• Why the need for the Barbero-Immirzi field? Let me briefly explain. We saw that LQG in the Holst formulation faces the serious problem that unless the Barbero-Immirzi parameter is promoted to a field, the three-dimensional action with the Barbero-Immirzi parameter:

cannot admit a 4-D uplifting of the reduced 3-D gauge-free spacetime compactified action:

to the 4-D Holst-action:

and that is because the total 3-D action with the Barbero-Immirzi parameter:

is invariant under rescaling symmetry and translational symmetry, which destroy the time-gauge accessibility of the theory and 4-D-uplifting. Let us see whether and how promoting the Barbero-Immirzi parameter to a field and using the Nieh–Yan topological invariant can ameliorate our crises. In Lagrangian Holst theory, a Hilbert–Palatini action can always be generalized to contain the Holst term and promotes the Barbero–Immirzi parameter to a field via:

with:

the determinant of the LQG-tetrad, and:

being the Riemannian curvature corresponding to:

Clearly,

is not equivalent to a Hilbert-Palatini action, since the first Cartan equation is affected by the BI-field: a torsion trace contributes depending on the derivative of the BI-field. Letting

be the covariant Lorentz spin-valued connection-derivative, with spin-connection

and

the torsion tensor.

The Bianchi cyclic equation is then:

and so the Holst term does not vanish, thus

is a constant.

Now, given the trace vector:

and the identity:

it follows that:

reduces to:

with

and

is the torsion-less metric-compatible covariant derivative. By solving,

induces contortion spin-connections, and hence:

generalizes to:

Thus, the second integral is the Nieh-Yan topological invariant and connects to the Holst term, yielding

Now, one varies the action with respect to the irreducible components of:

to obtain:

Inserting into:

one gets the effective action:

giving us an equivalence with the Hilbert-Palatini torsion-free action and thus solving the gauge-free accessibility problem as well as the 4-D uplifting problem caused by invariance under rescaling symmetry and translational symmetry: the proof is straightforward.

• Since the phase-space has symplectic structure:

and

It thus follows that the total BI-field Hamiltonian:

with

the Lagrange multipliers, obeys:

where

is the Poisson bracket satisfying:

with

being the time-evolution of the BI-field and

an arbitrary field. Q.E.D…