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

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 $(\theta ,A,\kappa )$, where $\theta$ 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 $\gamma \mathrm{\ne }0$ and constitutes the classical field theory which is then quantized in Loop Quantum Gravity. The choice of a real Immirzi parameter $\beta$ 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 $(\beta ,\gamma )$. 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 $\theta$ are equivalent to the vacuum Einstein Field Equations, while the equations for $A$ and $\kappa$ give the same constraint equation for any $\beta \in \mathbb{R}$, namely that $A+\kappa$ must be the Levi–Civita connection induced by $\theta$. We also prove that these results are valid for any value of $\gamma \mathrm{\ne }0$, meaning that the choice of parameters $(\beta ,\gamma )$ has no impact on the classical theory in a vacuum and, in particular, there is no need to set $\beta =\gamma$.