}</math> when acting on the Minkowski metric <math>\eta_{IJ}</math> vanishes, implying the connection is anti-symmetric in its internal indices <math>I,J</math>.
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| As with the first order tetradic Palatini action where <math>e^{\alpha}_{\ I}</math> and <math>A_{\alpha\beta}^{\ \ \ IJ}</math> are taken to be independent variables, variation of the action with respect to the connection <math>A_{\alpha\beta}^{\ \ \ IJ}</math> (assuming it to be torsion-free) implies the curvature <math>F_{\alpha\beta}^{\ \ \ IJ}</math> be replaced by the usual (mixed index) curvature tensor <math>R_{\alpha\beta}^{\ \ \ IJ}</math> (see article [[tetradic Palatini action]] for definitions). Variation of the first term of the action with respect to the tetrad <math>e^{\alpha}_{\ I}</math> gives the (mixed index) [[Einstein tensor]] and variation of the second term with respect to the tetrad gives a quantity that vanishes by symmetries of the [[Riemann tensor]] (specifically the first Bianchi identity), together these imply Einstein's vacuum field equations hold.
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| == Applications ==
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| The canonical 3+1 Hamiltonian formulation of the Holst action with <math>\alpha = i</math> happens to correspond to [[Ashtekar variables]] which formulates (complex) 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 (see article [[self-dual Palatini action]]). | |
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| The canonical 3+1 Hamiltonian formulation of the Holst action for real <math>\alpha</math> 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 (so we are dealing with real General Relativity). This Hamiltonian formulation is the classical starting point of [[loop quantum gravity]] (LQG)<ref name="Holst"/> which imports non-perturbative techniques from [[lattice gauge theory]].<ref>Modern Canonical Quantum General Relativity by Thomas Thiemann</ref> The parameter defined by <math>\beta := 1 / \alpha</math> is usually referred to as the Barbero-Immirzi parameter<ref>''Real Ashtekar Variables for Lorentzian Signature Space-times'', J. Fernando, G. Barbero, Phys.Rev.D51:5507-5510,1995.</ref><ref>''Real and complex connections for canonical gravity'', Giorgio Immirzi, Class.Quant.Grav.14:L177-L181,1997.</ref> The Holst action finds application in most recent versions of [[spin foam]] models,<ref>Engle, J., Pereira, R., and Rovelli, C. (2009). Phys. Rev. Lett. 99, 161301.</ref><ref>Freidal, L. and Krasnov, K. (2008) ''Clas. Quan. Grav.'' 25, 125018.</ref> which can be considered [[path integral]] versions of LQG.
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| == References ==
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| <references /> | |
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| [[Category:General relativity]]
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