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…

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