Barbero--Immirzi--Holst Lagrangian with Spacetime Barbero--Immirzi Connections

arXiv2210.15367MaRDI QIDQ6415304

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Publication date: 27 October 2022

Abstract: We carry out the complete variational analysis of the Barbero--Immirzi--Holst Lagrangian, which is the Holst Lagrangian expressed in terms of the triad of fields $(heta,A,kappa)$, where $heta$ is the solder form/spin frame, $A$ is the spacetime Barbero--Immirzi connection, and $kappa$ is the extrinsic spacetime field. The Holst Lagrangian depends on the choice of a real, non zero Holst parameter $gammaeq0$ and constitutes the classical field theory which is then quantized in Loop Quantum Gravity. The choice of a real Immirzi parameter $$ sets up a one-to-one correspondence between pairs $(A,kappa)$ and spin connections $omega$ on spacetime. The variation of the Barbero--Immirzi--Holst Lagrangian is computed for an arbitrary pair of parameters $$. We develop and use the calculus of vector-valued differential forms to improve on the results already present in literature by better clarifying the geometric character of the resulting Euler--Lagrange equations. The main result is that the equations for $heta$ are equivalent to the vacuum Einstein Field Equations, while the equations for $A$ and $kappa$ give the same constraint equation for any $$, namely that $A+kappa$ must be the Levi--Civita connection induced by $heta$. We also prove that these results are valid for any value of $gammaeq0$, meaning that the choice of parameters $$ has no impact on the classical theory in a vacuum and, in particular, there is no need to set $$.

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