Loop Quantum Gravity: the Barbero-Immirzi Parameter

By George Shiber Tuesday, October 11, 2016Friday, October 26, 2018 In Cosmology, Grand Unification Physics, Loop Quantum Gravity, Mathematical Physics, Mathematics, Philosophy Of Science, Physics, Super String Theory, Time Barbero-Immirzi Parameter, Holst Action

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:

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

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:

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:

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:

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:

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