# Finite Orthoalgebras without Two-valued Probability Measures

## Abstract

The Kochen-Specker theorem in quantum mechanics motivates the following combinatorial problem: how to construct a finite orthoalgebra which does not admit a two-valued probability measure? For example, the so-called 'Penrose's dodecahedron' (a projective configuration in C4) generates such an orthoalgebra. In this report one describes a new infinite family of examples of atomic coherent orthoalgebras with the mentioned properties which is intimately related to the geometry of the group's E8. The important features of the construction are the following: (1) the atomic elements are naturally indexed by the elements of a disjoint union of linear manifolds of codimension 1 of an N-dimensional vector space over F2; (2) the description of the orthogonality relation involves a pair of Z/2Z-valued functions on Z/4Z; (3) the symmetry of the construction is described in terms of an extension of GL(N, F2); (4) the whole construction works only if N is divisible by 4 (a phenomenon of periodicity)

- Authors:

- Faculteit Ingenieurswetenschappen, Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussel (Belgium)

- Publication Date:

- OSTI Identifier:
- 21054918

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: AIP Conference Proceedings; Journal Volume: 889; Journal Issue: 1; Conference: 4. international conference on foundations of probability and physics, Vaexjoe (Sweden), 4-9 Jun 2006; Other Information: DOI: 10.1063/1.2713487; (c) 2007 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGEBRA; GEOMETRY; PERIODICITY; PROBABILITY; QUANTUM MECHANICS; SPACE; SYMMETRY; VECTORS

### Citation Formats

```
Ruuge, Artur E.
```*Finite Orthoalgebras without Two-valued Probability Measures*. United States: N. p., 2007.
Web. doi:10.1063/1.2713487.

```
Ruuge, Artur E.
```*Finite Orthoalgebras without Two-valued Probability Measures*. United States. doi:10.1063/1.2713487.

```
Ruuge, Artur E. Wed .
"Finite Orthoalgebras without Two-valued Probability Measures". United States.
doi:10.1063/1.2713487.
```

```
@article{osti_21054918,
```

title = {Finite Orthoalgebras without Two-valued Probability Measures},

author = {Ruuge, Artur E.},

abstractNote = {The Kochen-Specker theorem in quantum mechanics motivates the following combinatorial problem: how to construct a finite orthoalgebra which does not admit a two-valued probability measure? For example, the so-called 'Penrose's dodecahedron' (a projective configuration in C4) generates such an orthoalgebra. In this report one describes a new infinite family of examples of atomic coherent orthoalgebras with the mentioned properties which is intimately related to the geometry of the group's E8. The important features of the construction are the following: (1) the atomic elements are naturally indexed by the elements of a disjoint union of linear manifolds of codimension 1 of an N-dimensional vector space over F2; (2) the description of the orthogonality relation involves a pair of Z/2Z-valued functions on Z/4Z; (3) the symmetry of the construction is described in terms of an extension of GL(N, F2); (4) the whole construction works only if N is divisible by 4 (a phenomenon of periodicity)},

doi = {10.1063/1.2713487},

journal = {AIP Conference Proceedings},

number = 1,

volume = 889,

place = {United States},

year = {Wed Feb 21 00:00:00 EST 2007},

month = {Wed Feb 21 00:00:00 EST 2007}

}