Embedding quantum logics in Hilbert space

We show that an orthomodular lattice is embeddable in a Hilbert space if and only if states of a certain kind exist. A physical motivation for the existence of such states is given and a connection is provided between the quantum logic, algebraic, and operational approaches to quantum mechanics.     

Independence of automorphism group,center, and state space of quantum logics

We prove that quantum logics (=orthomodular posets) admit full independence of the attributes important within the foundations of quantum mechanics. Namely, we present the construction of quantum logics with given sublogics (=physical subsystems), automorphism groups, centers (=classical parts of the systems), and state spaces. Thus, all these parameters are independent. Our result is rooted in the line of investigation carried out by Greechie; Kallus and Trnková; Kalmbach; and Navara and Pták; and considerably enriches the known algebraic methods in orthomodular posets.     

Minimal supports in quantum logics and Hilbert space

It is shown that if a fully atomic, complete orthomodular lattice satisfies a minimal support condition (m.s.c.), then it satisfies Piron's axioms, and is thereby shown to be the projection lattice of a generalized Hilbert space. It is shown, conversely, that m.s.c. holds in Hilbert space subspace lattices. The physical justification for m.s.c. is provided in the context of a property lattice (A, ) for a realistic entity (A, ) in the sense of Foulis-Piron-Randall. This context provides a clear focus on key issues in the debate over the appropriateness of requiring quantum logics to be represented over Hilbert spaces.     

On compatibility in quantum logics

We offer a variant of the intrinsic definition of compatibility in logics. We shown that any compatible subset can be embedded into a Boolean -algebra, we show how the algebra is constructed, and we demonstrate that our definition cannot be weakened unless we put additional assumptions on the logic.     

On blocks in quantum logics

Let L be a quantum logic, Ω(L) the convex set of states on L and M a property, i.e. a convex subset of Ω(L). For any P?L we define A_M(P)={pεL∣μ, vεM and μ|P=v|P?μ(p)=v(p)}. The subset A_M(P)?L is orthomodular and A_M is a closure operator on the subsets of L. We call P?LM-dense, provided A_M(P)=L.We show that a non-classical quantum logic satisfying the chain condition and having a full and unital property M has no block which is M-dense. We also prove that a quantum logic with a property M for which every counter is expectational and no block is M-dense necessarily has uncountably many blocks. In this setting we then discuss projection lattices of von Neumann algebras.     

On the observables on quantum logics

Two postulates concerning observables on a quantum logic are formulated. By Postulate 1 compatibility of observables is defined by the strong topology on the set of observables. Postulate 2 requires that the range of the sum of observables ought to be contained in the smallestC-closed sublogic generated by their ranges. It is shown that the Hilbert space logicL(H) satisfies the two postulates. A theorem on the connection between joint distributions of types 1 and 2 on the logic satisfying Postulate 2 is proved.     

On a characterization of the state space of quantum mechanics

A characterization of state spaces of Jordan algebras by Alfsen and Shultz is improved to a form with more physical appeal (proposed by Wittstock) in the simplified case of a finite dimension.On leave from Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606, Japan     

Symmetries in quantum logics

A symmetry in the quantum logic (L, M) is defined as a pair of bijections :L L andv :M M such that the probabilities are preserved. Some properties of the symmetries are investigated.     

Axiomatization of quantum logics

Starting with a quantum logic (a -orthomodular poset)L, a set of probabilistically motivated axioms is suggested to identifyL with a standard quantum logicL(H) of all closed linear subspaces of a complex, separable, infinite-dimensional Hilbert space. Attention is paid to recent results in this field.     

Automorphisms of quantum logics

Automorphisms of quantum logics are studied. If a quantum logic, i.e. an orthomodular complete lattice of propositions concerning a physical system, is represented as the lattice of all projections in a von Neumann algebra, then each automorphism of the logic can be represented as a Jordan automorphism in the algebra. Groups of transformations of a physical system are represented as groups of ¹-automorphisms in a von Neumann algebra, provided certain continuity conditions are fulfilled.     

