Ashtekar–Barbero Variables for Loop Quantization

The real Ashtekar–Barbero connection repackages spatial geometry and extrinsic curvature into an $SU(2)$ gauge pair. Its holonomies along curves and the densitized triad’s fluxes across surfaces remain meaningful without choosing a background metric, which is why they are the elementary variables of loop quantum geometry. This page applies the classical construction to its holonomy–flux use; the full classical canonical derivation is treated separately.

Required background. Variational and Symplectic Geometry supplies canonical brackets and constraints; Frames, Tetrads, and Spinors supplies coframes, local Lorentz gauge, and spin connections.

Helpful background. Flat-Spacetime Quantum Field Theory supplies canonical quantization; Einstein Equations supplies the dynamics to recover; Quantum Information supplies Hilbert-space language; and Quantum Gravity supplies the common regime distinctions.

Connection and densitized triad

Let $e_a^i$ be an oriented spatial coframe with $h_{ab}=e_a^ie_b^j\delta_{ij}$. Define the densitized inverse triad and triadic extrinsic curvature by

$$E^a_i=\sqrt h\,e^a_i, \qquad K_a^i=K_{ab}e^{bi}.$$

We use $K_{ab}=h_a{}^\mu h_b{}^\nu\nabla_\mu n_\nu$ for the future-directed unit normal. Let $\Gamma_a^i[E]$ be the torsion-free $SU(2)$ spin connection compatible with the triad. For real nonzero Barbero–Immirzi parameter $\gamma$,

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

The Ashtekar–Barbero Connection Variables page gives the rigorous classical canonical construction. Here the distinct task is to show why this pair supports loop quantization. Barbero introduced the real connection and its modified Hamiltonian in Barbero 1995, §§II–III.

Constraints in real variables

The Gauss constraint

$$\mathcal G_i=D_aE^a_i=0$$

generates local $SU(2)$ rotations. Spatial diffeomorphisms are generated, up to a Gauss term, by $\mathcal V_a=E^b_iF_{ab}^i$. The Hamiltonian constraint contains both curvature and extrinsic-curvature pieces,

$$\mathcal H_\perp= \frac{E^a_iE^b_j}{16\pi G\sqrt h} \left[\epsilon^{ij}{}_kF_{ab}^k -2(1+\gamma^2)K_{[a}^iK_{b]}^j\right] \simeq0,$$

Here $K_{[a}^iK_{b]}^j=(K_a^iK_b^j-K_b^iK_a^j)/2$, $F_{ab}^i=2\partial_{[a}A_{b]}^i+\epsilon^i{}_{jk}A_a^jA_b^k$, and the displayed $\mathcal H_\perp$ is a density of weight one, smeared with an ordinary scalar lapse. The formula assumes vacuum with $\Lambda=0$; a cosmological constant adds $\Lambda\sqrt h/(8\pi G)$. At $\gamma=\pm i$ the second term simplifies in the complex self-dual formulation, but reality conditions become difficult. Real $\gamma$ keeps the kinematical $SU(2)$ representation transparent while leaving the nonpolynomial dynamics.

The transformation is canonical only when the symplectic structure, Gauss constraint, and boundary terms are translated together. Treating $A$ as an arbitrary Yang–Mills connection while retaining the old metric momentum would give the wrong bracket. The dependence on $\gamma$ also provides a useful check: classical vacuum equations are unchanged, whereas positive kinematical geometric spectra acquire an overall $\lvert\gamma\rvert$ factor after quantization.

The curvature is recovered from a small loop $\alpha$ whose linear coordinate size is $\varepsilon$ and whose oriented area bivector is $\delta S^{ab}=O(\varepsilon^2)$:

$$h_\alpha[A]=\mathbf 1+\frac12F_{ab}^i\tau_i\,\delta S^{ab}+O(\varepsilon^3).$$

We use anti-Hermitian generators $\tau_i=-i\sigma_i/2$, where $\sigma_i$ are Pauli matrices. The generic cubic remainder includes derivatives of curvature; an area-squared remainder would require additional symmetry or constancy assumptions. This limit is later used to regulate the Hamiltonian, but its removal cannot be assumed merely because the classical expression is correct.

