Some general principles There are some important differences between quantum and classical reasoning which reflect the different mathematical structure of the two theories. The most precise classical description of a mechanical system is provided by a point in the classical phase space, while the most precise quantum description is a ray or onedimensional subspace of the Hilbert space. This in itself is not an important difference. What is more significant is the fact that two distinct points in a classical phase space represent mutually exclusive properties of the physical system: if one is a true description of the sytem, the other must be false. In quantum theory, on the other hand, properties are mutually exclusive in this sense only if the corresponding projectors are mutually orthogonal. Distinct rays in the Hilbert space need not be orthogonal to each other, and when they are not orthogonal, they do not correspond to mutually exclusive properties. As explained in Sec. 4.6, if the projectors corresponding to the two properties do not commute with one another, and are thus not orthogonal, the properties are (mutually) incompatible. The relationship of incompatibility means that the properties cannot be logically compared, a situation which does not arise in classical physics. The existence of this nonclassical relationship of incompatibility is a direct consequence of assuming (following von Neumann) that the negation of a property corresponds to the orthogonal complement of the corresponding subspace of the Hilbert space; see the discussion in Sec. 4.6.
Some general principles There are some important differences between quantum and classical reasoning which reflect the different mathematical structure of the two theories. The most precise classical description of a mechanical system is provided by a point in the classical phase space, while the most precise quantum description is a ray or onedimensional subspace of the Hilbert space. This in itself is not an important difference. What is more significant is the fact that two distinct points in a classical phase space represent mutually exclusive properties of the physical system: if one is a true description of the sytem, the other must be false. In quantum theory, on the other hand, properties are mutually exclusive in this sense only if the corresponding projectors are mutually orthogonal. Distinct rays in the Hilbert space need not be orthogonal to each other, and when they are not orthogonal, they do not correspond to mutually exclusive properties. As explained in Sec. 4.6, if the projectors corresponding to the two properties do not commute with one another, and are thus not orthogonal, the properties are (mutually) incompatible. The relationship of incompatibility means that the properties cannot be logically compared, a situation which does not arise in classical physics. The existence of this nonclassical relationship of incompatibility is a direct consequence of assuming (following von Neumann) that the negation of a property corresponds to the orthogonal complement of the corresponding subspace of the Hilbert space; see the discussion in Sec. 4.6.
Precise rules are developed in order to formalize the reasoning processes involved in standard non-relativistic quantum mechanics, with the help of analogies from classical physics. A classical or quantum description of a mechanical system involves a {\it framework}, often chosen implicitly, and a {\it statement} or assertion about the system which is either true or false within the framework with which it is associated. Quantum descriptions are no less ``objective'' than their classical counterparts, but differ from the latter in the following respects: (i) The framework employs a Hilbert space rather than a classical phase space. (ii) The rules for constructing meaningful statements require that the associated projectors commute with each other and, in the case of time-dependent quantum histories, that consistency conditions be satisfied. (iii) There are incompatible frameworks which cannot be combined, either in constructing descriptions or in making logical inferences about them, even though any one of these frameworks may be used separately for describing a particular physical system. A new type of ``generalized history'' is introduced which extends previous proposals by Omnès, and Gell-Mann and Hartle, and a corresponding consistency condition which does not involve density matrices or single out a direction of time. Applications which illustrate the formalism include: measurements of spin, two-slit diffraction, and the emergence of the classical world from a fully quantum description.
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.
