Affine group representation formalism for four dimensional, Lorentzian, quantum gravity

Within the context of the Ashtekar variables, the Hamiltonian constraint of four-dimensional pure General Relativity with cosmological constant, $Λ$, is reexpressed as an affine algebra with the commutator of the imaginary part of the Chern-Simons functional, $Q$, and the positive-definite volume element. This demonstrates that the affine algebra quantization program of Klauder can indeed be applicable to the full Lorentzian signature theory of quantum gravity with non-vanishing cosmological constant; and it facilitates the construction of solutions to all of the constraints. Unitary, irreducible representations of the affine group exhibit a natural Hilbert space structure, and coherent states and other physical states can be generated from a fiducial state. It is also intriguing that formulation of the Hamiltonian constraint or Wheeler-DeWitt equation as an affine algebra requires a non-vanishing cosmological constant; and a fundamental uncertainty relation of the form $\frac{Δ{V}}{<{V}>}Δ{Q}\geq 2πΛL^2_{Planck}$ (wherein $V$ is the total volume) may apply to all physical states of quantum gravity.

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