Effectively calculable quantum mechanics

TL;DR

This paper explores the constructiveness of quantum mechanics, arguing that it should be effectively calculable and proposing a reformulation that avoids non-constructive elements, potentially involving hypercomputation or superselection rules.

Contribution

It introduces a constructive reformulation of quantum mechanics that maintains its formalism while ensuring all elements are computable, addressing non-constructiveness in the standard theory.

Findings

Quantum formalism permits non-constructive, undecidable subsets of Hilbert space.

A constructive reformulation can be achieved without hypercomputation.

Superselection rules can enforce constructiveness without hypercomputation.

Abstract

According to mathematical constructivism, a mathematical object can exist only if there is a way to compute (or "construct") it; so, what is non-computable is non-constructive. In the example of the quantum model, whose Fock states are associated with Fibonacci numbers, this paper shows that the mathematical formalism of quantum mechanics is non-constructive since it permits an undecidable (or effectively impossible) subset of Hilbert space. On the other hand, as it is argued in the paper, if one believes that testability of predictions is the most fundamental property of any physical theory, one need to accept that quantum mechanics must be an effectively calculable (and thus mathematically constructive) theory. With that, a way to reformulate quantum mechanics constructively, while keeping its mathematical foundation unchanged, leads to hypercomputation. In contrast, the proposed in the paper superselection rule, which acts by effectively forbidding a coherent superposition of quantum states corresponding to potential and actual infinity, can introduce computable constructivism in a quantum mechanical theory with no need for hypercomputation.