Examples of non-constructive proofs in quantum theory

TL;DR

This paper discusses the importance of constructivism in physical theories, especially quantum mechanics, and presents examples where non-constructive proofs are used to demonstrate macroscopic quantum states, raising questions about their physical applicability.

Contribution

It provides the first explicit examples of non-constructive proofs in quantum theory, highlighting their implications for the physical interpretation of macroscopic quantum states.

Findings

Non-constructive proofs exist in quantum mechanics.

Such proofs rely on logical principles over infinite sets.

Implications for the physical applicability of macroscopic quantum states.

Abstract

Unlike mathematics, in which the notion of truth might be abstract, in physics, the emphasis must be placed on algorithmic procedures for obtaining numerical results subject to the experimental verifiability. For, a physical science is exactly that: algorithmic procedures (built on a certain mathematical formalism) for obtaining verifiable conclusions from a set of basic hypotheses. By admitting non-constructivist statements a physical theory loses its concrete applicability and thus verifiability of its predictions. Accordingly, the requirement of constructivism must be indispensable to any physical theory. Nevertheless, in at least some physical theories, and especially in quantum mechanics, one can find examples of non-constructive statements. The present paper demonstrates a couple of such examples dealing with macroscopic quantum states (i.e., with the applicability of the standard quantum formalism to macroscopic systems). As it is shown, in these examples the proofs of the existence of macroscopic quantum states are based on logical principles allowing one to decide the truth of predicates over an infinite number of things.