Summary Foundational studies in quantum theory have consisted in large part of attempts to avoid paradoxical situations in describing properties of a physical system. Quantum logic addresses some of the perplexing problems that plague interpretations of quantum theory by invoking a non-classical propositional calculus for the set of propositions of a quantum system. Historically, quantum logic derives from von Neumann’s observation that the set of projection operators on a Hilbert space constitute a ‘logic’ of experimental propositions. More than two decades passed from the pioneering work of Birkho and von Neumann before interest in quantum logic was re-ignited by Mackey’s probabilistic analysis of quantum theory. Piron’s axiomatization provided a signicant
Summary Foundational studies in quantum theory have consisted in large part of attempts to avoid paradoxical situations in describing properties of a physical system. Quantum logic addresses some of the perplexing problems that plague interpretations of quantum theory by invoking a non-classical propositional calculus for the set of propositions of a quantum system. Historically, quantum logic derives from von Neumann’s observation that the set of projection operators on a Hilbert space constitute a ‘logic’ of experimental propositions. More than two decades passed from the pioneering work of Birkho and von Neumann before interest in quantum logic was re-ignited by Mackey’s probabilistic analysis of quantum theory. Piron’s axiomatization provided a signicant
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 modular lattices respectively. We then derive these descriptions from what we claim are basic properties of any physical theory - i.e. the notion that a quantity in such a theory may be analysed into parts and that the results of this analysis may be treated in languages with an underlying boolean structure. We shall see that in the course of constructing a model of a theory with these properties different indistinguishable possibilities will arise for how the elements of the model may be named, that is to say different possibilities arise for how they can be associated with points from Set. Taking a particular collection of possibilities gives the usual boolean lattice of the propositions of classical physics. Taking all possibilities - in a sense, the set of all things that may be described by physical theories - gives the lattice of quantum mechanical propositions. This gives an interpretation of quantum mechanics as the complete set of such possible descriptions, the complete physical description of the world.
Quantum logic has been introduced by Birkhoff and von Neumann as an attempt to base the logical primitives, the propositions and the relations and operations among them, on quantum theoretical entities, and thus on the related empirical evidence of the quantum world. We give a brief outline of quantum logic, and some of its algebraic properties, such as nondistributivity, whereby emphasis is given to concrete experimental setups related to quantum logical entities. A probability theory based on quantum logic is fundamentally and sometimes even spectacularly different from probabilities based on classical Boolean logic. We give a brief outline of its nonclassical aspects; in particular violations of Boole-Bell type consistency constraints on joint probabilities, as well as the Kochen-Specker theorem, demonstrating in a constructive, finite way the scarcity and even nonexistence of two-valued states interpretable as classical truth assignments. A more complete introduction of the author can be found in the book "Quantum Logic" (Springer, 1998)
To my friend and mentor, Professor Attilio Rigamonti. Unperformed experiments have no results. Asher Peres Introduction More than a century after its birth, quantum mechanics (QM) remains mysterious. We still don't have general principles from which to derive its remarkable mathematical framework, as happened for the amazing Lorentz transformations, which were rederived by Einstein from the invariance of physical laws in inertial frames and from the constancy of the speed of light. Despite the utmost relevance of the problem of deriving QM from operational principles, research efforts in this direction have been sporadic. The deepest of the early attacks on the problem were the works of Birkhoff, von Neumann, Jordan, and Wigner, attempting to derive QM from a set of axioms with as much physical significance as possible. The general idea in Ref. is to regard QM as a new kind of prepositional calculus , a proposal that spawned the research line of quantum logic , which is based on von Neumann's observation that the two-valued observables – represented in his formulation of QM by orthogonal projection operators – constitute a kind of “logic” of experimental propositions. After a hiatus of two decades of neglect, interest in quantum logic was revived by Varadarajan, and most notably by Mackey, who axiomatized QM within an operational framework, with the single exception of an admittedly ad hoc postulate, which represents the propositional calculus mathematically in the form of an orthomodular lattice. The most significant extension of Mackey's work is the general representation theorem of Piron.
By quoting extensively from unpublished letters written by John von Neumann to Garret Birkhoff during the preparatory phase (in 1935) of their ground-breaking 1936 paper that established quantum logic, the main steps in the thought process leading to the 1936 Birkhoff–von Neumann paper are reconstructed. The reconstruction makes it clear why Birkhoff and von Neumann rejected the notion of quantum logic as the projection lattice of an infinite dimensional complex Hilbert space and why they postulated in their 1936 paper that the quantum propositional system should be isomorphic to an abstract projective geometry. Looking at the paper now I see, that I forgot to say this, which should be said somewhere in the first §: That while common logics did apply to quantum mechanics, if the notion of simultaneous measurability is introduced as an auxiliary notion, we wished to construct a logical system, which applies directly to quantum mechanics – without any extraneous secondary notions like simultaneous measurability. And in order to have such a consequent, one-piece system of logics, we must change the classical class calculus of logics. (J. von Neumann to G. Birkhoff, November 21, 1935)
A set of statements about the properties of a quantum system is looked at as at a partially ordered set of subspaces of finite or infinite dimensional Hilbert space. The operation of ordering is performed on a set of propositions comparing the truth values of these propositions and on the set of subspaces as the opera- tion of inclusion. Based on the required properties these structures are translated into operations on the lattice. The correspondence with Heisenberg uncertainty principle is shown there. Furthermore, it is shown that the lattices correspond- ing to the subspaces of infinite dimensional Hilbert space are not modular. This property is replaced with weaker property of orthomodularity, when operation of the negation is added. Following the work of G. Birkhoff and J. von Neumann, the structure of quantum logic is looked for in projective spaces, which are in- troduced either arithmetically or axiomatically. The examples of quantum logic, their physical implementation and eventual implementation in projective spaces are analysed. 1
Quantum logic as it is presented in Birkhoff-von Neumann’s 1936 paper arose within the framework of the mathematical elaboration of the foundations of quantum mechanics. Birkhoff-von Neumann’s axiomatic was an alternative of the traditional formulation of foundations of quantum mechanics proceeding from the technique of the Hilbert space. It provided more economic formulation of the foundations of the theory. In Mackay’s writings it has been transformed into the axiomatic of quantum theory of probability. However, Mackay came back to the concept of Hilbert space as the fundamental for quantum mechanics. He also adopted the rules of super-selections in his presentation of the foundations of quantum mechanics. The contemporary approach to Birkhoff-von Neumann’s axiomatic is twofold: the criticism of their too rigid construction from the point of view of physics and the development in connection with mathematical intuitionism.