Related papers: Density conditions for quantum propositions

The geometric form of standard quantum mechanics is compatible with the two postulates: 1) The laws of physics are invariant under the choice of experimental setup and 2) Every quantum observation or event is intrinsically statistical.…

Quantum theory's Hilbert space apparatus in its finite-dimensional version is nearly reconstructed from four simple and quantum-mechanically motivated postulates for a quantum logic. The reconstruction process is not complete, since it…

The quantum logic program originated in a 1936 article by G. Birkhoff and J. von Neumann. This program is generally disregarded due to no-go theorems restricting the existence of the tensor product of elementary quantum logics and, above…

General relativity required the abandonment of Euclidean geometry. Here we show that quantum theory requires the abandonment of classical logic. We show that the Hilbert space representation of quantum theory is logically inevitable. There…

Standard particle theory is based on quantized matter embedded in a classical geometry. Here, a complementary model is proposed, based on classical matter -- massive bodies, without quantum properties -- embedded in a quantum geometry. It…

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…

Lawvere showed that generalised metric spaces are categories enriched over $[0, \infty]$, the quantale of the positive extended reals. The statement of enrichment is a quantitative analogue of being a preorder. Towards seeking a logic for…

The real plane with its set of orientations or angles in $[0,\pi)$ is the simplest non trivial example of a (projective) Hilbert space and provides nice illustrations of quantum formalism. We present some of them, namely covariant integral…

This paper addresses the question why quantum mechanics is formulated in a unitary Hilbert space, i.e. in a manifestly complex setting. Investigating the linear dynamics of real quantum theory in a finite-dimensional Euclidean Hilbert space…

We take points and planes as fundamental, lines as derived, in an axiomatic formulation of three-dimensional projective space, the self-dual nature of which formulation renders automatic the principle of duality.

In quantum mechanics, symmetry groups can be realized by projective, as well as by ordinary unitary, representations. For the permutation symmetry relevant to quantum statistics of N indistinguishable particles, the simplest properly…

Based on a recent proof of free choices in linking equations to the experiments they describe, I clarify relations among some purely mathematical entities featured in quantum mechanics (probabilities, density operators, partial traces, and…

Sequential propositional logic deviates from ordinary propositional logic by taking into account that during the sequential evaluation of a propositional statement,atomic propositions may yield different Boolean values at repeated…

This paper deals with the foundations of quantum mechanics. We start by outlining the characterisation, due to Birkhoff and Von Neumann, of the logical structures of the theories of classical physics and quantum mechanics, as boolean and…

We consider a proper propositional quantum logic and show that it has multiple disjoint lattice models, only one of which is an orthomodular lattice (algebra) underlying Hilbert (quantum) space. We give an equivalent proof for the classical…

Quantum logic was introduced in 1936 by Garrett Birkhoff and John von Neumann as a framework for capturing the logical peculiarities of quantum observables. It generalizes, and on 1-dimensional Hilbert space coincides with, Boolean…

So far, none of attempts to quantize gravity has led to a satisfactory model that not only describe gravity in the realm of a quantum world, but also its relation to elementary particles and other fundamental forces. Here, we outline the…

Since the pioneering work of Birkhoff and von Neumann, quantum logic has been interpreted as the logic of (closed) subspaces of a Hilbert space. There is a progression from the usual Boolean logic of subsets to the "quantum logic" of…

The fundamental algebraic concepts of quantum mechanics, as expressed by many authors, are reviewed and translated into the framework of the relatively new non-distributive system of Boolean fractions (also called conditional events or…

We investigate the first-order theory of closed subspaces of complex Hilbert spaces in the signature $(\lor,\perp,0,1)$, where `$\perp$' is the orthogonality relation. Our main result is that already its quasi-identities are undecidable:…