• I will derive a crucial property of loop quantum cosmology it shares with string/M-theory and asymptotically free quantum gravity theory, namely, that the associated Wigner-Moyal-Groenewold operator-formalism entails that the Holst-Barbero-Immirzi 4-spinfold has the property of spacetime uncertainty that I derived for string/M-theory, an essential property if loop quantum gravity is to be a valid quantum gravity theory. As I showed, in 4-D spacetime, the general relativistic starting point for canonical loop quantum gravity is given by:

where the dynamical variables are the tetrad one-form fields:

and the $SL\left( {2,\mathbb{C}} \right)$-valued connection $\omega _\mu ^{IJ}$ whose curvature is:

and is a connection on the holonomy-flux algebra for a homogeneous isotropic Friedmann–Lemaître–Robertson–Walker 'space'

Hence, we have the two-form:

with:

and ${\rm{Tr}}$ is the Killing form on the Lie algebra $SL\left( {2,\mathbb{C}} \right)$:

with

the totally antisymmetric tensor given by:

and from the Ashtekar variables, our action is:

• I will derive the following: the Nieh–Yan action, in the context of Barbero-Immirzi Hamiltonian analysis, allows the phase-space of General Relativity to be determined by Ashtekar-Barbero variables, and as to why this is deep and crucial for the viability of LQG is a topic for another post. Recall how the Barbero–Immirzi field action:

with

and

the torsion-less metric-compatible covariant derivative, induces contortion spin-connections by solving:

and hence:

generalizes to:

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

After varying the action with respect to the irreducible components of:

we obtain:

Inserting into: