Barbero's Hamiltonian derived from a generalized Hilbert-Palatini action

Abstract

Barbero recently suggested a modification of Ashtekar's choice of canonical variables for general relativity. Although leading to a more complicated Hamiltonian constraint this modified version, in which the configuration variable still is a connection, has the advantage of being real. In this article we derive Barbero's Hamiltonian formulation from an action, which can be considered as a generalization of the ordinary Hilbert-Palatini action.Comment: 6 page

SO (4, C )-covariant Ashtekar-Barbero gravity and the Immirzi parameter

“…It is remarkable that the second class constraints (12) and (14) don't depend on β that proves consistency of the constraints in different formulations. One can calculate…”

“…In the work [12] Holst has shown that in the "time" gauge this action reproduces Barbero's formulation [8] and the parameter β plays the role of the Immirzi parameter. Our aim is to investigate the action (1) without imposing any gauge fixing.…”

Darkness: What comprises empty space?

“…The third term, widely known as the Holst term [20], is not topological. When written down in EinsteinCartan language, it bears a close kinship to the Einstein-Hilbert term in the action.…”

Immirzi parameter, torsion, and discrete symmetries

“…The coefficients at both are even functions of 1/γ, which means that they effectively arise via even powers of P -and T -odd term in (1). Besides, the coefficient at V 2 is proportional to α 2 , i.e.…”

“…This explains rather unexpected symmetry properties of the induced effective 4-fermion interaction.As has been demonstrated in [1], the gravitational action can be written asHere I, J = 0, 1, 2, 3 are internal Lorentz indices, µ, ν = 0, 1, 2, 3 are space-time indices; e I µ is the tetrad field, e is its determinant, and e µ I is the object inverse to e I µ ; the curvature tensor isω IJ µ is the connection. As to the so-called Immirzi parameter γ, the common belief is that it does not enter equations of motion since it appears as a factor at the term that vanishes on mass shell.…”