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

Eyo Ita

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 inCQG30 (2013) 065013 - The second result has just been published inAnnals of PhysicsJournal Vol.343, pages 153-163, April 2014

31 July 2014

http://arxiv.org/abs/1306.1489

Affine group formulation of the Standard Model coupled to gravity

Ching-Yi Chou, Eyo Ita, Chopin Soo

(Submitted on 6 Jun 2013)

This work demonstrates that a complete description of the interaction of matter and all forces, gravitational and non-gravitational, can in fact be realized within a quantum affine algebraic framework. Using the affine group formalism, we construct elements of a physical Hilbert space for full, Lorentzian quantum gravity coupled to the Standard Model in four spacetime dimensions. Affine algebraic quantization of gravitation and matter on equal footing implies a fundamental uncertainty relation which is predicated upon a non-vanishing cosmological constant.

9 pages, published inAnnals of Physics(April 2014)

http://inspirehep.net/record/1237226

http://inspirehep.net/author/profile/E.E.Ita.1

http://arxiv.org/abs/arXiv:1207.7263

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

Chou Ching-Yi, Eyo Ita, Chopin Soo

(Submitted on 30 Jul 2012)

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 (ΔV/⟨V⟩)ΔQ ≥ 2πΛL^{2}_{Planck}(wherein V is the total volume) may apply to all physical states of quantum gravity.

13 page, published inClassical and Quantum Gravity(March 2013)

http://inspirehep.net/record/1124330

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