Related papers: Local tomography and the Jordan structure of quant…
In 1934, Jordan et al. gave a necessary algebraic condition, the Jordan identity, for a sensible theory of quantum mechanics. All but one of the algebras that satisfy this condition can be described by Hermitian matrices over the complexes…
In quantum mechanics, often it is important for the representation of quantum system to study the structure-preserving bijective maps of the quantum system. Such maps are also called isomorphisms or automorphisms. In this note, using the…
A scheme for constructing quantum mechanics is given that does not have Hilbert space and linear operators as its basic elements. Instead, a version of algebraic approach is considered. Elements of a noncommutative algebra (observables) and…
In this survey paper we give an overview over constructions of geometries associated to Jordan structures (algebras, triple systems and pairs), featuring analogs of these constructions with the Lie functor on the one hand and with the…
The principal goal of this paper is to pass all quantum probability formulas to the projective space associated to the complex Hilbert space of a given quantum system, providing a more complete geometrization of quantum theory. Quantum…
Categorical quantum mechanics studies quantum theory in the framework of dagger-compact closed categories. Using this framework, we establish a tight relationship between two key quantum theoretical notions: non-locality and…
Quantum theory is notoriously counterintuitive, and yet remains entirely self-consistent when applied universally. Here we uncover a new manifestation of its unusual consequences. We demonstrate, theoretically and experimentally (by means…
We study real and integral structures in the space of solutions to the quantum differential equations. First we show that, under mild conditions, any real structure in orbifold quantum cohomology yields a pure and polarized tt^*-geometry…
Different versions of the notion of a state have been formulated for various so-called quantum structures. In this paper, we investigate the interplay among states on synaptic algebras and on its sub-structures. A synaptic algebra is a…
Quantum many-body systems are typically endowed with a tensor product structure. This structure is inherited from probability theory, where the probability of two independent events is the product of the probabilities. The tensor product…
We construct a class of quantum field theories depending on the data of a holomorphic Poisson structure on a piece of the underlying spacetime. The main technical tool relies on a characterization of deformations and anomalies of such…
Quantum-mechanical observables for spatial and spacetime localization are considered from a lattice-theoretic perspective. It is shown that when replacing the lattice of all complex orthogonal projections underlying the Born rule by the…
In the following we undertake to derive quantum theory as a stochastic low-energy and coarse-grained theory from a more primordial discrete and basically geometric theory living on the Planck scale and which (as we argue) possibly underlies…
Non-commutative propositions are characteristic of both quantum and non-quantum (sociological, biological, psychological) situations. In a Hilbert space model states, understood as correlations between all the possible propositions, are…
The absolute sets of local systems on a smooth complex algebraic variety are the subject of a conjecture of N. Budur and B. Wang based on an analogy with special subvarieties of Shimura varieties. An absolute set should be the…
We consider the action of symplectic monodromy on chain-level enhancements of quantum cohomology. First, we construct a family version of $A_\infty$-structure on quantum cohomology (this should morally correspond to Hochschild cohomology of…
In this paper we describe all subalgebras and automorphisms of simple noncommutative Jordan superalgebras $K_3(\alpha,\beta,\gamma)$ and $D_t(\alpha,\beta,\gamma)$ and compute the derivations of the nontrivial simple finite-dimensional…
We propose a toy-model theory, that mimics various characteristic features of quantum mechanics. Unlike the toy-models previously studied in the literature, our toy-model allows for an observer to have a full knowledge of a system's real…
The quantum mechanical transition probability is symmetric. A probabilistically motivated and more general quantum logical definition of the transition probability was introduced in two preceding papers without postulating its symmetry, but…
We show that, assuming that quantum mechanics holds locally, the finite speed of information is the principle that limits all possible correlations between distant parties to be quantum mechanical as well. Local quantum mechanics means that…