Ashtekar–Barbero Variables and Connection Dynamics

After time gauge, the real Ashtekar–Barbero formulation replaces the spatial metric by a densitized triad $E^a_i$ and, for $\gamma\in\mathbb R\setminus\{0\}$, an $su(2)$-valued connection $A_a^i=\Gamma_a^i+\gamma K_a^i$. In Lorentzian signature, $\gamma=\pm i$ instead gives a complex (anti-)self-dual connection: the gauge algebra is complexified, so this is not the real $SU(2)$ configuration used in standard loop quantum gravity. The extrinsic-curvature term in the scalar constraint then disappears, but recovering real geometry requires reality conditions Ashtekar 1987, p. 1587.

Required background. Canonical Constraints, Dirac Observables, and Constraint Algebras supplies the ADM benchmark; Bundle Connections, Curvature, Gauge Transformations, and Bianchi Identities supplies connection geometry.

Helpful background. Hamiltonian Group Actions and Moment Maps supplies Gauss constraints; Parallel Transport and Holonomy supplies holonomies.

Triad and connection phase space

For co-triad $e_a^i$ and spatial metric $q_{ab}=e_a^ie_b^j\delta_{ij}$, define

$$E^a_i=\sqrt q\,e^a_i,\qquad A_a^i=\Gamma_a^i(E)+\gamma K_a^i .$$

$\Gamma_a^i$ is the torsion-free spin connection and $K_a^i=K_{ab}e^{bi}$. With $\kappa=8\pi G$, the gravitational symplectic potential becomes

$$\Theta_{\mathrm{grav}} =\frac1\kappa\int_\Sigma E^a_i\,\delta K_a^i =\frac1{\kappa\gamma}\int_\Sigma E^a_i\,\delta A_a^i -\frac1{\kappa\gamma}\int_\Sigma E^a_i\,\delta\Gamma_a^i.$$

The last term is a boundary integral. It vanishes for closed $\Sigma$; with a boundary, compatible boundary conditions or an explicit boundary symplectic contribution are required. Under either of those conditions, the resulting bulk symplectic form gives

$$\{A_a^i(x),E^b_j(y)\} =8\pi G\gamma\,\delta_a^b\delta^i_j\delta^3(x-y).$$

For real nonzero $\gamma$, this is a real canonical chart on the nondegenerate-triad sector Barbero G. 1995, pp. 5507–5510, eqs. (4)–(8), Open PDF. Continuing to $\gamma=\pm i$ instead passes to a complex phase space subject to the reality conditions below.

First application: derive Gauss and vector constraints

Internal triad rotations produce

$$G_i=D_aE^a_i =\partial_aE^a_i+\epsilon_{ij}{}^kA_a^jE^a_k\approx0.$$

Normalize the smeared Gauss constraint as

$$G[\lambda] =\frac{1}{\kappa\gamma} \int_\Sigma\mathrm d^3x\,\lambda^iD_aE^a_i.$$

Then

$$\{A_a^i,G[\lambda]\}=-D_a\lambda^i, \qquad \{E^a_i,G[\lambda]\}=\epsilon_{ij}{}^k\lambda^jE^a_k.$$

Assume that $\Sigma$ is closed, or that the smearings and boundary terms make the generators differentiable. It is important to distinguish the gauge-covariant vector constraint from the generator of ordinary spatial diffeomorphisms. For a shift $N^a$, define

$$V[N] =\frac{1}{\kappa\gamma} \int_\Sigma\mathrm d^3x\,N^aF_{ab}^iE^b_i,$$

and

$$D[N] =V[N]-G[N^aA_a] =\frac{1}{\kappa\gamma} \int_\Sigma\mathrm d^3x\,N^a \left(F_{ab}^iE^b_i-A_a^iG_i\right).$$

For the connection, the curvature identity gives

$$\{A_a^i,V[N]\} =N^bF_{ba}^i =\mathcal L_NA_a^i-D_a(N^bA_b^i), \qquad \{A_a^i,D[N]\}=\mathcal L_NA_a^i.$$

For $E^a_i$, the smearing $\lambda^i=N^bA_b^i$ is field dependent, so the fixed-smearing Gauss formula is not the whole bracket:

$$\{E^a_i,G[N^bA_b]\} =\epsilon_{ij}{}^k(N^bA_b^j)E^a_k-N^aG_i.$$

Consequently,

$$\{E^a_i,V[N]\} =\mathcal L_NE^a_i +\epsilon_{ij}{}^k(N^bA_b^j)E^a_k -N^aG_i,$$

where

$$\mathcal L_NE^a_i =N^b\partial_bE^a_i-E^b_i\partial_bN^a +E^a_i\partial_bN^b$$

is the Lie derivative of a vector density of weight $+1$. Subtracting the full field-dependent Gauss bracket gives

$$\{E^a_i,D[N]\}=\mathcal L_NE^a_i.$$

Thus $D[N]$ generates ordinary spatial diffeomorphisms on both canonical variables. On the Gauss surface $V[N]$ and $D[N]$ define the same constraint surface; off shell, their flows differ by the Hamiltonian flow of the field-dependent functional $G[N^aA_a]$, not merely by a fixed internal rotation Ashtekar and Lewandowski 2004, § II.C.3, p. 18, eqs. (2.27)–(2.30), Open PDF.

