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Loop Quantum Gravity: the Barbero-Immirzi Parameter

I showed that in 4-D spacetime, the general relativistic starting point for canonical loop quantum gravity is given by:

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

with 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:

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

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:

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

allowing us to write down the Holst action as:

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

The Ashtekar-Barbero connection enters the picture in the following way: the phase space is parametrized by an \widetilde {S{U_{a\lg }}}(2)-valued connection and its conjugate triad field: and that is exactly the Ashtekar-Barbero connection! Moreover, the compactness of the gauge group ensures that the quantization leads to a mathematically exact kinematical Hilbert space. A good angle at seeing how the Barbero-Immirzi parameter comes in is via gauge-free spacetime compactification which is equivalent to a 3-D dimensional reduction of the 4-D Holst action:

    \[\begin{array}{*{20}{l}}{{S^{{\rm{Red}}}} = - \int_{{S^1}} {\rm{d}} {x^3}\int_{{M_3}} {{{\rm{d}}^3}} x{\varepsilon ^{\mu \nu \rho }}\left( {\frac{1}{2}} \right.}\\{{\varepsilon _{IJKL}}e_3^Ie_\mu ^JF_{\nu \rho }^{KL}\left. { + \frac{1}{\gamma }e_3^Ie_\mu ^J{F_{\nu \rho }}_{IJ}} \right)}\end{array}\]

with \mu = 0,1,2 the three-dimensional spacetime index and

    \[{{\rm{d}}^3}x{\varepsilon ^{\mu \nu \rho }}\]

is the local volume form on

    \[{M_3}\]

with

    \[{x^I} \equiv e_3^I\]

One then recovers the total three-dimensional action with the Barbero-Immirzi parameter:

    \[\begin{array}{l}S\left[ {e,x,\omega } \right] = {\int_{{M_3}} {\rm{d}} ^3}x{\varepsilon ^{\mu \nu \rho }}\left( {\frac{1}{2}} \right.{\varepsilon _{IJKL}}\\{x^I}e_\mu ^JF_{\nu \rho }^{KL}\left. { + \frac{1}{\gamma }{x^I}e_\mu ^J{F_{\nu \rho }}_{IJ}} \right)\end{array}\]

A major problem for LQG is that the full theory does not imply that the above total three-dimensional action implies three-dimensional gravity, as it ought given compactification and time-gauging: that would be a no-go theorem for up-lifting LQG to 4-D spacetime, and LQG theory would simply be false. Moreover, the variable x is an additional degree of freedom, and thus must be eliminable. Furthermore, the internal gauge group is not the correct one:

    \[SU(1,1)\]

of Lorentzian three-dimensional gravity, but the wrong one:

    \[SL(2,\mathbb{C})\]

I overlooked these problems in part one and instead noted that:

    \[\begin{array}{l}S\left[ {e,x,\omega } \right] = {\int_{{M_3}} {\rm{d}} ^3}x{\varepsilon ^{\mu \nu \rho }}\left( {\frac{1}{2}} \right.{\varepsilon _{IJKL}}\\{x^I}e_\mu ^JF_{\nu \rho }^{KL}\left. { + \frac{1}{\gamma }{x^I}e_\mu ^J{F_{\nu \rho }}_{IJ}} \right)\end{array}\]

is invariant under the action of

    \[SL(2,\mathbb{C})\]

and admits the infinite-dimensional gauge group:

    \[{\bar G^\varpi } \equiv {C^\infty }\left( {{{\tilde M}_4},SL(2,\mathbb{C})} \right)\]

as a symmetry group. A member \Lambda \in {\bar G^\varpi } is hence an SL(2,\mathbb{C})-valued function on {M_3} and acts on the dynamical variables according to the transformation rules:

    \[\left\{ {\begin{array}{*{20}{c}}{{e_\mu } \to \Lambda \cdot {e_\mu }}\\{x \to \Lambda \cdot x}\\{{\omega _\mu } \to {\rm{A}}{{\rm{d}}_\Lambda }\left( {{\omega _\mu }} \right) - {{\not \partial }_\mu }\Lambda {\Lambda ^{ - 1}}}\end{array}} \right.\]

