Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 709In (4.182), γ is a constant, denoted the Immirzi parameter, that canbe chosen arbitrarily (it will enter the Hamiltonian constraint). Differentchoices for γ yield different versions of the formalism, all equivalentin the classical domain. If we choose γ to be equal to the imaginary unit,γ = √ −1, then A is the standard Ashtekar connection, which can be shownto be the projection of the self–dual part of the 4D spin connection on theconstant time surface. If we choose γ = 1, we get the real Barbero connection.The Hamiltonian constraint of Lorentzian general relativity hasa particularly simple form in the γ = √ −1 formalism, while the Hamiltonianconstraint of Euclidean general relativity has a simple form whenexpressed in terms of the γ = 1 real connection. Other choices of γ areviable as well. In particular, it has been argued that the quantum theorybased on different choices of γ are genuinely physical inequivalent, becausethey yield ‘geometrical quanta’ of different magnitude [Rovelli (1998)]. Apparently,there is a unique choice of γ yielding the correct 1/4 coefficient inthe Bekenstein–Hawking formula.The spinorial version of the Ashtekar variables is given in terms of thePauli matrices σ i , i = 1, 2, 3, or the su(2) generators τ i = − i 2 σ i, byẼ a (x) = −i Ẽ a i (x) σ i = 2Ẽa i (x) τ i , (4.183)A a (x) = − i 2 Ai a(x) σ i = A i a(x) τ i . (4.184)Thus, A a (x) and Ẽa (x) are 2 × 2 anti–Hermitian complex matrices.The theory is invariant under local SU(2) gauge transformations, threedimensionaldiffeomorphisms of the manifold on which the fields are defined,as well as under (coordinate) time translations generated by the Hamiltonianconstraint. The full dynamical content of general relativity is capturedby the three constraints that generate these gauge invariances (see [Ashtekar(1991)]).4.13.4.3 Loop AlgebraCertain classical quantities play a very important role in the quantum theory.These are: the trace of the holonomy of the connection, which islabelled by loops on the three manifold; and the higher order loop variables,obtained by inserting the E field (in n distinct points, or ‘hands’)into the holonomy trace. More precisely, given a loop α in M and the points