Lying in the core of the conceptual scheme of current quantum theory, the notion of quantum superposition is a consequence of the quantum-mechanical superposition principle. However, relying on precise but not attainable space or time coordinates, quantum superposition makes little sense physically if we use it to describe or explain experimental results obtained by measurement. Space or time is usually modeled by a metric space endowed with the usual distance function. Irrelevant to instrument precision and measurement accuracy, unattainability of precise coordinates is a well-established mathematical fact following from properties of the corresponding metric topology, and hence rules out completely, in principle, the possibility for us to obtain precise coordinates by measurement in actually performed experiments. In classical mechanics, unattainability of precise coordinates is hardly noticeable, and measurement results obtained in classical physics can still be reasonably explained even though precise coordinates are unattainable. However, when we explain measurement results obtained in quantum physics, unattainability of precise coordinates should not be omitted. In the conceptual scheme of current quantum theory, omission of unattainability of precise coordinates is an unfortunate flaw, which is largely responsible not only for the invalidity of quantum superposition in description of the physical world but also for various ineligible applications of quantum mechanics. Nevertheless, in no sense does the flaw imply quantum mechanics as a successful theory failing to be correct. As shown in this article, the flaw is repairable. By taking into account unattainability of precise coordinates, we can reinterpret the meaning of quantum superposition to repair the flaw without affecting eligible applications of quantum mechanics while avoiding all the ineligible applications.
We analyze the difference between classical mechanics and quantum mechanics. We come to the conclusion that this difference can be found in the nature of the observables that are considered for the physical system under consideration. Classical mechanics can only describe a certain kind of what we called ‘‘classical observable.’’ Quantum mechanics can only describe another kind of observable; it cannot describe, however, classical observables. To perform this analysis, we use a theory where every kind of observable can be treated and which is in a natural way a generalization of both classical and quantum mechanics. If in a study of a physical system in this theory we restrict ourselves to the classical observables, we rediscover classical mechanics as a kind of first study of the physical system, where all the nonclassical properties are hidden. If we find that this first study is too rough we can also study the nonclassical part of the physical system by a theory which is eventually quantum mechanics.
It is well known that the “quantum logic” approach to the foundations of quantum mechanics is based on the subspace ordering of projectors on a Hilbert space. In this paper, we show that this is a special case of an ordering on partial isometries, introduced by Halmos and McLaughlin. Partial isometries have a natural physical interpretation, however, they are notoriously not closed under composition. In order to take a categorical approach, we demonstrate that the Halmos-McLaughlin partial ordering, together with tools from both categorical logic and inverse categories, allows us to form a category of partial isometries. This category can reasonably be considered a “categorification” of quantum logic – we therefore compare this category with Abramsky and Coecke's “compact closed categories” approach to foundations and with the “monoidal closed categories” view of categorical logic. This comparison illustrates a fundamental incompatibility between these two distinct approaches to the foundations of quantum mechanics. 9.1 Introduction As early as 1936, von Neumann and Birkhoff proposed treating projectors on Hilbert space as propositions about quantum systems (Birkhoff and von Neumann 1936), by direct analogy with classical order-theoretic approaches to logic. Boolean lattices arise as the Lindenbaum-Tarski algebras of propositional logics, and as the set of all projectors on a Hilbert space also forms an orthocomplemented lattice, the operations meet, join , and complement were analogously interpreted as the logical connectives conjunction, disjunction , and negation .
A system of quantum reasoning for a closed system is developed by treating nonrelativistic quantum mechanics as a stochastic theory. The sample space corresponds to a decomposition, as a sum of orthogonal projectors, of the identity operator on a Hilbert space of histories. Provided a consistency condition is satisfied, the corresponding Boolean algebra of histories, called a framework, can be assigned probabilities in the usual way, and within a single framework quantum reasoning is identical to ordinary probabilistic reasoning. A refinement rule, which allows a probability distribution to be extended from one framework to a larger (refined) framework, incorporates the dynamical laws of quantum theory. Two or more frameworks which are incompatible because they possess no common refinement cannot be simultaneously employed to describe a single physical system. Logical reasoning is a special case of probabilistic reasoning in which (conditional) probabilities are 1 (true) or 0 (false). As probabilities are only meaningful relative to some framework, the same is true of the truth or falsity of a quantum description. The formalism is illustrated using simple examples, and the physical considerations which determine the choice of a framework are discussed. \textcopyright{} 1996 The American Physical Society.