with:

    \[{\left( {\Lambda \cdot \nu } \right)^I} = {\Lambda ^{IJ}}{\nu _J}\]

the fundamental action of \Lambda on any four-dimensional vector \nu and:

    \[{\rm{A}}{{\rm{d}}_\Lambda }\left( \xi \right) = \Lambda \xi {\Lambda ^{ - 1}}\]

the adjoint action of

    \[SL(2,\mathbb{C})\]

on any Lie algebra element

    \[\xi \in \widetilde {S{L_{a\lg }}}\left( {2,\mathbb{C}} \right)\]

and our theory is therefore invariant under spacetime diffeomorphisms, as it ought to be for General Relativity. Infinitesimal diffeomorphisms are generated by vector fields

    \[\nu = {\nu ^\mu }{\not \partial _\mu }\quad {\rm{on}}\quad {M_3}\]

So, their action on the dynamical variables is given by the Lie derivatives:

    \[\left\{ {\begin{array}{*{20}{c}}{e \to {L^{lie}}_\upsilon e}\\{x \to {L^{lie}}_\upsilon x = {\nu ^\mu }{{\not \partial }_\mu }x}\\{\omega \to {L^{lie}}_\upsilon \omega }\end{array}} \right.\]

where for any one-form \varphi, we have:

    \[{L^{lie}}_\upsilon \varphi = \left( {{\nu ^\nu }{{\not \partial }_\nu }{\varphi _\mu } + {\varphi _\nu }{{\not \partial }_\mu }{\nu ^\nu }} \right){\rm{d}}{x^\mu }\]

The symmetries mentioned above are expected of a theory of gravity in first order variables formulation. However, in 3-D, such symmetries alone imply a collapse to SL(2,\mathbb{C}) BF theory, which is not isomorphic to the model above, and problematically, this would imply that our Lagrangian admits additional symmetries, as can be seen by the fact that

    \[\begin{array}{l}S\left[ {e,x,\omega } \right] = {\int_{{M_3}} {\rm{d}} ^3}x{\varepsilon ^{\mu \nu \rho }}\left( {\frac{1}{2}} \right.{\varepsilon _{IJKL}}\\{x^I}e_\mu ^JF_{\nu \rho }^{KL}\left. { + \frac{1}{\gamma }{x^I}e_\mu ^J{F_{\nu \rho }}_{IJ}} \right)\end{array}\]

is invariant under rescaling symmetry and translational symmetry. This destroys the time gauge accessibility of the theory and any 4-D equivalence, and as mentioned above, constitutes a no-go theorem for any possible uplift of:

    \[\begin{array}{*{20}{l}}{{S^{{\rm{Red}}}} = - \int_{{S^1}} {\rm{d}} {x^3}\int_{{M_3}} {{{\rm{d}}^3}} x{\varepsilon ^{\mu \nu \rho }}\left( {\frac{1}{2}} \right.}\\{{\varepsilon _{IJKL}}e_3^Ie_\mu ^JF_{\nu \rho }^{KL}\left. { + \frac{1}{\gamma }e_3^Ie_\mu ^J{F_{\nu \rho }}_{IJ}} \right)}\end{array}\]

to 4-D Einstein-Minkowskian spacetime: and this is the crux behind superstring-theory main ‘truth’: the graviton does not have a Yukawa-coupling constant in an Einstein-Newtonian universalistic sense in 4-D spacetime. For all skeptics of ‘extra-dimensionality’, this ought to be a wake-up call!

Let us address some of the foundational issues raised so far. Aside, read this.

