Introduction to loop quantum gravity. The Holst’s action and the covariant formalism

Abstract

We review Holst formalism and dynamical equivalence with standard GR (in dimension $4$). Holst formalism is written for a spin coframe field $e_{\mu }^I$ and a Spin(3,1)-connection ${\omega }_{\mu }^{IJ}$ on spacetime $M$ and it depends on the Holst parameter$\gamma \in ℝ-\{0\}$. We show the model is dynamically equivalent to standard GR, in the sense that up to a pointwise Spin(3,1)-gauge transformation acting on (uppercase Latin) frame indices, solutions of the two models are in one-to-one correspondence. Hence, the two models are classically equivalent. One can also introduce new variables by splitting the spin connection into a pair of a Spin(3)-connection $A_{\mu }^i$ and a Spin(3)-valued 1-form $k_{\mu }^i$. The construction of these new variables relies on a particular algebraic structure, called a reductive splitting. A weaker structure than requiring that the gauge group splits as the products of two sub-groups, as it happens in Euclidean signature in the selfdual formulation originally introduced in this context by Ashtekar, and it still allows to deal with the Lorentzian signature without resorting to complexifications. The reductive splitting of $\text{SL}(2,ℂ)$ is not unique and it is parameterized by a real parameter $\beta$ which is called the Immirzi parameter. The splitting is here done on spacetime, not on space as it is usually done in the literature, to obtain a Spin(3)-connection $A_{\mu }^i$, which is called the Barbero–Immirzi connection on spacetime. One obtains a covariant model depending on the fields $(e_{\mu }^I,A_{\mu }^i,k_{\mu }^i)$ which is again dynamically equivalent to standard GR. Usually in the literature one sets $\beta =\gamma$ for the sake of simplicity. Here, we keep the Holst and Immirzi parameters distinct to show that eventually only $\beta$ will survive in boundary field equations.