Information H is a unique relationship between probabilities, based on the property of independence which is central to scientific methodology. Information Geometry makes the tempting but fallacious assumption that a local metric (conventionally based on information) can be used to endow the space of probability distributions with a preferred global Riemannian metric. No such global metric can conform to H, which is “from-to” asymmetric whereas geometrical length is by definition symmetric. Accordingly, any Riemannian metric will contradict the required structure of the very distributions which are supposedly being triangulated. It follows that probabilities do not form a metric space. We give counter-examples in which alternative formulations of information, and the use of information geometry, lead to unacceptable results.
Information H is a unique relationship between probabilities, based on the property of independence which is central to scientific methodology. Information Geometry makes the tempting but fallacious assumption that a local metric (conventionally based on information) can be used to endow the space of probability distributions with a preferred global Riemannian metric. No such global metric can conform to H, which is “from-to” asymmetric whereas geometrical length is by definition symmetric. Accordingly, any Riemannian metric will contradict the required structure of the very distributions which are supposedly being triangulated. It follows that probabilities do not form a metric space. We give counter-examples in which alternative formulations of information, and the use of information geometry, lead to unacceptable results.
Abstract Our purpose here is to draw attention to an interesting relationship that can be shown to hold between (a) information geometry, and (b) quantum geometry. By ‘information geometry’, we mean the natural geometry associated with families of probability distributions. This is a subject that goes back more than half a century to the pioneering work of Rao (1945), who showed that the Fisher information matrix associated with a parametrised family of probability dis tributions gives rise to a natural Riemannian metric on the parameter space. This then allows one to speak in a precise way of the ‘distance’ between two probability distributions, an idea that turns out to be useful in problems of statistical inference. An extensive literature has thus developed, following this line of enquiry, on applications of differential geometry to statistics (see, e.g., Amari (1985), Barndorff-Nielsen et al. (1986), Murray and Rice (1993)). By ‘quantum geometry’, on the other hand, we mean the geometry of the manifold of states associated with a given quantum mechanical system. In particular, we consider the manifold of ‘pure’ states associated with a given complex Hilbert space. This is the space of ‘rays’ through the origin in the Hilbert space, which has the structure of a complex projective space. It is known that the usual rules of quantum mechanics allow one to construct a natural metric on this space, namely the Fubini-Study metric, the specification of which is equivalent to all the familiar structure associated with ordinary quantum mechanics.
Abstract Local information objectivity, that local, independent observers can infer the same information about a model upon exchange of independently acquired experimental data, is fundamental to science. It is mathematically encoded via Čencov’s theorem: the Fisher information metric is the unique metric invariant under the assumptions of independent, identically distributed sampling and sufficient statistics. However, quantum gravity typically violates these assumptions, permitting contextual deviations from the Fisher metric that reflect the dynamical experimental and environmental configurations. This yields a possible extension of spacetime general covariance to information geometry. Since compatibility with the metric on probability spaces heavily restricts the form of the Born rule for quantum mechanics, deviations from the Fisher metric also can induce modifications of the Born rule, leading it to vary between observers. We explain some possible variations, advocate for experimental tests, and suggest a new quantum gravity approach based on generally covariant information geometry.
Abstract Information Geometry provides a powerful mathematical framework that applies differential geometry to the study of probability distributions, offering profound insights into the structure, dynamics, and flow of information. This volume, the thirty-fifth in the ICQER (Informational Constraints Quantum Event Realism) series, develops a comprehensive geometric theory of event spaces by treating events as points on statistical manifolds endowed with natural information-theoretic metrics. The work establishes rigorous foundations for embedding ICQER events into statistical manifolds, where the Fisher information metric serves as a canonical Riemannian structure quantifying event distinguishability and constraint sensitivity. We demonstrate how informational constraints—arising from quantum principles, causal structures, and contextual conditions—shape manifold curvature, topology, and geodesic flows, fundamentally determining how events interact, evolve, and give rise to emergent quantum-realistic phenomena. Key contributions include: (1) a formal framework for representing events as probability distributions on curved manifolds; (2) the characterization of constraint-induced geometric deformations and their influence on event dynamics; (3) the development of divergence-based projection methods for constraint satisfaction and event filtering; (4) geometric interpretations of quantum uncertainty, contextuality, and complementarity; and (5) computational algorithms for metric computation, geodesic path finding, and event dynamics simulation. Applications span constraint evolution analysis, information flow quantification, quantum event network optimization, and geometric approaches to phenomena including superconductivity and quantum artificial intelligence. The framework provides a unified geometric language bridging quantum mechanics, information theory, and emergent realism, establishing information geometry as an essential tool for understanding the informational architecture of quantum reality. This volume contributes to the ongoing development of ICQER by providing both the mathematical infrastructure and practical computational methods necessary for analyzing complex quantum informational systems through the lens of differential geometry.
The absence of a common mathematical structure underlying quantum theory and relativistic physics remains a central obstacle to unification. While relativity is formulated in terms of spacetime geometry and covariance, quantum theory is intrinsically probabilistic, relying on amplitudes, uncertainty, and statistical interpretation. Existing unification programs typically attempt to quantize geometry or to geometrize quantum fields, but the probabilistic structure itself is rarely treated as a fundamental geometric object. In this paper, we propose a unified description of relativistic and quantum dynamics based on probabilistic geometry. Probability densities are taken as primary variables and are endowed with an intrinsic information–geometric structure defined by the Fisher–Rao metric. Within this framework, physical motion is described as the covariant evolution of probability distributions on a statistical manifold rather than as trajectories in a preassigned phase space or Hilbert space. The central result is a unified probabilistic–geometric field equation in which the relativistic d’Alembertian acting on the probability density is coupled to the scalar curvature of the associated information manifold. In an appropriate nonrelativistic limit, this equation reproduces the quantum potential as a curvature term, establishing a direct correspondence between quantum effects and informational geometry. Relativistic covariance is preserved throughout, allowing quantum and relativistic structures to appear as different regimes of a single probabilistic equation. The proposed framework does not introduce new particles, interactions, or quantization postulates. Instead, it provides a structural unification in which quantum and relativistic features emerge from the geometry of probability itself.
It is shown that the geometry of quantum theory can be derived from geometrical structure that may be considered more fundamental. The basic elements of this reconstruction of quantum theory are the natural metric on the space of probabilities (information geometry), the description of dynamics using a Hamiltonian formalism (symplectic geometry), and requirements of consistency (Kähler geometry). The theory that results is standard quantum mechanics, but in a geometrical formulation that includes also a particular case of a family of nonlinear gauge transformations introduced by Doebner and Goldin. The analysis is carried out for the case of discrete quantum mechanics. The work presented here relies heavily on, and extends, previous work done in collaboration with M. J. W. Hall.