Ashtekar-Barbero Connection Variables

Ashtekar–Barbero variables rewrite the classical ADM phase space as a real $SU(2)$ connection and a densitized triad. The transformation requires an oriented nondegenerate triad and the time gauge. With the site’s convention $K_{ij}=-\tfrac12\mathcal L_n\gamma_{ij}$, the extrinsic-curvature term enters the connection with the opposite sign from formulas that use $K_{ij}=+\tfrac12\mathcal L_n\gamma_{ij}$.

Required background. Frames, Coframes, and Anholonomy supplies triads and internal rotations, The Gibbons-Hawking-York Boundary Term supplies the canonical action’s boundary completion, and Presymplectic Potentials and Currents supplies the criterion for a canonical transformation.

Helpful background. The Einstein Equation supplies the classical dynamics, Hypersurface Normals and Tangential Projectors supplies the spatial split, and Formulations of General Relativity locates these variables after ADM and Holst formulations.

Time gauge and the densitized triad

Fix an internal future timelike normal and impose the time gauge $e^0{}_i=0$ on a spatial slice. The residual internal gauge group is $SO(3)$, or its double cover $SU(2)$. Let $e^a{}_i$ be an oriented spatial triad, with $a=1,2,3$, such that

$$\gamma_{ij}=\delta_{ab}e^a{}_i e^b{}_j.$$

Write $q=\det(\gamma_{ij})$ on this page, so that $\gamma$ can denote only the Immirzi parameter. Choose $\epsilon^{123}=\epsilon_{123}=+1$ and $\det(e^a{}_i)\gt0$. The densitized inverse triad is

$$E^i{}_a =\frac12\epsilon^{ijk}\epsilon_{abc}e^b{}_j e^c{}_k =\sqrt q\,e^i{}_a.$$

It has density weight one and determines the inverse metric by

$$E^i{}_aE^j{}_b\delta^{ab}=q\gamma^{ij}.$$

The torsion-free spatial spin connection $\Gamma_i{}^a(E)$ is fixed by

$$\partial_{[i}e^a{}_{j]} +\epsilon^a{}_{bc}\Gamma_{[i}{}^b e^c{}_{j]}=0.$$

Deriving the sign of the real connection

The ADM momentum for the site’s extrinsic curvature is

$$\pi^{ij}=\frac{\sqrt q}{2\kappa} \left(K\gamma^{ij}-K^{ij}\right), \qquad \kappa=8\pi G.$$

Introduce the canonical extrinsic-curvature one-form

$$\mathcal K_i{}^a\equiv-K_{ij}e^{ja}.$$

The minus sign appears because the conventional canonical velocity is $+\tfrac12\mathcal L_n\gamma_{ij}$, whereas the site defines $K_{ij}$ with a minus. Up to an exact phase-space variation and the internal-rotation constraint, the ADM symplectic potential becomes

$$\begin{aligned} \int_\Sigma d^3x\,\pi^{ij}\delta\gamma_{ij} ={}&\frac1\kappa\int_\Sigma d^3x\, E^i{}_a\delta\mathcal K_i{}^a\\ &-\frac1\kappa\delta\!\int_\Sigma d^3x\, E^i{}_a\mathcal K_i{}^a. \end{aligned}$$

For real nonzero $\gamma$, define

$$\boxed{ A_i{}^a=\Gamma_i{}^a+\gamma\mathcal K_i{}^a =\Gamma_i{}^a-\gamma K_{ij}e^{ja} }.$$

Here and in this canonical transformation assume a closed slice, compactly supported variations, or boundary data that remove the surface term. The torsion-free compatibility equation gives

$$E^i{}_a\,\delta\Gamma_i{}^a\,d^3x =\frac12d(e_a\wedge\delta e^a).$$

Thus its integral is a boundary term, leaving

$$\{A_i{}^a(x),E^j{}_b(y)\} =\kappa\gamma\, \delta_i^j\delta^a_b\delta^{(3)}(x,y).$$

