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arXiv:gr-qc/0501045

Fluctuations, gravity, and the quantum potential

Robert Carroll

Published Jan 13, 2005• 7 citations
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Abstract

We show how the quantum potential arises in various ways and trace its connection to quantum fluctuations and Fisher information along with its realization in terms of Weyl curvature. It is a quantization factor for certain classical systems as well as an expression for quantum matter in gravity theories of Weyl-Dirac type. We extract theories and examples from the literature, providing connections and interpretations, and make a few new observations.

DOI
10.48550/arxiv.gr-qc/0501045

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Entropy•Sep 4, 2013•28 citations
We show how the quantum potential arises in various ways and trace its connection to quantum fluctuations and Fisher information along with its realization in terms of Weyl curvature. It is a quantization factor for certain classical systems as well as an expression for quantum matter in gravity theories of Weyl-Dirac type. We extract theories and examples from the literature, providing connections and interpretations, and make a few new observations.
Understanding quantum theory in terms of a geometric picture sounds great. There are different approaches to this idea. Here we shall present a geometric picture of quantum theory using the de-Broglie--Bohm causal interpretation of quantum mechanics. We shall show that it is possible to understand the key character of de-Broglie--Bohm theory, the quantum potential, as the conformal degree of freedom of the space--time metric. In this way, gravity should give the causal structure of the space--time, while quantum phenomena determines the scale. Some toy models in terms of tensor and scalar--tensor theories will be presented. Then a few essential physical aspects of the idea including the effect on the black holes, the initial Big--Bang singularity and non locality are investigated. We shall formulate a quantum equivalence principle according to which gravitational effects can be removed by going to a freely falling frame while quantum effects can be eliminated by choosing an appropriate scale. And we shall see that the best framework for both quantum and gravity is Weyl geometry. Then we shall show how one can get the de-Broglie--Bohm quantum theory out of a Weyl covariant theory. Extension to the case of many particle systems and spinning particles is discussed at the end.
The de Broglie–Bohm theory allows us to have got a satisfactory geometrodynamic interpretation of quantum mechanics. The fundamental element, which creates a geometrodynamic picture of the quantum world in the non-relativistic domain, a relativistic curved space-time background, and the quantum gravity domain, is the quantum potential. It is shown that, in the non-relativistic domain, the geometrodynamic nature of the quantum potential followsfrom the fact that it is an information potential containing a space-like active information on the environment; the geometric properties of the space expressed by the quantum potential determine non-local correlations between subatomic particles. Moreover, in the de Broglie–Bohm theory in a curved space-time, it is shown that the quantum, as well as the gravitational, effects of matter have geometric nature and are highly related: the quantum potential can be interpreted as the conformal degree of freedom of the space-time metric, and its presence is equivalent to the curved space-time. It is shown on the basis of some recent research that, in quantum gravity, we have a generalized geometric unification of gravitational and quantum effects of matter; Bohm's interpretation shows that the form of a quantum potential and its relation to the conformal degree of freedom of the space-time metric can be derived from the equations of motion.
The formulation of a consistent scheme for treating semiclassical systems is considered, giving particular emphasis to the case of gravity. A critical review of the semiclassical problem is first given, stressing the conceptual aspects of the topic; the theory usually adopted is found to be unsatisfactory, and the need to have an unambiguous interpretation of quantum mechanics before trying to give an alternative description is realized. Our way to arrive at such an interpretation is rather unusual, but it presents the advantage of being almost compelling in the choice to make: First, we reformulate the Schrodinger equation as a set of hydrodynamical equations involving quantities which formally play the role of mass density, velocity and pressure for a fluid; the problem of interpreting the wave function is thus reduced to that of interpreting these quantities. We then show how they can be derived from the Wigner distribution function exactly as in the usual formalism of kinetic theory, and discuss how this fact provides strong support in favour of the statistical interpretation of quantum theory, according to which the state vector describes only ensembles, and not individual systems. We also consider some implications of these results on the possible existence of a more fundamental theory underlying quantum mechanics. It is shown that reconsidering the semiclassical problem in the light of the statistical interpretation, one is led to distinguish between a strongly and a weakly semiclassical regime, which are essentially characterized by the size of the statistical dispersion induced in the observables of the classical subsystem by the coupling to the quantum one. It turns out that in the weakly semiclassical regime, in which this dispersion is not negligible, the concept of coupling equations cannot be successfully applied, and one has rather to define a probability distribution even for the values of the classical observables; an hypothesis which allows to specify such a distribution is enunciated. Several examples of the application of these general principles are considered, and it is shown how the treatment of semiclassical relativistic fields requires a much more sophisticated treatment of the quantum source. This is provided reformulating quantum theory in terms of a quasiprobability functional P[1] in the space of the histories of the system. It is shown how such a functional allows to reconstruct the usual phase space distributions when integrated over suitable sets of paths, in a way which clarify the relations between operator ordering, path integration and phase space treatment of quantum theory. The relativistic extension of p[f] is also constructed, and an explicitly covariant version of relativistic quantum theory is discussed in some details. It is shown how the latter allows, formally, to consider superpositions of different mass eigenstates, although such superpositions are not directly observable. Finally, the application of these new techniques to the treatment of semiclassical electromagnetism and gravity, as well as of a scalar field, are considered. It is shown how the usual semiclassical field equations, suitably reinterpreted in terms of averages of the field, are recovered either in linear cases or in the strongly semiclassical regime, but that they do not hold in general. Finally, some possible extensions and implications of the formalism are discussed.
A quantum entropy space is suggested as the fundamental arena describing the quantum effects. In the quantum regime the entropy is expressed as the superposition of many different Boltzmann entropies that span the space of the entropies before any measure. When a measure is performed the quantum entropy collapses to one component. A suggestive reading of the relational interpretation of quantum mechanics and of Bohm’s quantum potential in terms of the quantum entropy are provided. The space associated with the quantum entropy determines a distortion in the classical space of position, which appears as a Weyl-like gauge potential connected with Fisher information. This Weyl-like gauge potential produces a deformation of the moments which changes the classical action in such a way that Bohm’s quantum potential emerges as consequence of the non classical definition of entropy, in a non-Euclidean information space under the constraint of a minimum condition of Fisher information (Fisher Bohm- entropy). Finally, the possible quantum relativistic extensions of the theory and the connections with the problem of quantum gravity are investigated. The non classical thermodynamic approach to quantum phenomena changes the geometry of the particle phase space. In the light of the representation of gravity in ordinary phase space by torsion in the flat space (Teleparallel gravity), the change of geometry in the phase space introduces quantum phenomena in a natural way. This gives a new force to F. Shojai’s and A. Shojai’s theory where the geometry of space-time is highly coupled with a quantum potential whose origin is not the Schrödinger equation but the non classical entropy of a system of many particles that together change the geometry of the phase space of the positions (entanglement). In this way the non classical thermodynamic changes the classical geodetic as a consequence of the quantum phenomena and quantum and gravity are unified. Quantum affects geometry of multidimensional phase space and gravity changes in any point the torsion in the ordinary four-dimensional Lorenz space-time metric.