Informational Approaches Lead to Formulations of Quantum Mechanics on Poincaré Disks ^†

Abstract

A novel methodological approach requires the observer to investigate the information accessible outside the object under scrutiny. The object becomes an “hole” devoid of information, surrounded by a container that is no longer a passive structure. We use this container-framed attitude for a mathematical attempt to take a step towards the unexploited unification of general relativity and quantum mechanics. We show that the tenets of quantum mechanics, i.e., the observable A, the autostates ψa, and the Schrodinger equation for the temporal evolution of states, can be described in terms of oscillatory paths taking place on negative-curvature Poincaré disks.

1. Introduction

The ancient Aristotelian concepts of active and potential intellects (henceforward AI and PI) display a viable counterpart to the modern-day theory of information. Alexander of Aphrodisias, Al-Farabi, and Avicenna suggested that past, present, and future knowledge is fully endowed in omniscient and everlasting AI, which is fixed, supralunar, external to the body, and shared by all individuals [1]. Each individual displays their own PI, i.e., the personal ability to grab chunks of AI. The current concept of Shannon’s/Wheeler’s total cosmic information (henceforward TCI) is analogous: TCI is eternal and can be neither created nor annihilated. AI and TCI both have a problem with the principle of individuation: how is the thing A identified as distinguished from the thing B? If two files contain the same fixed quantity of bits, what is the difference between their qualitative (we could use the term “semantic”) content? This raises doubts as whether the tenet of TCI as an unlimited, eternal whole holds true. The comparison between AI/PI and TCI has two opposite paradoxical consequences: on one hand, it wipes out the concept of Holy, Divine knowledge and leaves just the quantitative concept of physical information; on the other hand, it reintroduces a metaphysical component, i.e., the presence of a vague, eternal, indefinable substance permeating the universe. It looks like TCI allows the metaphysical concept of God to sneak in the back of scientific issues. Do scientists extract the information endowed in the object, or do they build information not existing inside the object? Is our qualitative mental information discovered, or is it invented? Recent papers suggest that quantum mechanics is a reference-frame theory pertaining to observer-dependent relational properties. Amazingly, these radical relational formulations of quantum mechanics have been experimentally supported: it has been demonstrated that the properties of the physical world are dependent on the observer as opposed to the tenets of local realism [2]. Once our pars destruens against the concept of TCI is completed, a pars construens is strongly required. We suggest a new methodology according to which the observer does not investigate the information endowed in the thing, but rather investigates the information outside the thing [3]. The thing becomes a “hole” devoid of information inside a surrounding environment that is no longer a passive container, but rather an active structure enabling the examination of its content. In the sequel, we will use this approach to amend our critique of information: in particular, we will suggest a new approach to the curvatures, i.e., the “containers”, subtending quantum dynamics.

2. Pars construens: Hyperbolic Dynamics and Quantum Mechanics

We will tackle the issue of the unexploited unification of general relativity (GR) and quantum mechanics (QM), which prevent a proper understanding of the micro- and macroscopic world [4,5]. GR assumes a 3+1-dimensional pseudo-Riemannian manifold with tensor fields, whereas QM assumes a R⁴ projective Hilbert space with operator-valued fields [6]. The projective infinite-dimensional Hilbert space of the QM’s microscopic world displays countless bases that are the source of the counterintuitive physical effects experienced during experimental observation. Our approach, based on the evaluation of the “container” instead of the “content”, suggests a curvature-based mathematical approach to QM. We examine the possibility that the oscillatory dynamics described by QM might take place on microscopic negative-curvature, hyperbolic continuous manifolds. This novel manifold stands for the container encompassing the QM’s Hilbert spaces. In particular, we describe how the tenets of quantum mechanics, such as the observable A, the autostates ψa, and the Schrodinger equation for the temporal evolution of states, might work very well on a Poincaré disk equipped with rotational groups.

