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- W4307536700 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 $(theta, A, kappa)$, 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 $gamma neq 0$ 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 $gamma neq 0$, 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 $beta = gamma$." @default.
- W4307536700 created "2022-11-03" @default.
- W4307536700 creator A5054798455 @default.
- W4307536700 date "2022-10-27" @default.
- W4307536700 modified "2023-09-26" @default.
- W4307536700 title "Barbero--Immirzi--Holst Lagrangian with Spacetime Barbero--Immirzi Connections" @default.
- W4307536700 doi "https://doi.org/10.48550/arxiv.2210.15367" @default.
- W4307536700 hasPublicationYear "2022" @default.
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