Loop Quantum Cosmology and the Wigner-Moyal-Groenewold Phase Space

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:

    \[\begin{array}{l}{S_{4{\rm{D}}}}\left[ {e',\omega } \right] = \int_{\tilde M} {\left( {\frac{1}{2}} \right.} {\rm{tr}}\left( {e \wedge e \wedge F} \right)\\\left. { + \frac{1}{\gamma }{\rm{tr}}\left( {e \wedge e \wedge * F} \right)} \right)\end{array}\]

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

    \[{e^I} = e_\mu ^I{\rm{d}}{x^\mu }\]

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

    \[F = {\rm{d}}\omega + \omega \wedge '\omega \]

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

Hence, we have the two-form:

    \[\begin{array}{l}{F^{IJ}} = \left( {{{\not \partial }_\mu }} \right.\omega _\nu ^{IJ} - {{\not \partial }_\nu }\omega _\mu ^{IJ} + \omega _\mu ^{IK}{\omega _\nu }{K^J}\\\left. { - \omega _\nu ^{IK}{\omega _\mu }{K^J}} \right){\rm{d}}{x^\mu } \wedge '{\rm{d}}{x^\nu }\end{array}\]

with:

    \[ * {F^{IJ}} = \frac{1}{2}{\varepsilon ^{IJ}}_{KL}{F^{KL}}\]

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

    \[{\rm{Tr}}\left( {e \wedge e \wedge F} \right) = {\varepsilon _{IJKL}}{e^I} \wedge {e^J}{F^{KL}}\]

with

    \[{\varepsilon _{IJKL}}\]

the totally antisymmetric tensor given by:

    \[{\varepsilon ^{0123}} = + 1\]

Now, I can write down the Holst action more informatively:

    \[\begin{array}{*{20}{l}}{{S_{4D}}\left[ {e,\omega } \right] = \int_{{{\tilde M}_4}} {{{\rm{d}}^4}} x{\varepsilon ^{\mu \nu \rho \sigma }}\left( {\frac{1}{2}} \right.{\varepsilon _{IJKL}}}\\{e_\mu ^Ie_\nu ^JF_{\rho \sigma }^{KL}\left. { + \frac{1}{\gamma }e_\mu ^Ie_\nu ^J{F_{\rho \sigma }}_{IJ}} \right)}\end{array}\]

and from the Ashtekar variables, our action is:

    \[{{S_H} = \int {{d^3}} x\left\{ {{{\tilde E}^a}_B\dot A_a^B - \frac{1}{2}{\omega _{aBC}}{\varepsilon ^{BCD}}{t^a}{G_D} - {N^a}{C_a} - NH} \right\}}\]

    \[{\left\{ {A_a^B\left( x \right),\tilde E_A^b\left( y \right)} \right\} = \delta _a^b\delta _A^B\delta \left( {x,y} \right)}\]

Quantum Geometry, Emergence and Noncommutative Spacetime

I will show that any field theory that supervenes on noncommutative spacetime analytically leads to a quantum geometry embedding quantum gravity, and leads to an interpretation of classical Einsteinian spacetime and quantum gravity as emergent properties of symplectic noncommutative spacetime. Again, two good references are my last post and P. Aschieri.

I will start with matrix models, where T-duality reflects a change of dimensionality, given by:

    \[i{\not D_\mu } = i{\not \partial _\mu } + {A_\mu } \mathbin{\lower.3ex\hbox{$\buildrel\textstyle\rightarrow\over{\smash{\leftarrow}\vphantom{_{\vbox to.5ex{\vss}}}}$}} {\Phi ^a}\]

and applied to the U\left( {N \to \infty } \right) Yang-Mills theory in d dimensions:

    \[\begin{array}{l}{S_M} = - \frac{1}{{{G_s}}}\int {{d^d}} \tilde z\,{\rm{Tr}}\left( {\frac{1}{4}} \right.{F_{\mu \nu }}{F^{\mu \nu }} + \\\frac{1}{2}{{\not D}_\mu }{\Phi ^a}{{\not D}^\mu }{\Phi ^a} - \left. {\frac{1}{4}{{\left[ {{\Phi ^a},{\Phi ^b}} \right]}^2}} \right)\end{array}\]

we get the n-dimensional IKKT matrix model, in generic form:

    \[{S_{IKKT}} = - \frac{{2\pi }}{{{g_s}{\kappa ^2}}}{\rm{Tr}}\left( {\frac{1}{4}\left[ {{X^M},{X^N}} \right]\left[ {{X_M},{X_N}} \right]} \right)\]

and the n-dimensional BFSS matrix model in the case of the Lorentzian signature:

    \[{S_{BFSS}} = - \frac{1}{{{G_s}}}\int {dt} {\rm{Tr}}\left( {\frac{1}{2}{{\not D}_n}{\Phi ^a}{{\not D}^n}{\Phi ^a} - \frac{1}{4}{{\left[ {{\Phi ^a},{\Phi ^b}} \right]}^{n + 2}}} \right)\]

both of which, given the Morita-reduction to quantum mechanics, reflects the latter’s noncommutative phase space:

    \[\left[ {{x^a},{p_b}} \right] = i\hbar {\delta ^a}_b\]

leading to the Heisenberg’s uncertainty relation, and the large energy localization at Planck distances deform the background geometry according to the equivalence principle. This kind of backreaction of the background spacetime introduces a new kind of spacetime-uncertainties that reflect noncommutative spacetime,

thus, at Planck distances and ‘scales’, quantum mechanics necessarily implies noncommutative geometry

with space M derived from a symplectic structure