• 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:

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

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

    with the Gaussian constraint:

        \[{G_A} = {D_a}{\tilde E^a}A\]

    the diffeomorphism constraint:

        \[{C_a} = {\tilde E^b}{F_{ab}}^A + \frac{{\left( {1 + {\gamma ^2}} \right)}}{\gamma }{K_a}^A{G_A}\]

    and our Hamiltonian is given by:

        \[H = \frac{{8\pi G{\gamma ^2}}}{{\sqrt {\left| {\det \left( q \right)} \right|} }}{\tilde E^a}_A{\tilde E^b}_B\left[ {{\varepsilon ^{AB}}_C{F_{ab}}^C - \frac{{2\left( {1 + {\gamma ^2}} \right)}}{\gamma }{K_a}^A{K_b}^B} \right]\]

    The LQC Wigner-Moyal-Groenewold operator is the unique operator with the following properties:

        \[M\left( {\tau ,\theta } \right) = \left\langle {{\psi ^ * },\hat M\psi } \right\rangle \]

    \psi the LQC Holst-cylindrical functions and M\left( {\tau ,\theta } \right) the LQC characteristic function, the same as the Fourier transform of the the quasi probability density function of the group characters.

    It immediately follows from Fourier phase space symplecticity that the LQC Wigner-Moyal-Groenewold operator satisfies the following relation:

        \[{\hat M\left( {\tau ,\theta } \right) = {e^{i\frac{\tau }{a}\hat p}}{{\hat h}_\theta }{e^{i\frac{\tau }{a}\hat p}} = {e^{i\frac{\tau }{a}\hat p}}{e^{i\theta c}}{e^{i\frac{\tau }{a}\hat p}}}\]

        \[{\hat p \equiv - ia\frac{d}{{dc}}}\]

    and for the LQC characteristic function, we have:

        \[M\left( {\tau ,\theta } \right) = \int {{\psi ^ * }} \left( {c - \frac{\tau }{2}} \right){e^{i\theta c}}\psi \left( {c + \frac{\tau }{2}} \right)dc\]

    noting that any connection c is gauge and diffeomorphism invariant in homogeneous isotropic space.

    We can now define the holonomy-flux algebra for homogeneous isotropic Friedmann–Lemaître–Robertson–Walker space model via:

        \[\left[ {{{\hat N}_{\left( \mu \right)}},\hat p} \right] = - \frac{{8\pi G\hbar }}{3}\mu {\hat N_{\left( \mu \right)}}\]

    where the holonomy and the flux operators act as:

        \[{{\hat N}_{\left( \mu \right)}}\Psi \left( c \right) = {e^{i\mu c}}\]

        \[\hat p\Psi \left( c \right) = - i\frac{{8\pi \gamma G\hbar }}{3}\frac{{d\Psi }}{{dc}}\]

    The Hilbert space basis is given by the connection-lifter LQG spin-networks:

        \[{\hat N_{\left( \mu \right)}} = {e^{i\mu c}}\]

    with c the configuration variable corresponding to the connection, \mu the number of the Fourier fiducial cell repetition, and satisfy:

        \[\left\langle {{N_{\left( \mu \right)}},{N_{\mu '}}} \right\rangle = \left\langle {{e^{i\mu c}}{e^{i\mu 'c}}} \right\rangle = {\delta _{\mu ,\mu '}}\]

    and:

        \[\left[ {\hat p,\hat N} \right] = a\mu \hat N\]

    with a a constant satisfying:

        \[a = \frac{{4\pi \gamma G\hbar }}{3}\]

    Let us derive now the Wigner function and show that it satisfies the property that when integrated by one variable it reduces to the distribution density of the other variable. Define it as:

        \[F\left( {\mu ,c} \right) = \int {{\psi ^ * }} \left( {c - a\tau } \right){e^{ - 2ia\tau \mu }}\psi \left( {c - a\tau } \right)d\tau \]

    with:

        \[\psi \left( c \right) = \sum\limits_{n = 0}^N {{{\hat \Psi }_{{\mu _n}}}{e^{i{\mu _n}c}}} ,\quad {\mu _n} \in R\]

    For the distribution density function to be definable, the mutual quasi distribution function of \mu and c the following two equalities should be true:

        \[{\rho _c} = \int {F\left( {\mu ,c} \right)} \,dc = {\left| {\psi \left( c \right)} \right|^2}\]

        \[{\rho _\mu } = \int {F\left( {\mu ,c} \right)} \,dc = {\left| {\hat \Psi \left( c \right)} \right|^2}\]

    Hence, when integrating with respect to one variable it becomes the distribution density of the other one. The above equalities hold since our measures dc and d\mu satisfy:

        \[\int\limits_{{{\hat R}_b}} {{{\hat f}_\mu }d\mu } = \sum\limits_{\mu \in R} {{{\hat f}_\mu }} \]

    and:

        \[\int\limits_{{R_b}} {{e^{i\mu c}}} dc = {\delta _{\mu ,0}}\]

    with {\hat R_b} the Bohr dual space and {\delta _{\mu ,0}} a Kronecker delta.