Holonomies and fluxes

For an oriented edge $e$ and surface $S$,

$$h_e[A]=\mathcal P\exp\int_e A_a^i\tau_i\,dx^a, \qquad E_i(S)=\int_S n_aE^a_i\,d^2\sigma.$$

No background metric is needed to define the one-dimensional and two-dimensional smearings. Their Poisson bracket is supported at intersections. For one transverse interior intersection $p$,

$$\{E_i(S),h_e\} =-\sigma(e,S)\,8\pi G\gamma\, h_{e_1}\tau_i h_{e_2},$$

where $\sigma(e,S)=\operatorname{sgn}(n_a\dot e^a)$ at the intersection and the composition convention is $h_e=h_{e_1}h_{e_2}$. The minus sign follows from $\{E,A\}=-\{A,E\}$ and the positive exponential used for $h_e$. Endpoint intersections require a separate half-weight or one-sided convention; the displayed formula concerns an interior crossing. Ashtekar and Lewandowski derive the holonomy–flux construction in Ashtekar and Lewandowski 2004, §III; holonomy-exponential and generator signs must be translated together when comparing formulas.

With path ordering compatible with the stated product, under a local rotation $g(x)$ one has $h_e\mapsto g(s(e))h_eg(t(e))^{-1}$. The triad transforms pointwise in the adjoint representation. An integrated flux without parallel transports does not transform by a single adjoint matrix when $g(x)$ varies over the surface; a gauge-covariant flux referred to one point needs transporting the integrand to that point. Contracting edge representations with invariant intertwiners at vertices yields gauge-invariant spin networks.

Why no pointwise connection operator appears

In the Ashtekar–Lewandowski representation, shrinking an embedded edge does not make its nontrivial holonomy matrix elements approach the identity weakly. It is this missing continuity in the edge parameter, not a limit of the discrete spin label, that obstructs differentiation to a pointwise $A_a^i(x)$ operator. Holonomy operators are well defined, and fluxes act as invariant vector fields on copies of $SU(2)$ associated with graph edges.

This is a representation choice with a precise symmetry theorem, not a claim that spacetime is literally a fixed lattice. Graphs label cylindrical functions, and diffeomorphisms move their embeddings. Physical predictions still require the quantum constraints, a physical inner product, and a continuum or semiclassical limit. String and holographic constructions, semiclassical thermodynamics, and classical alternative gravities remain separate; this page does not replace their distinct variables or field equations.

Independent checks

For a short edge, the holonomy expands to the identity plus the line integral of $A$, fixing its orientation sign. Reversing $e$ gives $h_{e^{-1}}=h_e^{-1}$. Reversing $S$ changes the flux sign. The intersection bracket therefore changes sign under either reversal but not both. With coordinate lengths, $[A]=L^{-1}$ and $[E^a_i]=L^0$, so the line integral is dimensionless and the integrated flux $E_i(S)$ has dimension $L^2$. In the representation, the magnitude of its eigenvalue scale is set by $8\pi\lvert\gamma\rvert G$ multiplying a dimensionless Lie-algebra generator; the signed $\gamma$ remains in the oriented flux action, and no extra factor of $G$ multiplies the already integrated classical flux.

Exercises

An edge $e=e_1\circ e_2$ crosses $S$ once with positive orientation. Compute the flux action on a cylindrical function $f(h_e)$ and state what changes if the edge orientation is reversed.

Solution

The Poisson bracket inserts the Lie-algebra generator at the intersection:

$$\{E_i(S),f(h_e)\} =-8\pi G\gamma\, \left.\frac{d}{dt} f\!\left(h_{e_1}e^{t\tau_i}h_{e_2}\right)\right|_{t=0}.$$

Writing $X_i f=\left.d f(h_{e_1}e^{t\tau_i}h_{e_2})/dt\right|_{t=0}$, quantization gives $\widehat E_i(S)=-i8\pi G\gamma X_i$. Consequently $[\widehat E_i,\widehat f]=i\widehat{\{E_i,f\}}$, as required. Reversing $e$ replaces $h_e$ by $h_e^{-1}$ and reverses the intersection orientation; the insertion is translated to the corresponding left- or right-invariant vector field. In particular, the inverse-matrix rule is $\{E,h^{-1}\}=-h^{-1}\{E,h\}h^{-1}$, not merely a sign change of an otherwise unchanged matrix. This ties the algebraic sign to the geometry and to canonical quantization.

References

• Ashtekar, Abhay, and Jerzy Lewandowski. “Background Independent Quantum Gravity: A Status Report.” Classical and Quantum Gravity 21 (2004): R53–R152. arXiv:gr-qc/0404018.
• Barbero G., J. Fernando. “Real Ashtekar Variables for Lorentzian Signature Space-Times.” Physical Review D 51 (1995): 5507–5510. DOI.