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arXiv:quant-ph/9809017 · Physical Review A

Comment on “Consistency, amplitudes, and probabilities in quantum theory”

J. Finkelstein

Published Aug 1, 1999• 2 citations
arXivHTMLPDF

Abstract

In a recent article [Phys. Rev. A 57, 1572 (1998)] Caticha has concluded that ``nonlinear variants of quantum mechanics are inconsistent.'' In this paper we identify which assumptions made in that article lead to this conclusion.
DOI
10.1103/physreva.60.1723

Related papers

Consistency, amplitudes and probabilities in quantum theory

Yoel Tikochinsky, Stephen F Gull

Caticha (1998 Phys. Rev. A 57 1572) applies consistency arguments to derive the quantum mechanical rules for compounding probability amplitudes in much the same way as in earlier work by Tikochinsky (1988 J. Math. Phys. 29 398, 1988 Int. J. Theor. Phys. 27 543). These works are examined together to find the assumptions needed to obtain the most general results.

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The assumptions underlying the consistent-amplitude approach to quantum theory

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In this note I expand further on the main assumptions leading to the consistent-amplitude approach to quantum theory and I offer a reply to Jerry Finkelstein's recent comment (quant-ph/9809017) concerning my argument for the linearity of quantum mechanics (Phys. Rev. A57, 1572 (1998)).

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Consistency, amplitudes, and probabilities in quantum theory

Ariel Caticha

Quantum theory is formulated as the only consistent way to manipulate probability amplitudes. The crucial ingredient is a very natural consistency constraint: if there are two different ways to compute the amplitude for a given process, the two answers must agree. This constraint is expressed in the form of functional equations the solution of which leads to the usual sum and product rules for amplitudes. An immediate consequence is that the Schr\"odinger equation must be linear; i.e., nonlinear variants of quantum mechanics are inconsistent. The physical interpretation of the theory is given in terms of a single natural rule. This rule, which does not itself involve any probabilities, is used to obtain a proof of Born's statistical postulate: we show that the probability of a certain outcome in an experiment is given by the square of the modulus of the corresponding amplitude. Thus, consistency leads to indeterminism.

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Reply to “Comment on ‘Consistency, amplitudes, and probabilities in quantum theory’ ”

Ariel Caticha

I reply to Finkelstein’s comment [preceding paper, Phys. Rev. A 60, 1723 (1999)] and remark on the meanings conveyed by the word “amplitude.”Received 23 December 1998DOI:https://doi.org/10.1103/PhysRevA.60.1725©1999 American Physical Society

Physical Review A•Aug 1, 1999•0 citations

Reply to “Comment on ‘Nonlocality counterfactuals, and quantum mechanics’ ”

William George Unruh

Stapp [preceding Comment, Phys. Rev. A 60, 2595 (1999)] has argued that my criticism of his nonlocality proof is wrong. I point out that counterfactual arguments are fraught with dangers, and that any such claim should use them in a conservative manner. A reanalysis of his claims continues to lead me to the conclusion that the nonlocality of quantum mechanics remains unproven.

Physical Review A•Sep 1, 1999•2 citations

Comment on "Association between quantum paradoxes based on weak values and a realistic interpretation of quantum measurements"

Juan José Seoane, Xabier Oianguren-Asua, Albert Solé +1 more

In their paper (arXiv:2402.09879), Aredes and Saldanha analyze several paradoxes related to weak values and present a "general argument" that aims to show that "realistic interpretations ...of weak values lead to inconsistencies". Although we agree with the identified inconsistencies for the specific weak values analyzed there, in this Comment we demonstrate that the origin of these inconsistencies is not their general argument, which is formally incorrect. We use Bohmian mechanics as a counterexample to confirm that their conclusions are not valid for all weak values and quantum theories. In particular, we show that weak values postselected in position can in fact be interpreted within Bohmian mechanics as properties of quantum systems, detached from any measuring devices, in a consistent and meaningful way.

quant-pharXiv:2603.18713•Physical Review A•Mar 20, 2026•3 citations
Quantum theory is formulated as the only consistent way to manipulate probability amplitudes. The crucial ingredient is a very natural consistency constraint: if there are two different ways to compute the amplitude for a given process, the two answers must agree. This constraint is expressed in the form of functional equations the solution of which leads to the usual sum and product rules for amplitudes. An immediate consequence is that the Schr\"odinger equation must be linear; i.e., nonlinear variants of quantum mechanics are inconsistent. The physical interpretation of the theory is given in terms of a single natural rule. This rule, which does not itself involve any probabilities, is used to obtain a proof of Born's statistical postulate: we show that the probability of a certain outcome in an experiment is given by the square of the modulus of the corresponding amplitude. Thus, consistency leads to indeterminism.
In their paper (arXiv:2402.09879), Aredes and Saldanha analyze several paradoxes related to weak values and present a "general argument" that aims to show that "realistic interpretations ...of weak values lead to inconsistencies". Although we agree with the identified inconsistencies for the specific weak values analyzed there, in this Comment we demonstrate that the origin of these inconsistencies is not their general argument, which is formally incorrect. We use Bohmian mechanics as a counterexample to confirm that their conclusions are not valid for all weak values and quantum theories. In particular, we show that weak values postselected in position can in fact be interpreted within Bohmian mechanics as properties of quantum systems, detached from any measuring devices, in a consistent and meaningful way.