These notes provide a compact but systematic pedagogical derivation of the Hamiltonian formulation of general relativity and Ashtekar’s new variables. The presentation begins with the Dirac–Bergmann theory of constrained Hamiltonian systems, emphasizing how singular Lagrangians lead to primary and secondary constraints and why this structure is essential in canonical gravity. After establishing the notation, including Penrose’s abstract index convention and vielbein-based internal indices, the notes develop the 3+1 decomposition of spacetime in detail, introducing the lapse, shift, spatial metric, spatial projection, extrinsic curvature, spatial covariant derivative, and Lie evolution of spatial tensors. These ingredients are then used to derive the ADM canonical variables, the gravitational Hamiltonian, and the Hamiltonian and momentum constraints. The final part reformulates the canonical theory in terms of Ashtekar’s complex connection variables, replacing metric dynamics by connection dynamics through the canonical pair \ ( (Aⁱₐ, Eᵃᵢ) \). In this formulation, the gravitational constraints become the Gauss, vector, and scalar constraints, and the scalar constraint takes a polynomial form after the introduction of the densitized lapse. The notes conclude by clarifying the gauge-theoretic interpretation of the Ashtekar formulation and the role of the reality conditions required to recover real Lorentzian general relativity.
Qian-Rui Lee (Fri,) studied this question.