Abstract
Entropic dynamics (ED) are a general framework for constructing indeterministic dynamical models based on entropic methods. ED have been used to derive or reconstruct both non-relativistic quantum mechanics and quantum field theory in curved space-time. Here we propose a model for a quantum scalar field propagating in dynamical space-time. The approach rests on a few key ingredients: (1) Rather than modelling the dynamics of the fields, ED models the dynamics of their probabilities. (2) In accordance with the standard entropic methods of inference, the dynamics are dictated by information encoded in constraints. (3) The choice of the physically relevant constraints is dictated by principles of symmetry and invariance. The first of such principle imposes the preservation of a symplectic structure which leads to a Hamiltonian formalism with its attendant Poisson brackets and action principle. The second symmetry principle is foliation invariance, which, following earlier work by Hojman, Kuchař, and Teitelboim, is implemented as a requirement of path independence. The result is a hybrid ED model that approaches quantum field theory in one limit and classical general relativity in another, but is not fully described by either. A particularly significant prediction of this ED model is that the coupling of quantum fields to gravity implies violations of the quantum superposition principle.
Highlights
Without any empirical matter of fact, and none likely on the near horizon, quantum gravity (QG)research has largely split off into distinct channels, each reflecting a different set of attitudes, and yes, philosophies directed towards the problem at hand. (See e.g., [1] for a recent overview of some feasible experimental proposals)
We have introduced nμ, which is the unit normal to the surface that is determined by the μ μ μ conditions nμ nμ = −1 and nμ Xix = 0), and where we have introduced Xix = ∂ix Xx, which are the space-time components of three-vectors tangent to σ
Using the family of H⊥ x s that we identified in Equation (79) and the super-momentum Hix in Equation (51) we can compute all of the necessary Poisson brackets
Summary
Without any empirical matter of fact, and none likely on the near horizon, quantum gravity (QG). A sharper understanding of the deep role played by isometries and symplectic symmetries in QT (see, e.g., [36,37,38,39,40]) suggested another path wherein symplectic and metric structures take a more fundamental place in the ED approach [23] Issues, such as the single-valued nature of the wave function Ψ, or more importantly, the linearity of quantum time evolution, are clarified from this perspective as resulting from the marriage of symmetry principles with the probabilistic structure of ED. A significant result of our present ED reconstruction of a relativistic QFT coupled to gravity is that the dynamics are fundamentally nonlinear Does this imply violations of the quantum superposition principle, but it brings into question the very reason for Hilbert spaces.
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