2.1. Before Measuring

Every quantum physical system is described by a (topologically) separable complex Hilbert space of states H with inner product, in which each vector can be decomposed in the linear combination of other vectors [7]. Every observable A in the space H is associated with an hermitian linear self-adjoint operator and with an orthonormal basis of vectors, i.e., the base of its autostates ψa [8,9]. The autovalue a of the operator A displays one or more autostates ψa. Since each autovalue a depicts a possible value of A, ψa forms a compete set, i.e., a wave function, standing for the superposition of all the ψa. Before the measurement of the observable A, all the autostates ψa are superimposed, giving rise to a wave function that completely describes the state of the system:

A ψa = a ψa

(1)

In search of relationships between the dynamics of quantum oscillations and the movements on a hyperbolic manifold, we suggest that the features of the Hilbert space, the observable A, and the autostates ψa before and after measuring could be described in terms of their negative-curvature counterparts on the Poincaré disk. Since the spectrum of an operator corresponds to the set of its autovalues that can be discrete or continuous, the continuous manifold of the Poincaré disk is split in tessellated areas, giving rise to countless discrete tiles. The physical pattern occurring before the experimental measurement can be illustrated in terms of the virtually uncountable number of ψa, each one filling a single tile of the Poincaré disk (Figure 1A). From the standpoint of the observer, most autostates ψa are difficult to access, since the most of them are in the periphery of the Poincaré sphere. In turn, from the standpoint of the observer, it is much easier to assess the largest tiles in the center of the Poincaré disk. We shall see in the sequel that this observation has considerable implications after the experimental measurement. The QM wavefunction evolves in time according to the time-dependent Schrödinger equation [10]. Considering this postulate, one of the most important features of our QM hyperbolic model is that the continuous QM Poincaré disk is not at all a static manifold; rather, it is equipped with rotations that provide the tiles with effortless physical movements. This permit shifts of the tessellated areas from the center to the periphery of the disk surface, and vice versa (Figure 1B). Once we have described the dynamic tiles detectable by the observer before measuring, we will describe what happens to the tiles after the measurement.

2.2. After Measuring

After the experimental measurement of the observable A, the system collapses towards one of its ψa autostates, chosen based on probability. Every ψa corresponds to a physical state in which, after the measurement of A, one finds one of the values a_n or a_(f), so that just a single autostate ψa becomes detectable (Figure 1C). We suggest that the very operation of measurement selects a single tile located inside the rotating Poicaré disk. Incidentally, the movements must be random, because QM, according to another of its postulates, is described by statistical results of continuous probability, where the square module of the vector coefficients stand for probabilities. Before the measurement, the tiles are free to move on the surface of the Poincaré disk, following hyperbolic paths. In turn, the experimental measurement “freezes” the movements of the tessellated areas so that, after the experiment, a single motionless tile can be detected by the standpoint of the observer. The experimental procedure of measurement generates a stationary window of observation located at the center of the Poincaré disk, where the largest tile stands for the tessellation that is the most accessible to the observer. To provide an analogy, think to the inspection of a single “frozen” frame extracted from a long video sequence. Paraphrasing [11], pathologies associated with the possible existence of many unitarity inequivalent representations on the hyperbolic disk disappear, since the whole operation of QM measurement is realized by choosing only one representation, i.e., the ψa located in the central tile.

3. Conclusions

Several attempts have been provided to describe QM in terms of hyperbolic manifolds. See, e.g., [12,13,14,15,16]. Though these authors claim that hyperbolic manifolds are just methodological tools to approach QM, our container-framed attitude suggests that hyperbolic manifolds are part of the description of QM. We suggest that QM might really take place on non-Euclidean hyperbolic manifolds, in the same way that GR really takes place on Euclidean manifolds. In this new physical framework, the macroscopic Euclidean world is like a container encompassing sub-microscopic regions where a negative-curvature fabric includes the waves and paths described by QM.