Coupling of quantum logics

A quantum logic is a couple (L, M), whereL is a logic andM is a quite full set of states onL. A tensor product in the category of quantum logics is defined and a comparison with the definition of free orthodistributive product of orthomodular σ lattices is given. Several physically important cases are treated.     

Probabilistic forcing in quantum logics

It is shown that an orthomodular lattice can be axiomatized as an ortholattice with aunique operation of identity (bi-implication) instead of the operation of implication, and a corresponding algebraic unified quantum logic is formulated. A statisticalyes-no physical interpretation of the quantum logical propositions is then provided to establish a support for a novelyes-no representation of quantum logic which prompts a conjecture about a possible completion of quantum logic by means of probabilistic forcing.     

Irreversible evolution in quantum logics

The notion of dynamical semigroup is introduced in the quantum logic scheme on the set of the states. Under suitable nonempty mathematical assumptions it is shown that a Heisenberg picture exists equivalent to the Schrödinger one and having many aspects similar to those of the Hilbert case.     

The physical state space of quantum electrodynamics

Starting from the fact that electrically charged particles are massive, we derive a criterion which characterizes the state space of quantum electrodynamics. This criterion clarifies the special role of the electric charge amongst the uncountably many superselection rules in quantum electrodynamics and provides a basis for a general analysis of the infrared problem. Within this framework we establish the existence of asymptotic electromagnetic fields in all charge-sectors, find a general characterization of infra-particles and introduce a notion of asymptotic completeness.     

Convexity and finite quantum logics

The notion of a superposition of a set of states and that of a Jauch-Piron state are geometrically interpreted in the context of the facial structure of the state space of a finite quantum logic.     

Partial and unsharp quantum logics

The total and the sharp character of orthodox quantum logic has been put in question in different contexts. This paper presents the basic ideas for a unified approach to partial and unsharp forms of quantum logic. We prove a completeness theorem for some partial logics based on orthoalgebras and orthomodular posets. We introduce the notion of unsharp orthoalgebra and of generalized MV algebra. The class of all effects of any Hilbert space gives rise to particular examples of these structures. Finally, we investigate the relationship between unsharp orthoalgebras, generalized MV algebras, and orthomodular lattices.     

Spectral theory in quantum logics

If one supposes a quantum logicL to be a -orthocomplete, orthomodular partially ordered set admitting a set of -orthoadditive functions (called states) fromL to the unit intervals [0, 1] such that these states distinguish the ordering and orthocomplement onL, then the observables onL are identified withL-valued measures defined on the Borel subsets of the real line. In this structure (and without the aid of Hilbert space formalism) the author shows that (1) the spectrum of an observable can be completely characterised by studying the observable (A–)^–1, and (2) corresponding to every observableA there is a spectral resolution uniquely determined byA and uniquely determiningA.     

Quantum logics and hilbert space

Starting with a quantum logic (a -orthomodular poset) L. a set of probabilistically motivated axioms is suggested to identify L with a standard quantum logic L(H) of all closed linear subspaces of a complex, separable, infinite-dimensional Hilbert space.     

Automorphisms and symmetries of quantum logics

Given an amalgam of groups then every quantum logicQ ₀ = (L ₀,M ₀) (L ₀ is aσ-orthomodular poset,M ₀ is a full set of states on it) satisfying some reasonable conditions can be embedded in a quantum logicQ = (L, M), in which (1) all the automorphisms ofL form a group ∼-G ₁, (2) all the automorphisms ofM form a group ∼-G ₂, and (3) all the symmetries ofQ form a group ∼-G ₀. The quantum logic of all closed subspaces of a Hilbert spaceH and all its measures satisfies the conditions required fromQ ₀; hence, enlarging it, one can obtain “anything.”