    Now, the characters of the compactified line {R_b} are the functions {h_\mu }\left( c \right) = {e^{i\mu c}}, hence the Fourier transform of the function on {R_b} is given by:

        \[{\hat f_\mu } = \int {f\left( c \right)} {h_{ - \mu }}\left( c \right)dc\]

    which is an isomorphism of:

        \[{L^2}\left( {{R_b},c} \right) \to {L^2}\left( {{{\hat R}_b},d\mu } \right)\]

    and {e^{i\mu c}} comprise the basis of:

        \[H = {L^2}\left( {{R_b},dc} \right)\]

    We need to prove the above equalities. First, substitute the expression:

        \[F\left( {\mu ,c} \right) = \int {{\psi ^ * }} \left( {c - a\tau } \right){e^{ - 2ia\tau \mu }}\psi \left( {c - a\tau } \right)d\tau \]

    of F\left( {\mu ,c} \right) and the expression:

        \[\psi \left( c \right) = \sum\limits_{n = 0}^N {{{\hat \Psi }_{{\mu _n}}}{e^{i{\mu _n}c}}} ,\quad {\mu _n} \in R\]

    for \psi \left( c \right) into:

        \[{\rho _c} = \int {F\left( {\mu ,c} \right)} \,dc = {\left| {\psi \left( c \right)} \right|^2}\]

    giving us:

        \[\begin{array}{l}\int {F\left( {\mu ,c} \right)} \,d\mu = \int {\int {\sum\limits_{n = 0}^N {\sum\limits_{k = 0}^K {\hat \Psi _{{\mu _n}}^ * } } } } {e^{ - ia{\mu _n}c}}\\{e^{ia{\mu _n}\tau }}{{\hat \Psi }_{{\mu _k}}}{e^{ia{\mu _k}c}}{e^{ia{\mu _k}\tau }}{e^{ - 2ia\tau \mu }}d\tau d\mu \end{array}\]

    with \tau \in {R_b},\;\mu \in R, and since integration with respect to \mu is just a sum as \mu is discrete, we have:

        \[\begin{array}{l}\int {F\left( {\mu ,c} \right)} \,d\mu = \sum\limits_{\mu \in R} {\sum\limits_{n = 0}^N {\sum\limits_{k = 0}^K {\int {\hat \Psi _{{\mu _n}}^ * } } } } {e^{ - ia{\mu _n}c}}\\{e^{ia{\mu _n}\tau }}{{\hat \Psi }_{{\mu _n}}}{e^{ia{\mu _n}c}}{e^{ia{\mu _n}\tau }}{e^{ - 2ia\tau \mu }}d\tau \end{array}\]

    Now, using:

        \[\int\limits_{{R_b}} {{e^{i\mu c}}} dc = {\delta _{\mu ,0}}\]

    and integrating with respect to \tau, we can derive:

        \[\int {{e^{ia{\mu _n}\tau }}{{\hat \Psi }_{{\mu _n}}}{e^{ia{\mu _k}\tau }}{e^{ - 2ia\tau \mu }}d\tau } = {\delta _{2\mu ,{\mu _k} + {\mu _n}}}\]

    Given \mu \in R, it follows that summation by \mu makes the terms with 2\mu \ne {\mu _k} + {\mu _n} equal to zero and the terms with \mu = {\mu _k} + {\mu _n} equal to one and all terms with \tau and \mu vanish from the sum. Hence, by using:

        \[\psi \left( c \right) = \sum\limits_{n = 0}^N {{{\hat \Psi }_{{\mu _n}}}{e^{i{\mu _n}c}}} ,\quad {\mu _n} \in R\]

    we derive:

        \[\begin{array}{l}\int {F\left( {\mu ,c} \right)} \,d\mu = \sum\limits_{n = 0}^N {\sum\limits_{k = 0}^K {\hat \Psi _{{\mu _n}}^ * {e^{ - ia{\mu _n}c}}} } \\{{\hat \Psi }_{{\mu _k}}}{e^{ia{\mu _k}c}} = {\psi ^ * }\left( c \right)\psi \left( c \right) = {\left| {{\psi ^ * }\left( c \right)} \right|^2}\end{array}\]

    and to prove the equality:

        \[{\rho _\mu } = \int {F\left( {\mu ,c} \right)} \,dc = {\left| {\hat \Psi \left( c \right)} \right|^2}\]

    we substitute the expression: