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arXiv:1605.03125

Intuitionistic interpretation of quantum mechanics

Arkady Bolotin

Published May 10, 2016• 1 citation
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Abstract

In the present paper, the decision problem of the Schrödinger equation (asking whether or not a given Hamiltonian operator has the nonempty solution set) is represented as a logical statement. As it is shown in the paper, the law of excluded middle would be applicable to the introduced statement if and only if quantum fundamentalism (asserting that everything in the universe is ultimately describable in quantum-mechanical terms) held. But, since the decision problem of the Schrödinger equation is in general undecidable, such a statement is allowed to be other than true or false, explicitly, it may fail to have truth values at all. This makes possible to abandon the law of excluded middle together with quantum fundamentalism in the proposed intuitionistic interpretation of quantum mechanics.

DOI
10.48550/arxiv.1605.03125

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In the present paper, the decision problem of the Schrödinger equation (asking whether or not a given Hamiltonian operator has the nonempty solution set) is represented as a logical statement. As it is shown in the paper, the law of excluded middle would be applicable to the introduced statement if and only if quantum fundamentalism (asserting that everything in the universe is ultimately describable in quantum-mechanical terms) held. But, since the decision problem of the Schrödinger equation is in general undecidable, such a statement is allowed to be other than true or false, explicitly, it may fail to have truth values at all. This makes possible to abandon the law of excluded middle together with quantum fundamentalism in the proposed intuitionistic interpretation of quantum mechanics.
We argue that limitations arise when using non-constructive mathematics (NCM) as a foundation for quantum mechanics, since the Boolean logical substrate of NCM fits a distributive propositional structure, but fits less naturally into the non-distributive lattice of quantum propositions. We distinguish three logical structures relevant to this discussion: Boolean algebras (classical, distributive, complemented), Heyting algebras (constructive, distributive, not necessarily complemented), and orthomodular lattices (quantum, non-distributive). The linear structure of quantum state space is compatible with NCM, but the propositional structure of quantum mechanics forms a non-distributive lattice rather than a Boolean algebra. Topos theory, one of the models of constructive mathematics, offers a way forward: the Döring–Isham framework provides an explicit bridge—the daseinization map—from quantum propositions to a topos whose internal logic is intuitionistic. Although some topoi carry Boolean logic, in general, topoi are not constrained to it. We argue that this bridged framework is more suitable for the mathematical foundation of quantum mechanics than one that retains the law of excluded middle (LEM) without such a bridge. Our discussion centers on the pivotal role of the LEM.
What is the nature of reality? How should be an answer to this question? At this level, we are so deep that all our concepts are obscure. Quantum theory (QT) is at this level. The quest for interpreting it fails because the clarity of our actual fundamental concepts in ordinary language cannot match the precision of the theory's mathematical formalism. The interpretation of quantum mechanics should define and bring clarity to fundamental concepts; not simply rely over them. Quantum theory requires a new worldview, the one that will make it understandable. We discuss on the formulation and interpretation of closed, and final theories, in other words, completely axiomatic theories that define their own conceptual framework, and closed theories that cannot be made simpler. We claim that everything we can properly say about a theory follows from its interpretation, and that interpreting is saying the meaning of each of its mathematical statements. We argue that quantum mechanics is a closed theory; and, therefore, that part of it is part of the final or fundamental theory of physics (FTP). We propose a new entirely axiomatic formulation of part of quantum theory (including all its mathematical framework) in which new powerful and useful elements are added to the existing formalism. Our interpretation is straightforward: it allows us reading or saying the meaning of each of the theory's axiom, theorem or element. We state that the proposed theory is the FTP. Its interpretation may dissolve most of the existing paradoxes around QT. The principle of excluded middle is not valid in general. Both epistemic and ontic elements are present in the theory since it is a theory of knowledge and truth. This and other fundamental concepts are defined by the theory, which can represent both reality and our knowledge about it. Aesthetics is one of its key features.
Within the many versions of the Copenhagen interpretation numerous hints can be found about how to reason about quantum mechanics. However, strict guidelines are often missing. The standard quantum logic is sometimes opted as providing such a guideline but, in itself, it is again puzzling due to the failure of the law of distributivity. In this paper I take an instrumentalist approach inspired by the Copenhagen interpretation towards the interpretation of propositions in a physical theory. Starting with so-called elementary propositions for a physical theory I introduce logical connectives in accordance with this approach and relate the thereby expanding set of propositions to mathematical objects within the theory in a consistent way. This is first done for classical mechanics and then for quantum mechanics. In both cases this approach leads to the development of intuitionistic logics rather than classical ones. It also follows that the obtained intuitionistic quantum logic is a proper generalization of the obtained intuitionist logic for classical mechanics. Finally, some hints are provided in how these logics are to be used in practical situations.
In the absence of a satisfactory interpretation of quantum theory, physical law lacks physical basis. This paper reviews the orthodox, or Dirac–von Neumann interpretation, and makes explicit that Hilbert space describes propositions about measurement results. Kets are defined as conditional clauses referring to measurements in a formal language. It is seen that these clauses are elements of a Hilbert space, such that addition is logical disjunction, the dual space consists of consequent clauses, and the inner product is a set of statements in the subjunctive mood. The probability interpretation gives truth values for corresponding future tense statements when the initial state is actually prepared and the final state is to be measured. The mathematical structure of quantum mechanics is formulated in terms of discrete measurement results at finite level of accuracy and does not depend on an assumption of a substantive, or background, space–time continuum. A continuum of kets, ∣x〉 for x ∈ ℝ3, is constructed from linear combinations of kets in a finite basis. The inner product can be expressed either as a finite sum or as an integral. Discrete position functions are uniquely embedded into smooth wave functions in such a way that differential operators are defined. It is shown that the choice of basis has no effect on underlying physics (quantum covariance). The Dirac delta has a representation as a smooth function. Operators do not in general have an integral form. The Schrödinger equation is shown from the requirements of the probability interpretation. It is remarked that a formal construction of quantum electrodynamics (qed) avoiding divergence problems has been completed using finite dimensional Hilbert space. I conclude that quantum mechanics makes statements about the world with clear physical meaning, such that space is emergent from particle interactions and has no fundamental role.