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Physical Hilbert space for the affine group formulation of 4D, gravity of Lorentzian signature.

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The authors have revealed a fundamental structure which has been hidden within the Wheeler-DeWitt (WDW) constraint of four dimensional General Relativity (GR) of Lorentzian signature in the Ashtekar self-dual variables. The WDW equation can be written as the commutator of two geometric entities, namely the imaginary part of the Chern-Simons functional Q and the local volume element V(x) of 3-space. Upon quantization with cosmological constant, the WDW equation takes on the form of the Lie algebra of the affine group of transformations of the straight line, with Q and V(x) playing the role of the generators for the Lie algebra. The generators are Hermitian, which addresses the issue of the implementation of the reality conditions of GR at the quantum level. Additionally, the irreducible unitary representations (IUR) implement the positivity of the spectrum of the volume operator V(x) at the quantum level This development has led to the existence of elements of the physical Hilbert space for four dimensional gravity of Lorentzian signature, the full theory, in the form of irreducible, unitary representations of the affine group of transformations of the straight line. The affine Lie algebraic structure of the WDW equation remains intact even in the presence of nongravitational fields. This feature has led to the extension of the affine group formulation to elements of the physical Hilbert space for gravity coupled to the full Standard Model of particle physics, quantized on equal footing. Work on the physical interpretation of the states with respect to gauge-diffeomorphism invariant observables, and spacetime geometries solving the Einstein equations is in progress. The journal reference for these results are as follows: - The first result has been published in CQG 30 (2013) 065013 - The second result has just been published in Annals of Physics Journal Vol.343, pages 153-163, April 2014