First, note that the gauge group SL(2,\mathbb{C}) is broken, via fixing in:

    \[\begin{array}{l}S\left[ {e,x,\omega } \right] = {\int_{{M_3}} {\rm{d}} ^3}x{\varepsilon ^{\mu \nu \rho }}\left( {\frac{1}{2}} \right.{\varepsilon _{IJKL}}\\{x^I}e_\mu ^JF_{\nu \rho }^{KL}\left. { + \frac{1}{\gamma }{x^I}e_\mu ^J{F_{\nu \rho }}_{IJ}} \right)\end{array}\]

the field {x^I} to \left( {0,0,0,1} \right), into the subgroup SU(1,1), and the rescaling symmetry:

    \[\left\{ {\begin{array}{*{20}{c}}{e_\mu ^I \to \alpha e_\mu ^I}\\{{x^I} \to \frac{1}{\alpha }{x^I}}\end{array}} \right.\]

fixes the norm of x to one. Hence we have a correspondence between the isotropy group of x and SU(1,1). One naturally decomposes the connection {\omega ^{IJ}} into its \widetilde {S{U_{a\lg }}}(1,1)-components, denoted by {\omega ^i} and the compliment by {\omega ^{(3)i}}.

We can now infer:

    \[\left\{ {\begin{array}{*{20}{c}}{{\omega ^i} \equiv \frac{1}{2}{\varepsilon ^i}_{jk}{\omega ^{ij}}}\\{{\omega ^{(3)i}} \equiv {\omega ^{)I = 3)i}}}\end{array}} \right.\]

Also, {F^{IJ}} decomposes into \widetilde {S{U_{a\lg }}}(1,1)-components:

    \[{F^i} = {\varepsilon ^i}_{jk}{F^{jk}}/2\]

and

    \[{F^{(3)i}}\]

Droping the ‘i’, we get the following relations:

    \[\begin{array}{c}{F_{\mu \nu }} = {{\not \partial }_\mu }{\omega _\nu } - {{\not \partial }_\nu }{\omega _\mu } - {\omega _\mu } \times {\omega _\nu }\\ - \omega _\mu ^{\left( 3 \right)} \times \omega _\nu ^{\left( 3 \right)}\end{array}\]

as well as

    \[\begin{array}{c}F_{\mu \nu }^{(3)} = {{\not \partial }_\mu }\omega _\nu ^{(3)} - {{\not \partial }_\nu }\omega _\mu ^{(3)} - {\omega _\mu } \times \omega _\nu ^{(3)}\\ + {\omega _\nu } \times \omega _\mu ^{\left( 3 \right)}\end{array}\]

Thus, the action:

    \[\begin{array}{l}S\left[ {e,x,\omega } \right] = {\int_{{M_3}} {\rm{d}} ^3}x{\varepsilon ^{\mu \nu \rho }}\left( {\frac{1}{2}} \right.{\varepsilon _{IJKL}}\\{x^I}e_\mu ^JF_{\nu \rho }^{KL}\left. { + \frac{1}{\gamma }{x^I}e_\mu ^J{F_{\nu \rho }}_{IJ}} \right)\end{array}\]

metaplectically reduces to:

    \[\begin{array}{l}S = \int_{{M_3}} {\rm{d}} x{\varepsilon ^{\mu \nu \rho }}{e_\mu } \cdot \\\left( {{F_{\nu \rho }} + \frac{1}{\gamma }F_{\nu \rho }^{(3)}} \right)\end{array}\]

And the the canonical \widetilde {S{U_{a\lg }}}(1,1)-connection reduces to:

    \[A_a^i \equiv - \left( {\omega _a^i + \frac{1}{\gamma }\omega _a^{(3)i}} \right)\]

and occurs in the action through its curvature {\tilde F_c} as:

    \[\begin{array}{l}S = - {\int_{{M_3}} {\rm{d}} ^3}x{\varepsilon ^{\mu \nu \rho }}{e_\mu } \cdot \\\left( {{{\tilde F}_c}_{_{\nu \rho }} + \left( {1 + \frac{1}{{{\gamma ^2}}}} \right)\omega _\nu ^{(3)} \times \omega _\rho ^{(3)}} \right)\end{array}\]

which takes the form of an SU(1,1)-BF action with quadratic uplift term in {\omega ^{(3)}}.

