This work employs the empirical logic approach of D. Foulis and C. Randall to compare non-relativistic, conservative, non-compound physical systems (universes) in classical and quantum mechanics. We present definitions of test spaces, propositions, proposition logics, statistical states, observables, and dynamical systems. Then we try to identify their mathematical representatives for both, classical and quantum universes. Except for the case of the dynamical system, which is not defined for classical universes, the definitions prove to be general enough to apply to both cases. Necessary mathematical tools not generally known to physicists are also treated in a concise fashion. These include some basic theory of partially ordered sets (posets) and lattices, in particular boolean and orthomodular lattices.
This work employs the empirical logic approach of D. Foulis and C. Randall to compare non-relativistic, conservative, non-compound physical systems (universes) in classical and quantum mechanics. We present definitions of test spaces, propositions, proposition logics, statistical states, observables, and dynamical systems. Then we try to identify their mathematical representatives for both, classical and quantum universes. Except for the case of the dynamical system, which is not defined for classical universes, the definitions prove to be general enough to apply to both cases. Necessary mathematical tools not generally known to physicists are also treated in a concise fashion. These include some basic theory of partially ordered sets (posets) and lattices, in particular boolean and orthomodular lattices.
The term proposition usually denotes in quantum mechanics (QM) an element of (standard) quantum logic (QL). Within the orthodox interpretation of QM the propositions of QL cannot be associated with sentences of a language stating properties of individual samples of a physical system, since properties are nonobjective in QM. This makes the interpretation of propositions problematical. The difficulty can be removed by adopting the objective interpretation of QM proposed by one of the authors (semantic realism, or SR, interpretation). In this case, a unified perspective can be adopted for QM and classical mechanics (CM), and a simple first order predicate calculus L(x) with Tarskian semantics can be constructed such that one can associate a physical proposition (i.e., a set of physical states) with every sentence of L(x). The set Pf of all physical propositions is partially ordered and contains a subset PTf of testable physical propositions whose order structure depends on the criteria of testability established by the physical theory. In particular, PTf turns out to be a Boolean lattice in CM, while it can be identified with QL in QM. Hence the propositions of QL can be associated with sentences of L(x), or also with the sentences of a suitable quantum language LTQ(x), and the structure of QL characterizes the notion of testability in QM. One can then show that the notion of quantum truth does not conflict with the classical notion of truth within this perspective. Furthermore, the interpretation of QL propounded here proves to be equivalent to a previous pragmatic interpretation worked out by one of the authors, and can be embodied within a more general perspective which considers states as first order predicates of a broader language with a Kripkean semantics.
According to quantum mechanics, the pure states of a microsystem are represented by vectors in a Hilbert Space. Sentences of the form, "x є L" (where x is the state vector for a system, L a subspace of the appropriate Hilbert space), may be called Q-propositions: such sentences serve to summarise our information about the results of possible experiments on the system. Quantum logic investigates the relations which hold among the Q-propositions about a given physical sys tem. These logical relations correspond to algebraic relations among the subspaces of Hilbert space. The algebra of this set of subspaces is non-Boolean, and may be regarded either as an orthomodular lattice or as a partial Boolean algebra. With each type of structure we can associate a logic. A general approach to the semantics for such a logic is provided in terms of interpretations of a formal language within an algebraic structure; an interpretation maps sentences of the language homomorphically onto elements of the structure. When the structure in question is a Boolean algera, the resulting logic is classical; here we develop a semantics for the logic associated with partial Boolean algebras. Two systems of proof, based on the natural deduction systems of Gentzen, are shown for this logic. With respect to the given sematics, these calculi are sound and weakly complete. Strong completeness is conjectured. Quantum logic deals with the logical relations between sentences, and so is properly called a logic. However, it is the logic appropriate to a limited class of sentences: proposals that it should replace classical logic wherever the latter is used should be viewed with suspicion.
This paper develops some of the work of Foulis, Randall, Aerts, and Piron in the fields of empirical science and quantum logic from an algebraic point of view. More specifically, it begins with three axioms of what is called a 'subtraction algebra,' and generates various theorems associated with properties which are useful in empirical science. After a foundation is established, it moves on to define the term manual. This term is defined as a dominated, atomic, semi-Boolean algebra which satisfies an additional condition called condition M. Several properties of the manual are discussed, and different types of manuals are given: classical semi-classical and non-classical. The paper defines operational complements, operational perspectively, atoms, events, and tests, before moving on to define a logic, and how it is derived from a manual. Properties of the logic are discussed, including a subtraction operation, a partial order, and an ortho complement. Next, a computer program is presented. Its purpose is to take a finite semi-Boolean algebra and decide if the algebra is a manual. This is followed by a brief non-classical probabilistic discussion, which includes topics such as weights, pure states, and dispersion-free states. Aerts' and Piron's work with properties, states, and questions is briefly discussed before moving on to several examples, some of them arising from navigation problems. The examples include the hook, the square, the Wright Triangle, and the free algebra. Empirical techniques are demonstrated on these examples. The examples comprise the bulk of this paper.
This introductory chapter sets out a general structure of physical theories which is applicable to both classical and quantum mechanics. We start with measurable properties of a given physical system, be it classical or quantum. These properties are called observables . We then introduce a definition of the state of a physical system in terms of measured values of a sufficiently large set of observables. A theory to describe the system should consists of four basic components: Basic mathematical framework This comprises a set of elements endowed with some specific mathematical structure and properties. In mathematics such a set is generally known as a space . Description of states States are described by elements of the space in the chosen mathematical framework. For this reason the space is called the state space of the system. Description of observables Observables are to be described by quantities defined on the state space. The description should yield all possible values of observables. The relationship between observables and states should be explicitly stated. The following two cases are of particular interest: 2 For a deterministic theory like classical mechanics a state should determine the values of all observables. For a probabilistic theory like quantum mechanics a state should determine the probability distribution of the values of all observables. Description of time evolution (dynamics).
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 mechanics is usually described in the terminology of probability theory even though the properties of the probability spaces associated with it are fundamentally different from the standard ones of probability theory. For example, Kolmogorov's axioms are not general enough to encompass the non-commutative situations that arise in quantum theory. There have been many attempts to generalise these axioms to meet the needs of quantum mechanics. The focus of these attempts has been the observation, first made by Birkhoff and von Neumann (1936), that the propositions associated with a quantum-mechanical system do not form a Boolean σ-algebra. There is almost universal agreement that the probability space associated with a quantum-mechanical system is given by the set of subspaces of a separable Hilbert space, but there is disagreement over the algebraic structure that this set represents. In the most popular model for the probability space of quantum mechanics the propositions are assumed to form an orthocomplemented lattice (Mackey (1963), Jauch (1968)). The fundamental concept here is that of a partial order, that is a binary relation that is reflexive and transitive but not symmetric. The partial order is interpreted as embodying the logical concept of implication in the set of propositions associated with the physical system. Although this model provides an acceptable mathematical expression of the probabilistic structure of quantum mechanics in that the subspaces of a separable Hilbert space give a representation of an ortho-complemented lattice, it has several deficiencies which will be discussed later.