On the general covariance and strong equivalence principles in quantum general relativity

Prugovečki, Eduard

doi:10.1007/bf02054648

Cited by 1 publication

(4 citation statements)

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“…This leads us to the conjecture: ( 189) " (312), (313) if the left hand side of this equation is evaluated in terms of the MCP, instead of the GNS, Hilbert space. While this conjecture is quite heuristic, it seems to be a legitimate candidate for a W ˚-geometric analogue of the Daubechies-Klauder propagator formula (282). A development of a suitable stochastic calculus allowing for an exact mathematical treatment of (312), as well as the proof that the proposed construction of MCP Hilbert space is well defined, are the necessary conditions to approach the problem of proving this conjecture.…”

Section: W ˚-Geometric Quantum Historiesmentioning

confidence: 97%

“…In addition, we replace an affine function ℎp𝑧p𝑡qq in the Daubechies-Klauder formula (282), which is corresponding to a Killing hamiltonian vector field and is generated by a coherent state expectation value of a self-adjoint hamiltonian operator, by any smooth function on ℳp𝒩 q, understood as a hamiltonian function on a BLP manifold. This leads us to ask how one can relate local entropic and local hamiltonian dynamics in the histories context.…”

Section: Local Information Geometry In Quantum Historiesmentioning

confidence: 99%

“…[294,234] and references therein for a comparative discussion of these two approaches). The above construction of geodesic propagation of 'quantum particles' is partially influenced by the works of Drechsler [87,88,89] and Prugovečki [278,279,280,281,282,283,285,284] (see also [123,124]). As opposed to them, we do not require any pseudo-riemannian metric on the base manifold, so we do not introduce soldered Poincaré frame bundles, and we also consider the GNS Hilbert spaces (which may be unitarily inequivalent, if 𝜔 R 𝒩 ‹0 ) varying over the base manifold instead of pasting fibre bundle from identical copies of a single Hilbert space.…”

Section: Local Gauge and Geodesic Propagationmentioning

confidence: 99%

“…Both formulations admit introducing additional local gauge and source terms, so they can be used to study various applied models. Taking a closer look at the Daubechies-Klauder formula (282), one may note that the left side of this equation is formulated without taking into consideration the possible changes of the GNS representation along the states, because the coherent vector states are considered to be defined in a single Hilbert space. If such changes would be considered (as we do it here), then an operator e ´i𝐻𝑠 in (282) should be multiplied from left by a corresponding standard unitary transition operator.…”

Section: W ˚-Geometric Quantum Historiesmentioning

confidence: 99%

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Local quantum information dynamics

Kostecki

2016

Preprint

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This paper is intended to: 1) show how the local smooth geometry of spaces of normal quantum states over W ˚-algebras (generalised spaces of density matrices) may be used to substantially enrich the description of quantum dynamics in the algebraic and path integral approaches; 2) provide a framework for construction of quantum information theories beyond quantum mechanics, such that quantum mechanical linearity holds only locally, while the nonlocal multi-user dynamics exhibits some similarity with general relativity. In the algebraic setting, we propose a method of incorporating nonlinear Poisson and relative entropic local dynamics, as well as local gauge and local source structures, into an effective description of local temporal evolution of quantum states by using fibrewise perturbations of liouvilleans in the fibre bundle of Hilbert spaces over the quantum state manifold. We apply this method to construct an algebraic generalisation of Savvidou's action operator. In the path integral setting, motivated by the Savvidou-Anastopoulous analysis of the role of Kähler space geometry in the Isham-Linden quantum histories, we propose to incorporate local geometry by means of a generalisation of the Daubechies-Klauder coherent state phase space propagator formula. Finally, we discuss the role of Brègman relative entropy in the Jaynes-Mitchell-Favretti renormalisation scheme. Using these tools we show that: 1) the propagation of quantum particles (in Wigner's sense) can be naturally explained as a free fall along the trajectories locally minimising the quantum relative entropy; 2) the contribution of particular trajectories to the global path integral is weighted by the local quantum entropic prior, measuring user's lack of information; 3) the presence of nonlinear quantum control variables results in the change of the curvature of the global quantum state space; 4) the behaviour of zero-point energy under renormalisation of local entropic dynamics is maintained by local redefinition of information mass (prior), which encodes the curvature change. We conclude this work with a proposal of a new framework for nonequilibrium quantum statistical mechanics based on quantum Orlicz spaces, quantum Brègman distances and Banach Lie algebras.

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Section: W ˚-Geometric Quantum Historiesmentioning

confidence: 97%

Section: Local Information Geometry In Quantum Historiesmentioning

confidence: 99%