FLAP Journal 2024 Journal Article
Projection-algebras and Quantum Logic
- Daniel Lehmann
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 com- mutative 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 se- quences 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.