Related papers: All Quantum Probability viewed in Complex Projecti…
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…
A new type of gauge quantum theory (superrelativity) has been proposed. This differs from ordinary gauge theories in sense that the affine connection of our theory is constructed from first derivatives of the Fubini-Study metric tensor.…
Joint measurability of sharp quantum observables is determined pairwise, and so can be captured in a graph. We prove the converse: any graph, whose vertices represent sharp observables, and whose edges represent joint measurability, is…
We study the quantum Riemannian geometry of quantum projective spaces of any dimension. In particular we compute the Riemann and Ricci tensors, using previously introduced quantum metrics and quantum Levi-Civita connections. We show that…
This note is devoted to the study of the links between the Hilbert function of a subscheme X of the projective space, and its geometric properties. We will assume that X is arithmetically Cohen-Macaulay, which allows us to characterize its…
The basic methods of constructing the sets of mutually unbiased bases in the Hilbert space of an arbitrary finite dimension are discussed and an emerging link between them is outlined. It is shown that these methods employ a wide range of…
We propose a special relativistic framework for quantum mechanics. It is based on introducing a Hilbert space for events. Events are taken as primitive notions (as customary in relativity), whereas quantum systems (e.g. fields and…
There has been a body of works deriving the complex Hilbert space structure of quantum theory from axioms/principles/postulates to deepen our understanding about quantum theory and to reveal ways to go beyond it to resolve foundational…
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…
Information geometry provides differential geometric concepts like a Riemannian metric, connections and covariant derivatives on spaces of probability distributions. We discuss here how these concepts apply to quantum field theories in the…
Principle of ``Superrelativity'' has been proposed in order to avoid the contradiction between principle of relativity and foundations of quantum theory. Solutions of a newly derived non-linear Klein-Gordon equation presumably may be…
Building on earlier work, we further develop a formalism based on the mathematical theory of frames that defines a set of possible phase-space or quasi-probability representations of finite-dimensional quantum systems. We prove that an…
We give a survey of our joint ongoing work with Ali Chamseddine, Slava Mukhanov and Walter van Suijlekom. We show how a problem purely motivated by "how geometry emerges from the quantum formalism" gives rise to a slightly noncommutative…
Cirelli, Mani\`{a} and Pizzocchero generalized quantum mechanics by K\"{a}hler geometry. Furthermore they proved that any unital C$^{*}$-algebra is represented as a function algebra on the set of pure states with a noncommutative…
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…
A quantum physical projector is proposed for generally covariant theories which are derivable from a Lagrangian. The projector is the quantum analogue of the integral over the generators of finite one-parameter subgroups of the gauge…
Observable properties of a classical physical system can be modelled deterministically as functions from the space of pure states to outcomes; dually, states can be modelled as functions from the algebra of observables to outcomes. The…
Recently, much research has been carried out on Hamiltonians that are not Hermitian but are symmetric under space-time reflection, that is, Hamiltonians that exhibit PT symmetry. Investigations of the Sturm-Liouville eigenvalue problem…
We present a simple geometric construction linking geometric to deformation quantization. Both theories depend on some apparently arbitrary parameters, most importantly a polarization and a symplectic connection, and for real polarizations…
We explain why and how the Hilbert space comes about in quantum theory. The axiomatic structures of vector space, of scalar product, of orthogonality, and of the linear functional are derivable from the statistical description of quantum…