Related papers: Geometry and the Quantum
The purpose of this contribution is to give an introduction to quantum geometry and loop quantum gravity for a wide audience of both physicists and mathematicians. From a physical point of view the emphasis will be on conceptual issues…
We present a heuristic derivation of Born's rule and unitary transforms in Quantum Mechanics, from a simple set of axioms built upon a physical phenomenology of quantization. This approach naturally leads to the usual quantum formalism,…
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…
In this paper and a companion paper, we show how the framework of information geometry, a geometry of discrete probability distributions, can form the basis of a derivation of the quantum formalism. The derivation rests upon a few…
Exactly soluble models can serve as excellent tools to explore conceptual issues in non-perturbative quantum gravity. In perturbative approaches, it is only the two radiative modes of the linearized gravitational field that are quantized.…
We consider Hilbert's problem of the axioms of Physics at a qualitative or conceptual level. This issue is more pressing than ever as we seek to understand how both General Relativity and quantum theory could emerge from some deeper theory…
With approaching quantum/noncommutative models for the deep microscopic spacetime in mind, and inspired by our recent picture of the (projective) Hilbert space as the model of physical space behind basic quantum mechanics, we reformulate…
This is a self-contained introduction to quantum Riemannian geometry based on quantum groups as frame groups, and its proposed role in quantum gravity. Much of the article is about the generalisation of classical Riemannian geometry that…
In any attempt to build a quantum theory of gravity, a central issue is to unravel the structure of space-time at the smallest scale. Of particular relevance is the possible definition of coordinate functions within the theory and the study…
In this paper, we show how information geometry, the natural geometry of discrete probability distributions, can be used to derive the quantum formalism. The derivation rests upon three elementary features of quantum phenomena, namely…
We review quantum gravity model building using the new formalism of `quantum Riemannian geometry' to construct this on finite discrete spaces and on fuzzy ones such as matrix algebras. The formalism starts with a `differential structure' as…
A new functional calculus, developed recently for a fully non-perturbative treatment of quantum gravity, is used to begin a systematic construction of a quantum theory of geometry. Regulated operators corresponding to areas of 2-surfaces…
We propose a new method to define theories of random geometries, using an explicit and simple map between metrics and large hermitian matrices. We outline some of the many possible applications of the formalism. For example, a…
Over the last two years, the canonical approach to quantum gravity based on connections and triads has been put on a firm mathematical footing through the development and application of a new functional calculus on the space of gauge…
The geometry of the symplectic structures and Fubini-Study metric is discussed. Discussion in the paper addresses geometry of Quantum Mechanics in the classical phase space. Also, geometry of Quantum Mechanics in the projective Hilbert…
Many basis sets for electronic structure calculations evolve with varying external parameters, such as moving atoms in dynamic simulations, giving rise to extra derivative terms in the dynamical equations. Here we revisit these derivatives…
We generalize the Hamiltonian picture of General Relativity coupled to classical matter, known as geometrodynamics, to the case where such matter is described by a Quantum Field Theory in Curved Spacetime, but gravity is still described by…
In this article the geometry of quantum gravity is quantized in the sense of being noncommutative (first quantization) but it is also quantized in the sense of being emergent (second quantization). A new mechanism for quantum geometry is…
The turn of the millennium was a time of optimism about an approach to noncommutative geometry inspired by rich mathematical objects called `quantum groups' and its applications to quantum spacetime. This would model quantum gravity effects…
The exploration of the Riemannian structure of the Hilbert space has led to the concept of quantum geometry, comprising geometric quantities exemplified by Berry curvature and quantum metric. While this framework has profoundly advanced the…