The same sign follows directly from the Holst action. In time gauge, $\omega_i{}^{0a}=\mathcal K_i{}^a$ and $\omega_i{}^{ab}=-\epsilon^{ab}{}_c\Gamma_i{}^c$. Its surface potential on a slice therefore reduces to

$$\begin{aligned} \Theta_\Sigma^{\rm H} &=\frac1{2\kappa}\int_\Sigma \left(\Sigma_{IJ}-\gamma^{-1}e_I\wedge e_J\right) \wedge\delta\omega^{IJ}\\ &=\frac1\kappa\int_\Sigma d^3x\, E^i{}_a\left(\delta\mathcal K_i{}^a +\gamma^{-1}\delta\Gamma_i{}^a\right)\\ &=\frac1{\kappa\gamma}\int_\Sigma d^3x\, E^i{}_a\delta A_i{}^a. \end{aligned}$$

Both $\Gamma$ and $\mathcal K$ have now been tied to the same Lorentz connection. This prevents changing one sign in the action while retaining an incompatible canonical bracket.

Barbero’s real transformation and constraints are derived in Barbero 1995, equations (1)–(18), pp. 3–6, PDF; their origin in the generalized Palatini action is shown in Holst 1996, equations (5)–(14), pp. 2–5, PDF.

Gauss, diffeomorphism, and scalar constraints

The curvature of the real connection is

$$F^a{}_{ij} =2\partial_{[i}A_{j]}{}^a +\epsilon^a{}_{bc}A_i{}^bA_j{}^c.$$

The new first-class constraint associated with the triad redundancy is the Gauss constraint

$$\mathcal G_a=D_i^{A}E^i{}_a=0.$$

A generator of spatial diffeomorphisms is, up to its normalization and boundary term,

$$\mathcal V_i=E^j{}_aF^a{}_{ij}-A_i{}^a\mathcal G_a.$$

For vacuum Lorentzian gravity with $\Lambda=0$, a convenient scalar constraint is

$$\mathcal C_\perp =\frac{E^i{}_aE^j{}_b}{2\kappa\sqrt q} \left[ \epsilon^{ab}{}_cF^c{}_{ij} -2(1+\gamma^2) \mathcal K_{[i}{}^a\mathcal K_{j]}{}^b \right]=0,$$

modulo Gauss-proportional rearrangements. The second term is the Lorentzian correction. It is why a real connection does not retain the exceptionally simple polynomial scalar constraint of complex self-dual variables. Because $\Gamma(E)$ and $\sqrt q$ occur, the real scalar constraint remains nonpolynomial in the elementary pair.

Time gauge hides boost covariance rather than abolishing it. $A_i{}^a$ is an $SU(2)$ connection on the slice; for generic real $\gamma$ it is not simply the pullback of a four-dimensional Lorentz connection. Classical equivalence also presumes a nondegenerate triad and the same boundary sector as ADM.

From classical variables to loop quantization

The canonical bracket makes holonomies of $A$ and fluxes of $E$ natural variables for loop quantization. Their operator representation, spectra, and the status of different $\gamma$ values are developed on Ashtekar–Barbero Variables for Loop Quantization. None of those quantum conclusions follows merely from the classical canonical transformation derived here.

Exercise: check the extrinsic-curvature sign

For a spatially flat slicing $\gamma_{ij}=a(t)^2\delta_{ij}$ with unit lapse and zero shift, compute $K_{ij}$, $E^i{}_a$, $\Gamma_i{}^a$, and $A_i{}^a$.

Solution

The site convention gives

$$K_{ij}=-\frac12\dot\gamma_{ij} =-H\gamma_{ij}, \qquad H=\frac{\dot a}{a}.$$

Choose $e^a{}_i=a\delta^a_i$. Then

$$E^i{}_a=a^2\delta^i_a, \qquad \Gamma_i{}^a=0,$$

and

$$\mathcal K_i{}^a=-K_{ij}e^{ja} =Ha\delta_i^a.$$

Therefore

$$A_i{}^a=\gamma Ha\delta_i^a.$$

Using $A=\Gamma+\gamma K_{ij}e^{ja}$ instead would reverse this result and would be appropriate only after simultaneously changing the convention for $K_{ij}$ or the canonical bracket.

References

• Barbero G., J. Fernando. “Real Ashtekar Variables for Lorentzian Signature Space-Times.” Physical Review D 51, no. 10 (1995): 5507–5510. DOI. Open PDF.
• Holst, Sören. “Barbero’s Hamiltonian Derived from a Generalized Hilbert–Palatini Action.” Physical Review D 53, no. 10 (1996): 5966–5969. DOI. Open PDF.