Loops, Knots, Gauge Theories and Quantum Gravity 2026, 32 - Tetradic General Relativity

To introduce new variables for the new Hamiltonian formulation, a tetrad is formed, which is a vector basis in terms on a spacetime looks locally flat g_ab = e^I_ae^J_bη_IJ where η is the Minkowski metric (Eq. 7.23). The local flatness will be sufficient for the same reason that it works in general relativity. Tetrads can reconstruct the metric of space-time, with extra degrees of freedom, which reflects on the canonical formalism (7.3.1). The Einstein action in terms of tetrads is derived through the covariant derivative as S(e, ω) = ∫ d⁴x e e_I^ae_J^bΩ_ab^IJ (Eq. 7.24) where e is the determinant of the tetrad. By rewriting Ω_ab^IJ = R_ab^IJ + ∇_[aC_b]^IJ + C_[a^IMC_b]M^J (Eq. 7.25), the action is simplified for computation. The equation of motion also gives e_I^cR_cb^IJ - 1/2 R_cd^MNe_M^ce_N^de_b^J = 0 (Eq. 7.26), which implies that the Einstein tensor of the metric in terms of the tetrad vanishes (7.3.2). To allow the inclusion of Ahtekar variables, the tetradic formalism needs to be reconstructed with a change: The connection ω_a^IJ is replaced by its self-dual part with respect to the internal indices, which requires the connection to be complex. By repeating the calculations for the self-dual case, the self-dual action wrt. A_a^IJ results in a torsion-free connection that annihilates the triad (7.3.3). Decomposing the self-dual action requires the 3+1 split through the vector t^a = Nn^a + N^a. Applying the canonical variables to the action recovers the Poisson bracket relations (Eq. 7.34). This simplifies the constraint equations heavily. The theory's reality condition can be eliminated to recover a canonical theory, though this implies a different dimension at quantum level (7.3.4). The theory can be quantized through the regular procedure. The difference between this theory and the Yang-Mills case is due to the connection having a complex value. The wavefunctions are holomorphic functions of the connection and the functional derivative independently acts on the connection and its complex conjugate. The choice in the representation of the canonical algebra promotes the constraint equations to operatorial equations (7.4.1). Ordering the triads to the right, the constraints contribute to gauge transformations and diffeomorphisms on the wavefunctions. A potential problem arises due to the fact that the algebra of constraint is not a true algebra, and the constant in the commutator from the Hamiltonian would have to appear to the right of the algebra's commutator. The commutator of two Hamiltonians might not vanish automatically, and this characteristic needs to be checked explicitly. Wilson loops alone are not solutions within the diffeomorphism constraint, as diffeomorphisms displace Wilson loops, and don't annihilate under the constraint. The Wilson loops form an overcomplete basis through which any physical state should be expandable. When the Wilson loops act with the Hamiltonian constraint, if the loop has no kinks or intersections, the closed portion shrinks to a point. Divergent terms are cancelled. For regularization in order to apply the theory beyond a formal level, one could use flux tubes, or point-splitting the functional derivatives of the Hamiltonian constraint.

When ordering the constraints with the triads to the left, the diffeomorphism constraint failing to generate diffeomorphisms on the wave-functions becomes a problem. However, a diffeomorphism constraint be added to ensure the constraint generates diffeomorphisms. The constraint can be solved through wavefunctionals of the connection Ψ[A] invariant under diffeomorphisms. Wilson loops still don't solve the Hamiltonian constraint, but the Chern-Simons form via the Ashtekar connection exhibits invariance under small transformations and diffeomorphisms, and it annihilates with the constraints.