This can be strengthened: since a solution to:

    \[\delta S/\delta {\omega ^{(3)i}} = 0\]

entails the on-shell-vanishing of {\omega ^{(3)}}, thus, our theory becomes equivalent to an SU(1,1)-BF theory.

The canonical analysis proceeds now by splitting the spacetime indices \mu ,\nu ,... \in \left\{ {0,1,2} \right\} into spatial indices a,b,... \in \left\{ {1,2} \right\} with \mu = 0 the time-direction. Under such conditions, the action has the following canonical form:

    \[\begin{array}{l}S = \int_\mathbb{R} {\rm{d}} t{\int_\Sigma {\rm{d}} ^2}x\left( {{E^a}} \right. \cdot {{\not \partial }_0}{A_a} + {A_0} \cdot \\G + {e_0} \cdot H + 2\left( {1 + \frac{1}{{{\gamma ^2}}}} \right)\left. {\omega _0^{(3)} \cdot \Phi } \right)\end{array}\]

with the electric field:

    \[{E^a} = {\varepsilon ^{ba}}{e_b}\]

introduced. We now have the following Lagrangian conditions:

    \[G = {\not \partial _a}{E^a} + {A_a} \times {E^a} \simeq 0\]

    \[H = {\varepsilon ^{ab}}\left( {{{\tilde F}_{{c_{ab}}}} + \left( {1 + \frac{1}{{{\gamma ^2}}}} \right)\omega _a^{(3)} \times \omega _b^{(3)}} \right) \simeq 0\]

and

    \[\Phi = {E^a} \times \omega _a^{(3)} \simeq 0\]

The canonical action entails that the only dynamical variables are the electric field {E^a} and its canonically conjugated connection {A_a}. Yet, in 4-D canonical analysis, \omega _a^{(3)} is also a dynamical variable: let {\pi ^a} be its conjugate momenta with the condition:

    \[{\pi ^a} \simeq 0\]

enforced by the Lagrange multiplier {\mu _a}.

Hence, the symplectic structure is completely characterized by the following Poisson brackets:

    \[\begin{array}{l}\left\{ {E_i^a(x),A_b^j(y)} \right\} = \delta _b^a\delta _i^j(x - y)\\ = \left\{ {\pi _i^a(x),\omega _b^{(3)i}(y)} \right\}\end{array}\]

and the time-evolution

    \[{\not \partial _{{0_t}}}\varphi \]

for any field

    \[\varphi \]

is completely determined by the total Hamiltonian:

    \[\begin{array}{l}{H_{{\rm{tot}}}} = - {\int_\Sigma {\rm{d}} ^2}x\left( {{A_0}} \right. \cdot G + {e_0} \cdot H + 2\\\left( {1 + \frac{1}{{{\gamma ^2}}}} \right)\omega _0^{(3)} \cdot \Phi + \left. {{\mu _a} \cdot {\pi ^a}} \right)\end{array}\]

obeying the following identity:

    \[{\not \partial _{{0_t}}}\phi = \left\{ {{H_{{\rm{tot}}}},\phi } \right\}\]

The condition that the time-evolution

    \[{\not \partial _{{0_t}}}\pi _i^a = \left\{ {{H_{{\rm{tot}}}},\pi _i^a} \right\}\]

vanishes yields the following equations:

    \[{P^a} \equiv {\varepsilon ^{ab}}\left( {\omega _b^{(3)} \times {e_0} - \omega _0^{(3)} \times {e_b}} \right) \simeq 0\]

Combining all of the above, one can write the critical equation:

    \[{\varepsilon ^{\mu \nu \rho }}{e_\nu } \times \omega _\rho ^{(3)} \simeq 0\]

and solving implies that the Barbero-Immirzi parameter disappears completely: ending up with the standard action of Lorentzian three-dimensional gravity, and that is a necessary condition for dimensional uplifting to the four-dimensional Ashtekar-Barbero phase space

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