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)}

Space-Time Uncertainty and Non-Locality in String-Theory

Among the many truly remarkable properties of M-theory, that it is a unified theory of all interactions, including quantum gravity, and gives a completely well-defined analytic S-matrix satisfying all the axioms for a physically acceptable theory entailing Lorentz invariance, macro-causality and unitarity is perhaps the deepest, and to boot, the only quantum gravity paradigm that has that essential feature. Here, I will discuss some key aspects of nonlocality and space-time uncertainty in string theory. Let us start with an action smoothly interpolating between the area preserving Schild action and the fully reparametrization invariant Nambu–Goto action:

I\left[ {\Phi ,X} \right] \equiv \frac{{{\mu _0}}}{2}\int\limits_\Sigma {{d^2}} \sigma \left[ {\frac{{\det \left( {{\gamma _{mn}}} \right)}}{{\Phi \left( \sigma \right)}} + \Phi \left( \sigma \right)} \right]

where \Phi \left( \sigma \right) is an auxiliary world-sheet field, {\gamma _{mn}} \equiv {\eta _{\mu \nu }}{\partial _m}{X^\mu }{\partial _n}{X^\nu } the induced metric on the string Euclidean world-sheet {x^\mu } = {X^\mu }\left( \sigma \right), and {\mu _0} \equiv 1/2\pi \alpha ' is the string tension. Combining, we get the Nambu-Goto-Schild action:

{S_{ngs}} = - \int\limits_\Sigma {{d^2}} \xi \left\{ {\frac{1}{e}\left[ { - \frac{1}{{2{{\left( {4\pi \alpha '} \right)}^2}}}{{\left( {{\varepsilon ^{ab}}{\partial _a}{X^\mu }{\partial _b}{X^\nu }} \right)}^2}} \right] + e} \right\}

And to make the Nambu-Goto-Schild action quadratic in space-time coordinates, we use the Virasoro constraint and an auxiliary field that transforms as a world-sheet scalar and as an anti-symmetric tensor with respect to the space-time indices:

\left\{ {\begin{array}{*{20}{c}}{{b_{\mu \nu }}\left( \xi \right)}\\{{P^2} + \frac{1}{{4\pi \alpha '}}{{\hat X}^2} = 0,\;P \cdot \hat X = 0}\end{array}} \right.

to yield:

Before proceeding, let us get some clarity.