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arXiv:quant-ph/9804012 · Physical Review A

Consistency, amplitudes, and probabilities in quantum theory

Ariel Caticha

Published Mar 1, 1998• 61 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.57.1572

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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.
Quantum theory is formulated as the uniquely consistent way to manipulate probability amplitudes. The crucial ingredient is a consistency constraint: if the amplitude of a quantum process can be computed in two different ways, the two answers must agree. The 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 Schrödinger equation must be linear: non-linear variants of quantum mechanics violate the requirement of consistency. PACS: 03.65.Bz, 03.65.Ca. The formalism of quantum mechanics is very robust; modifications are invariably unsuccessful. A number of non-linear alternative theories have been proposed, [1] experimentally tested, [2] and promptly discarded, but the question of whether new non-linear variants might prove successful is not settled. In fact, interest in the subject remains in spite of, or perhaps, partly because of
First steps are taken toward a formulation of quantum mechanics which avoids the use of probability amplitudes and is expressed entirely in terms of observable probabilities. Quantum states are represented not by state vectors or density matrices but by “probability tables,” which contain only the probabilities of the outcomes of certain special measurements. The rule for computing transition probabilities, normally given by the squared modulus of the inner product of two state vectors, is re-expressed in terms of probability tables. The new version of the rule is surprisingly simple, especially when one considers that the notion of complex phases, so crucial in the evaluation of inner products, is entirely absent from the representation of states used here.
The objective of the consistent-amplitude approach to quantum theory has been to justify the mathematical formalism on the basis of three main assumptions: the first defines the subject matter, the second introduces amplitudes as the tools for quantitative reasoning, and the third is an interpretative rule that provides the link to the prediction of experimental outcomes. In this work we introduce a natural and compelling fourth assumption: if there is no reason to prefer one region of the configuration space over another then they should be ‘weighted ’ equally. This is the last ingredient necessary to introduce a unique inner product in the linear space of wave functions. Thus, a form of the principle of insufficient reason is implicit in the Hilbert inner product. Armed with the inner product we obtain two results. First, we elaborate on an earlier proof of the Born probability rule. The implicit appeal to insufficient reason shows that quantum probabilities are not more objective than classical probabilities. Previously we had argued that the consistent manipulation of amplitudes leads to a linear time evolution; our second result is that time evolution must also be unitary. The argument is straightforward and hinges on the conservation of entropy. The only subtlety consists of defining the correct entropy; it is the array entropy, not von Neumann’s. After unitary evolution has been established we proceed to introduce the useful notion of observables and we explore how von Neumann’s entropy can be linked to Shannon’s information theory. Finally, we discuss how various connections among the postulates of quantum theory are made explicit within this approach. 1
The predictions of quantum mechanics are probabilistic. Quantum probabilities are extracted using a postulate of the theory called the Born rule, the status of which is central to the "measurement problem" of quantum mechanics. Efforts to justify the Born rule from other physical principles, and thus elucidate the measurement process, have involved lengthy statistical or information-theoretic arguments. Here we show that Bohm's deterministic formulation of quantum mechanics allows the Born rule for measurements on a single system to be derived, without any statistical assumptions. We solve a simple example where the creation of an ensemble of identical quantum states, together with position measurements on those states, are described by Bohm's quantum dynamics. The calculated measurement outcomes agree with the Born-rule probabilities, which are thus a consequence of deterministic evolution. Our results demonstrate that quantum probabilities can emerge from simple dynamical laws alone, and they support the view that there is no underlying indeterminism in quantum phenomena.
The Born rule, P(x)=∣ψ(x)∣2P(\mathbf{x})=|\psi(\mathbf{x})|^2P(x)=∣ψ(x)∣2 is one of the most successful relations in modern physics. Yet its physical meaning remains conceptually distinctive. Schrödinger's equation describes the continuous evolution of a complex wave amplitude, whereas an actual detector records discrete and localized events. The Born rule connects these two descriptions with remarkable precision, but the physical mechanism responsible for the quadratic relation between wave amplitude and event frequency is not contained explicitly in the Schrödinger equation itself. This paper revisits the historical and physical relationship between Schrödinger's wave mechanics and Born's statistical interpretation of scattering. The purpose is not to deny the empirical validity of the Born rule, but to identify the physical gap between continuous wave evolution and discrete detection statistics. We distinguish three levels of description: the preparation of localized quantum excitations, their continuous propagation and scattering, and the final microscopic detector event. This leads to a central question: \boxedtext{Why should the rate of discrete detection events be proportional to ∣ψ∣2|\psi|^2∣ψ∣2?} The paper argues that the answer cannot be obtained merely by defining ∣ψ∣2|\psi|^2∣ψ∣2 to be a probability density. Instead, a physical theory must explain how a continuously distributed complex amplitude produces a localized event rate. We investigate the possibility that the quadratic dependence emerges from a local interaction functional, energy-transfer mechanism, or bilinear response of the detector to the incident quantum amplitude. A particularly important distinction is made between the wave amplitude itself and the experimentally recorded event frequency. A single detection event is localized, whereas the Born distribution emerges only through repeated preparations and statistical accumulation. The paper therefore proposes that the Born rule may be viewed as the macroscopic statistical image of a microscopic event-generation process. No completed derivation of Born's rule is claimed here. Rather, the paper formulates the precise dynamical problem that such a derivation must solve and identifies several mathematical and experimental constraints that any successful mechanism must satisfy.
The predictions of quantum mechanics are probabilistic. Quantum probabilities are extracted using a postulate of the theory called the Born rule, the status of which is central to the problem of quantum mechanics. Efforts to justify the Born rule from other physical principles, and thus elucidate the measurement process, have involved lengthy statistical or information-theoretic arguments. Here we show that Bohm's deterministic formulation of quantum mechanics allows the Born rule for measurements on a single system to be derived, without any statistical assumptions. We solve a simple example where the creation of an ensemble of identical quantum states, together with position measurements on those states, are described by Bohm's quantum dynamics. The calculated measurement outcomes agree with the Born-rule probabilities, which are thus a consequence of deterministic evolution. Our results demonstrate that quantum probabilities can emerge from simple dynamical laws alone, and they support the view that there is no underlying indeterminism in quantum phenomena.