Before inflation ?
|Feb20-13, 08:37 PM||#18|
Before inflation ?
Continuing with http://arxiv.org/abs/1211.1354
An Extension of the Quantum Theory of Cosmological Perturbations to the Planck Era
Ivan Agullo, Abhay Ashtekar, William Nelson
Well this gives only a limited partial satisfaction. They introduce a kind of general Quantum Field Theory (QFT) sort of matter that participates in the bounce.
But they don't say what this matter IS. Is it electromagnetic radiation? Is it neutrinos? Or what?
Whatever it is, it introduces PERTURBATIONS in the Bunch-Davies vacuum that inflation scenarios postulate existed before inflation.
These MATTER quantum fields are what they label Qˆ, Tˆ. And their Hilbert space is what they call H1. These fields live on a quantum geometry, and its Hilbert space is what they call Ho. So the combination is the tensor product of the two Hilbert spaces. That is where the states of the combined system live.
==quote pages 39 and 40 of http://arxiv.org/abs/1211.1354 ==
Having constructed the dynamics of gauge invariant variables on the truncated phase space, we then used LQG techniques to construct quantum kinematics: the Hilbert space Ho of states of background quantum geometry, the Hilbert space H1 of gauge invariant quantum fields Qˆ, Tˆ representing perturbations and physically interesting operators on both these Hilbert spaces. The imposition of the quantum constraint on the homogeneous sector leads one to interpret the background scalar field φ as a relational or emergent time variable with respect to which physical degrees of freedom evolve. Furthermore, the background geometry is now represented by a wave function Ψo which encodes the probability amplitude for various FLRW geometries to occur. The physically interesting wave functions Ψo are sharply peaked, but the peak follows a bouncing trajectory, not a classical FLRW solution that originates at the big bang. In addition, Ψo has fluctuations about this bouncing trajectory. Quantum fields Qˆ, Tˆ, representing inhomogeneous scalar and tensor perturbations, propagate on this quantum geometry and are therefore sensitive not only to the major de- parture from the classical FLRW solutions in the Planck regime, but also to the quantum fluctuations around the bouncing trajectory, encoded in Ψo. Therefore at first the problem appears to be very complicated. However, a key simplification made it tractable: Within the test field approximation inherent to the truncation strategy, the propagation of Qˆ, Tˆ on the quantum geometry Ψo is completely equivalent to that of their propagation on a specific, quantum corrected FLRW metric g ̃ab. Although
Furthermore, away from the Planck regime, g ̃ab satisfies Einstein’s equations to an excellent approximation. In this sense, the standard quantum field theory of Qˆ, Tˆ emerges from the more fundamental description of these fields evolving on the quantum geometry Ψo with respect to the relational time φ. This exact relation between quantum fields Qˆ, Tˆ on the quantum geometry Ψo and those on the dressed, effective geometry of g ̃ab enabled us to carry over adiabatic regularization techniques from quantum field theory in curved space-times to those on quantum geometries Ψo. Together, all this structure provides us with a well-defined quantum theory of the truncated phase space we began with.
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