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Il Nuovo cimento della Società italiana di fisica. B/Il Nuovo cimento B

Can mathematics help solving the interpretational problems of quantum theory?

Luigi Accardi

Published May 1, 1995• 22 citations
Publisher
DOI
10.1007/bf02741475

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Realistic Quantum Theory: Solving the Interpretation Problem. English version.

Rudolf Lierenfeldt

In 2018, a new systematic approach to solving the interpretation problem of quantum theory was published under the title "Realistische Quantentheorie" (in German). In the present document, the main results of this approach are reported (in English), with the necessary mathematical proofs being provided in an appendix. The approach comprises a canonical modeling of quantum reality, a precise formulation of the deterministic and probabilistic laws, proof of logical consistency of these laws (thus solving the well-known no-go problems), unequivocal definition of the quantum state and clarification of its ontological status, solution to the problem of measurement, and clarification of the role played by the reduction of the quantum state. A more detailed description of this solution to the interpretation problem can be found in the summary (pp. 7-9 of this document). Ambiguities of the Copenhagen interpretation are avoided as are speculative concepts of many-worlds theories or the attempt to adhere to outdated particle concepts (Bohmian mechanics). Also avoided are pseudo-philosophical speculations about subjectivism or phenomenalism as expressed in the assertion that the world is "nothing but information". A postprint of the original publication is available in the Zenodo repository (zenodo.7750648) and in the Mathematical Physics Preprint Archive (mp_arc 23-6), see links below under "Related works" / "Is derived from". Remark: The present text mostly corresponds to the "Short Account" published earlier (in German), which is also available in the Zenodo repository (zenodo.8245998, see link below). However, the necessary proofs have been extracted from the original publication and added here in a translated version.

•Feb 15, 2026•1 citation
This book is a comprehensive text in the field of quantum mechanics, covering fundamental concepts including the state of a quantum mechanical system, operators, superposition principle and measurement postulate. The notion of an operator and the algebra of operators are introduced with the help of elementary concepts of mathematical analysis. Mathematical tools developed will help readers in understanding the difficulties encountered in classical physics while trying to explain the experimental results involving atomic spectra and other phenomena. The differential equations that arise while solving eigenvalue problems are solved rigorously, to make the text self-sufficient. The solutions are then physically interpreted and explained. The text offers solved examples, analogous and homework problems to help students in solving practical problems of physics requiring quantum mechanical treatment.
Recently we proposed "quantum language" (or, "the linguistic Copenhagen interpretation of quantum mechanics", "measurement theory") as the language of science.This theory asserts the probabilistic interpretation of science (=the linguistic quantum mechanical worldview), which is a kind of mathematical generalization of Born's probabilistic interpretation of quantum mechanics.In this paper, we consider the most fundamental problems in philosophy of science such as Hempel's raven paradox, Hume's problem of induction, Goodman's grue paradox, Peirce's abduction, flagpole problem, which are closely related to measurement.We believe that these problems can never be solved without the basic theory of science with axioms.Since our worldview (=quantum language) has the axiom concerning measurement, these problems can be solved easily.Thus we believe that quantum language is the central theory in philosophy of science.Hence there is a reason to assert that quantum language gives the mathematical foundations to science.
In 2018, a new systematic approach to solving the interpretation problem of quantum theory was published under the title "Realistische Quantentheorie" (in German). In the present document, the main results of this approach are reported (in English), with the necessary mathematical proofs being provided in an appendix. The approach comprises a canonical modeling of quantum reality, a precise formulation of the deterministic and probabilistic laws, proof of logical consistency of these laws (thus solving the well-known no-go problems), unequivocal definition of the quantum state and clarification of its ontological status, solution to the problem of measurement, and clarification of the role played by the reduction of the quantum state. A more detailed description of this solution to the interpretation problem can be found in the summary (pp. 7-9 of this document). Ambiguities of the Copenhagen interpretation are avoided as are speculative concepts of many-worlds theories or the attempt to adhere to outdated particle concepts (Bohmian mechanics). Also avoided are pseudo-philosophical speculations about subjectivism or phenomenalism as expressed in the assertion that the world is "nothing but information". A postprint of the original publication is available in the Zenodo repository (zenodo.7750648) and in the Mathematical Physics Preprint Archive (mp_arc 23-6), see links below under "Related works" / "Is derived from". Remark: The present text mostly corresponds to the "Short Account" published earlier (in German), which is also available in the Zenodo repository (zenodo.8245998, see link below). However, the necessary proofs have been extracted from the original publication and added here in a translated version.