For a scalar lapse $N$, the Lorentzian scalar constraint contains

$$H[N] =\frac{1}{2\kappa} \int_\Sigma\mathrm d^3x\, N\frac{E^a_iE^b_j}{\sqrt{\lvert\det E\rvert}} \left[ \epsilon^{ij}{}_kF_{ab}^k -2(1+\gamma^2)K_{[a}^iK_{b]}^j \right].$$

For $\gamma=\pm i$, the factor $1+\gamma^2$ vanishes. This simplification belongs to the complex (anti-)self-dual phase space, not the real $SU(2)$ chart. Moreover, with the ordinary scalar lapse displayed above the inverse-volume factor remains. The familiar polynomial form uses the densitized lapse $\widetilde N=N/\sqrt q$, of density weight $-1$, so that the self-dual integrand is proportional to $\widetilde N E^a_iE^b_j\epsilon^{ij}{}_kF_{ab}^k$ Ashtekar and Lewandowski 2004, §§ II.A and II.C, pp. 11, 15–19, especially eq. (2.31), Open PDF.

Reality and Immirzi dependence

With complex self-dual variables, recovering real Lorentzian geometry requires

$$E^a_i=\overline{E^a_i},\qquad A_a^i+\overline{A_a^i}=2\Gamma_a^i(E),$$

implemented together with a compatible inner product. Real $\gamma$ avoids these conditions classically. In the standard kinematical representation, area eigenvalues carry an overall $\lvert\gamma\rvert$, whereas volume eigenvalues carry $\lvert\gamma\rvert^{3/2}$, up to the chosen volume-operator prescription Ashtekar and Lewandowski 2004, §§ V.A–B, pp. 44–50, especially eqs. (5.18), (5.21), and (5.24), Open PDF.

Adversarial control: change variables at the quantum level

Repeat a spectrum calculation with $\gamma\to\gamma'$. Classical equations remain equivalent after the canonical transformation, while a fixed representation can yield rescaled spectra. For $\gamma=\pm i$, quoting the simpler constraint without solving reality conditions does not define a physical Hilbert space.

Classical equivalence of variables therefore does not imply unitary equivalence of quantizations. The subsequent loop construction must state $\gamma$, gauge group, domain, and constraint implementation.

Exercises

Why the Gauss correction matters

Use $N^bF_{ba}^i=\mathcal L_NA_a^i-D_a(N^bA_b^i)$ and the Gauss action on $A_a^i$ to show that $D[N]=V[N]-G[N^aA_a]$ generates an ordinary spatial Lie derivative. Repeat the calculation for $E^a_i$, taking account of the field dependence of $N^bA_b^i$.

Solution

The vector constraint gives

$$\{A_a^i,V[N]\} =\mathcal L_NA_a^i-D_a(N^bA_b^i).$$

For the field-dependent Gauss smearing $\lambda^i=N^bA_b^i$, the connection part of its action is $\{A_a^i,G[\lambda]\}=-D_a\lambda^i$. Therefore

$$\begin{aligned} \{A_a^i,D[N]\} &=\{A_a^i,V[N]\} -\{A_a^i,G[N^bA_b]\}\\ &=\mathcal L_NA_a^i-D_a(N^bA_b^i) +D_a(N^bA_b^i)\\ &=\mathcal L_NA_a^i. \end{aligned}$$

For the triad, field dependence supplies the term that a fixed-smearing calculation would miss:

$$\{E^a_i,G[N^bA_b]\} =\epsilon_{ij}{}^k(N^bA_b^j)E^a_k-N^aG_i,$$

while direct variation of the vector constraint gives

$$\{E^a_i,V[N]\} =\mathcal L_NE^a_i +\epsilon_{ij}{}^k(N^bA_b^j)E^a_k -N^aG_i.$$

Their subtraction cancels both the internal-rotation and constraint-proportional terms:

$$\{E^a_i,D[N]\} =\{E^a_i,V[N]\} -\{E^a_i,G[N^bA_b]\} =\mathcal L_NE^a_i.$$

The two constraints agree on gauge-invariant data at $G_i=0$, but their off-shell Hamiltonian actions are not interchangeable.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Ashtekar, Abhay. “New Hamiltonian Formulation of General Relativity.” Physical Review D 36, 1587–1602 (1987). DOI.
• Ashtekar, Abhay, and Jerzy Lewandowski. “Background Independent Quantum Gravity: A Status Report.” Classical and Quantum Gravity 21 (2004): R53–R152. DOI. Open PDF.
• Barbero G., J. Fernando. “Real Ashtekar Variables for Lorentzian Signature Space-times.” Physical Review D 51 (1995): 5507–5510. DOI. Open PDF.