We describe a scheme of quantum mechanics in which the Hilbert space and linear operators are only secondary structures of the theory. As primary structures we consider observables, elements of noncommutative algebra, and the physical states, the nonlinear functionals on this algebra, which associate with results of single measurement. We show that in such scheme the mathematical apparatus of the standard quantum mechanics does not contradict a hypothesis on existence of an objective local reality, a principle of a causality and Kolmogorovian probability theory.
Logical inference leads to one of the major interpretations of probability theory called logical interpretation, in which the probability is seen as a measure of the plausibility of a logical statement under incomplete information. In this paper, assuming that our usual inference procedure makes sense for every set of logical propositions represented in terms of commuting projectors on a given Hilbert space, we extend the logical interpretation to quantum mechanics and derive the Born rule. Our result implies that, from the epistemological viewpoints, we can regard quantum mechanics as a natur…
At the onset of quantum mechanics, it was argued that the new theory would entail a rejection of classical logic. The main arguments to support this claim come from the non-commutativity of quantum observables, which allegedly would generate a non-distributive lattice of propositions, and from quantum superpositions, which would entail new rules for quantum disjunctions. While the quantum logic program is not as popular as it once was, a crucial question remains unsettled: what is the relationship between the logical structures of classical and quantum mechanics? In this essay we answer this q…
We give an overview and conceptual discussion of some of our results on contextuality and non-locality. We focus in particular on connections with the work of Itamar Pitowsky on correlation polytopes, Bell inequalities, and Boole's "conditions of possible experience".
I show that the application of the quantum-mechanical (QM) which-way weak measurement scheme of Vaidman may lead to logical inconsistencies. To this end, I study weak values of projection operators. Weak values are (normalized) amplitudes, operationally defined by a weak measurement followed by postselection. Projector weak values have a direct physical significance. This allows formulating an inconsistency in classical terms, viz., the contradiction in the readings of several measuring devices. To arrive at the contradiction, I also invoke the property of weak measurements not to collapse the…
Nancy Cartwright and the Logic of Quantum Mechanics
本文论述了Nancy Cartwright在她的著作{ \it{How the Laws of Physics Lie} }中对量子力学中测量问题的看法。她不接受量子力学的逻辑。她提出的与量子力学的许多事实结果和认识论相矛盾的命题是她选择经典逻辑的结果，这导致她提出跃迁率作为量子力学的基本对象。我认为这是不正确的。南希·卡特赖特在波包约化问题上所捍卫的立场并没有解决根本问题，即量子和经典物体共存并相互作用的世界的对偶性。我认为，南希·卡特赖特的立场的主要问题是，她难以接受量子力学基础上的矛盾，即量子物体同时具有粒子性和波动性，是一个自然事实。本文所描述的最近的实验，对量子力学的基础和本文的主题有了新的认识。不矛盾原则的局限性进行了讨论，现代辩证唯物主义认为知识和现实之间的相互作用提供了一个有用的框架。
The concept of Logical Entropy, $S_L = 1- \sum_{i=1}^n p_i^2$, where the $p_i$ are normalized probabilities, was introduced by David Ellerman in a series of recent papers. Although the mathematical formula itself is not new, Ellerman provided a sound probabilistic interpretation of $S_L$ as a measure of the distinctions of a partition on a given set. The same formula comes across as a useful definition of entropy in quantum mechanics, where it is linked to the notion of purity of a quantum state. The quadratic form of the logical entropy lends itself to a generalization of the probabilities th…
This thesis develops the categorical proof theory for the non-compact multiplicative dagger linear logic, and investigates its applications to Categorical Quantum Mechanics (CQM). The existing frameworks of CQM are categorical proof theories of compact dagger linear logic, and are motivated by the interpretation of quantum systems in the category of finite dimensional Hilbert spaces. This thesis describes a new non-compact framework called Mixed Unitary Categories which can accommodate infinite dimensional systems, and develops models for the framework. To this end, it builds on linearly distr…
On generalized probabilities: correlation polytopes for automaton logic and generalized urn models, extensions of quantum mechanics and parameter cheats
Karl Popper published, in 1968, a paper that allegedly found a flaw in a very influential article of Birkhoff and von Neumann, which pioneered the field of "quantum logic". Nevertheless, nobody rebutted Popper's criticism in print for several years. This has been called in the historiographical literature an "unsolved historical issue". Although Popper's proposal turned out to be merely based on misinterpretations and was eventually abandoned by the author himself, this paper aims at providing a resolution to such historical open issues. I show that (i) Popper's paper was just the tip of an ic…
The extravagances of quantum mechanics never fail to enrich daily the debate around natural philosophy. Entanglement, non-locality, collapse, many worlds, many minds, and subjectivism have challenged generations of thinkers. Its approach can perhaps be placed in the stream of quantum logic, in which the "strangeness" of quantum mechanics is "measured" through the violation of Bell's inequalities and, from there, attempts an interpretative path that preserves realism yet ends up overturning it, restating the fundamental mechanisms of QM as a logical necessity for a strong realism.
In this paper we discuss limitions appearing while using Boolean Logic in Non-Constructive Mathematics (NCM) as a mathematical foundation for Quantum Mechanics. These limitations naturally vanish using Constructive Mathematics (CM) and Topos Theory which is one of its models. The line of reasoning is in agreement with recent theoretical work on Quantum Logic, and focusses on the pivotal role of the excluded middle law, and this will be addressed in more detail in our discussion.
Quantum mechanics predicts many surprising phenomena, including the two-slit interference of electrons. It has often been claimed that these phenomena cannot be understood in classical terms. But the meaning of "classical" is often not precisely specified. One might, for example, interpret it as "classical physics" or "classical logic" or "classical probability theory". Quantum mechanics also suffers from a conceptual difficulty known as the measurement problem. Early in his career, Hilary Putnam believed that modifications of classical logic could both solve the measurement problem and accoun…
The development of the new logic of partitions (= equivalence relations) dual to the usual Boolean logic of subsets, and its quantitative version as the new logical theory of information provide the basic mathematical concepts to describe distinctions/indistinctions, definiteness/indefiniteness, and distinguishability/indistinguishability. They throw some new light on the objective indefiniteness or literal interpretation of quantum mechanics (QM) advocated by Abner Shimony. This paper shows how the mathematics of QM is the math of indefiniteness and thus, literally and realistically interpret…
