• 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)}

  • The Nieh–Yan Action and Barbero-Immirzi Hamiltonian Analysis

    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:

    \begin{array}{l}{S^{{\gamma _f}}} = - \frac{1}{2}\int {{d^4}{\,^{(4)}}} e\left[ {e_a^\mu } \right.e_b^\nu {\overline {{R_{\mu \nu }}} ^{ab}}\\ + \frac{{{\gamma _f}}}{2}{{\bar \nabla }_\mu }{S^\mu } + \frac{1}{{24}}{S_\mu }{S^\mu } - \\\frac{2}{3}{T_\mu }{T^\mu } + \frac{{{\gamma _f}}}{3}{T_\mu }{S^\mu } + \frac{1}{2}{q_{\mu \nu \rho }}{q^{\mu \nu \rho }}\\ + \frac{{{\gamma _f}}}{2}{\varepsilon _{\mu \nu e\sigma }}{q_\tau }^{\mu \rho }\left. {{q^{\tau \nu \sigma }}} \right]\end{array}

    with

    {S_\mu } \equiv {\varepsilon _{\nu \rho \sigma \mu }}{T^{\nu \rho \sigma }}

    and

    {\bar \nabla _\mu }

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

    {\gamma _f}

    and hence:

    {S^{{\gamma _f}}}

    generalizes to:

    \begin{array}{l}S_{NY}^{{\gamma _f}} = - \frac{1}{2}\int {{d^4}} x{\,^{(4)}}e\,e_a^\mu e_b^\nu {R_{\mu \nu }}^{ab} - \\\frac{1}{4}\int {{d^4}} x{\,^{(4)}}e{\gamma _f}\left( {{\eta _{ab}}} \right.T_{\mu \nu }^aT_{\rho \sigma }^a - \\e_a^\mu e_b^\nu \varepsilon _{cd}^{ab}\left. {{R_{\mu \nu }}^{cd}} \right)\end{array}

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

    \begin{array}{l}^\dagger S_{NY}^{{\gamma _f}} = - \frac{1}{2}\int {{d^4}} x{\,^{(4)}}e\left[ {e_a^\mu } \right.e_b^\nu {\overline {{R_{\mu \nu }}} ^{ab}}\\ + \frac{{{\gamma _f}}}{2}{{\bar \nabla }_\mu }{S^\mu } + \frac{1}{{24}}{S_\mu }{S^\mu } - \frac{1}{3}{T_\mu }{T^\mu }\\ + \frac{1}{2}{q_{\mu \nu \rho }}\left. {{q^{_{\mu \nu \rho }}}} \right]\end{array}

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

    \left\{ {\begin{array}{*{20}{c}}{{S^\mu }}\\{{T^\nu }}\\{{q^{\rho \sigma \tau }}}\end{array}} \right.

    we obtain:

    \left\{ {\begin{array}{*{20}{c}}{{\partial _\mu }{\gamma _f} - \frac{1}{6} - {S_\mu } = 0}\\{{T_\mu } = 0}\\{{q_{\mu \nu \rho }} = 0}\end{array}} \right.

    Inserting into:

    ^\dagger S_{NY}^{{\gamma _f}}