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

Oct 27, 2022
43 pages
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Abstract: (arXiv)
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 (θ,A,κ)(\theta, A, \kappa), where θ\theta is the solder form/spin frame, AA is the spacetime Barbero--Immirzi connection, and κ\kappa is the extrinsic spacetime field. The Holst Lagrangian depends on the choice of a real, non zero Holst parameter γ≠0\gamma \neq 0 and constitutes the classical field theory which is then quantized in Loop Quantum Gravity. The choice of a real Immirzi parameter β\beta sets up a one-to-one correspondence between pairs (A,κ)(A, \kappa) and spin connections ω\omega on spacetime. The variation of the Barbero--Immirzi--Holst Lagrangian is computed for an arbitrary pair of parameters (β,γ)(\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 θ\theta are equivalent to the vacuum Einstein Field Equations, while the equations for AA and κ\kappa give the same constraint equation for any β∈R\beta \in \mathbb{R}, namely that A+κA + \kappa must be the Levi--Civita connection induced by θ\theta. We also prove that these results are valid for any value of γ≠0\gamma \neq 0, meaning that the choice of parameters (β,γ)(\beta, \gamma) has no impact on the classical theory in a vacuum and, in particular, there is no need to set β=γ\beta = \gamma.
Note:
  • 43 pages, 0 figures
  • field theory: classical
  • quantum gravity: loop space
  • space-time
  • spin
  • differential forms
  • Immirzi parameter
  • field equations
  • quantization
  • Einstein equation: vacuum
  • variational