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Projection-algebras and quantum logic

Daniel Lehmann

TL;DR

This work introduces P-algebras as a non-commutative, non-associative generalization of Boolean algebras tailored to quantum logic, modeling sequential measurement knowledge via a projection-like conjunction. It develops seven defining properties, defines a corresponding language with a unary negation and a binary dot, and proves a sound and complete deductive system for this logic, linking syntax to semantics through interpretation on P-algebras. The framework yields an orthomodular poset structure and recovers Boolean algebras in the commutative case, while connecting to Hilbert-space subspaces through atomic P-algebras and projections. The paper also sketches avenues for extending to probabilistic aspects and richer quantum models, including representation, topology on atoms, and potential tensor-like constructions across P-algebras.

Abstract

P-algebras are a non-commutative, non-associative generalization of Boolean algebras that are for quantum logic what Boolean algebras are for classical logic. P-algebras have type <X, 0, ', .> where 0 is a constant, ' is unary and . is binary. Elements of X are called features. A partial order is defined on the set X of features by x <= y iff x.y = x. Features commute, i.e., x.y = y.x iff x.y <= x. Features x and y are said to be orthogonal iff x.y = 0 and orthogonality is a symmetric relation.The operation + is defined as the dual of . and it is commutative on orthogonal features. The closed subspaces of a separable Hilbert space form a P-algebra under orthogonal complementation and projection of a subspace onto another one.P-algebras are complemented orthomodular posets but they are not lattices. Existence of least upper bounds for ascending sequences is equivalent to the existence of least upper bounds for countable sets of pairwise orthogonal elements. Atomic algebras are defined and their main properties are studied. The logic of P-algebras is then completely characterized. The language contains a unary connective corresponding to the operation ' and a binary connective corresponding to the operation ".". It is a substructural logic of sequents where the Exchange rule is extremely limited. It is proved to be sound and complete for P-algebras.

Projection-algebras and quantum logic

TL;DR

This work introduces P-algebras as a non-commutative, non-associative generalization of Boolean algebras tailored to quantum logic, modeling sequential measurement knowledge via a projection-like conjunction. It develops seven defining properties, defines a corresponding language with a unary negation and a binary dot, and proves a sound and complete deductive system for this logic, linking syntax to semantics through interpretation on P-algebras. The framework yields an orthomodular poset structure and recovers Boolean algebras in the commutative case, while connecting to Hilbert-space subspaces through atomic P-algebras and projections. The paper also sketches avenues for extending to probabilistic aspects and richer quantum models, including representation, topology on atoms, and potential tensor-like constructions across P-algebras.

Abstract

P-algebras are a non-commutative, non-associative generalization of Boolean algebras that are for quantum logic what Boolean algebras are for classical logic. P-algebras have type <X, 0, ', .> where 0 is a constant, ' is unary and . is binary. Elements of X are called features. A partial order is defined on the set X of features by x <= y iff x.y = x. Features commute, i.e., x.y = y.x iff x.y <= x. Features x and y are said to be orthogonal iff x.y = 0 and orthogonality is a symmetric relation.The operation + is defined as the dual of . and it is commutative on orthogonal features. The closed subspaces of a separable Hilbert space form a P-algebra under orthogonal complementation and projection of a subspace onto another one.P-algebras are complemented orthomodular posets but they are not lattices. Existence of least upper bounds for ascending sequences is equivalent to the existence of least upper bounds for countable sets of pairwise orthogonal elements. Atomic algebras are defined and their main properties are studied. The logic of P-algebras is then completely characterized. The language contains a unary connective corresponding to the operation ' and a binary connective corresponding to the operation ".". It is a substructural logic of sequents where the Exchange rule is extremely limited. It is proved to be sound and complete for P-algebras.
Paper Structure (27 sections, 30 theorems, 45 equations, 2 figures)

This paper contains 27 sections, 30 theorems, 45 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Von Neumann's doubts about the Hilbert space formalism
  4. Background
  5. P-algebras
  6. Preliminaries
  7. Definition of P-algebras
  8. Properties of P-algebras
  9. The operation $+$ and duality
  10. Language and interpretation
  11. The language and its interpretation
  12. Propositions and Sequents
  13. Interpretation
  14. The deductive system $\cal R$
  15. A sound deductive system
  16. ...and 12 more sections

Key Result

Theorem 1

In a P-algebra, for any $x , y , z \in X$,

Figures (2)

  • Figure 1: Deductive system R
  • Figure 2: Derived rules

Theorems & Definitions (76)

  • Definition 1
  • Definition 2
  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • Theorem 2
  • proof
  • Definition 3
  • Theorem 3
  • ...and 66 more