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arXiv:quant-ph/9903031 · Journal of Physics A Mathematical and General

Consistency, amplitudes and probabilities in quantum theory

Yoel Tikochinsky, Stephen F Gull

Published Jul 28, 2000• 4 citations
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Abstract

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.

DOI
10.1088/0305-4470/33/31/314

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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.
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.
Barros discusses that [Jose Acacio de Barros, Int. J. Theor. Phys. 50, 1828 (2011)] Nagata derives inconsistencies from quantum mechanics [K. Nagata, Int. J. Theor. Phys. 48, 3532 (2009)]. Barros considers that the inconsistencies do not come from quantum mechanics, but from extra assumptions about the reality of observables. Here we discuss the fact that there is a contradiction within the quantum theory. We discuss the fact that only one expected value in a spin-1/2 pure state 〈σx〉rules out the reality of the observable. We do not accept extra assumptions about the reality of observables. We use the actually measured results of quantum measurements (raw data). We use a single Pauli observable. We stress that we can use the quantum theory even if we give up the axiomatic system for the quantum theory.
Jose Acacio de Barros discusses as follows [{Jose Acacio de Barros, Int. J. Theor. Phys. {\bf 50}, 1828 (2011)}]. Nagata claims to derive inconsistencies from quantum mechanics [{K. Nagata, Int. J. Theor. Phys. {\bf 48}, 3532 (2009)}]. Jose Acacio de Barros considers that the inconsistencies do not come from quantum mechanics, but from extra assumptions about the reality of observables. Here we discuss the fact that there is a contradiction within the quantum theory. We discuss the fact that only one expected value in a spin-1/2 pure state ⟨σx⟩\langle\sigma_x\rangle⟨σx​⟩ rules out the reality of the observable. We do not accept extra assumptions about the reality of observables. We use the actually measured results of quantum measurements (raw data). We use a single Pauli observable. We stress that we can use the quantum theory even if we give up the axiomatic system for the quantum theory.