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arXiv:quant-ph/0406005

Quantum Mechanics without amplitudes

J. M. A. Figueiredo

Published Jun 1, 2004• 0 citations
arXivHTMLPDF

Abstract

Amplitudes are the major logical object in Quantum Theory. Despite this fact they presents no physical reality and in consequence only observables can be experimetally checked. We discuss the possibility of a theory of Quantum Probabilities capable of give full account to quantum phenomena. Advanteges of this formulation are the evidence of physical processes not described by the orotodox formulation using amplitudes and the possibility of a full algoritimization of Quantum Mechanics.

DOI
10.48550/arxiv.quant-ph/0406005

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Amplitudes are the major logical object in Quantum Theory. Despite this fact they presents no physical reality and in consequence only observables can be experimetally checked. We discuss the possibility of a theory of Quantum Probabilities capable of give full account to quantum phenomena. Advanteges of this formulation are the evidence of physical processes not described by the orotodox formulation using amplitudes and the possibility of a full algoritimization of Quantum Mechanics.
Abstract In the 1950s and 1960s it was causing interest among physicists that in the formalism of quantum mechanics (complex-valued) ‘probability amplitudes’ obeyed laws similar to the laws obeyed by probabilities in the ordinary probability calculus. But this did not then lead decisively to the claim that probabilities should be represented by complex numbers. It became fashionable instead to regard probability amplitudes as an abstract concept from which actual probabilities could be derived by taking the squared moduli of the amplitudes. In this monograph, however, we make another attempt to show how probability amplitudes might after all be identified with actual probabilities. To do this, probability itself is interpreted in a rational Bayesian manner (i.e. as an extension of logic) and a new (complex-valued) probability theory is formulated that incorporates the uncertainty principle (i.e. that takes account of the fact that acquisition of knowledge of a quantum mechanical process generally interferes with it). Taking this probability theory as the new logic of science, and assuming certain physical laws and properties of matter, an interpretation of non-relativistic quantum mechanics is built up. It is consistent with the usual quantum mechanical formalism but allows a clear distinction to be made between the physical world and our knowledge of it.
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.
Based on a thought experiment the Einstein-Podolsky-Rosen paper’s authors concluded that in quantum mechanics the mathematical objects called wave functions fail to describe the physical reality and consequently quantum theory is incomplete. They left open the question of whether or not a complete description of the physical reality exists, but expressed their belief that such a complete theory is possible. In this paper I show that indeed a theory that gives a complete description of the physical reality exists.
Since its emergence, quantum mechanics has been a challenge for an understanding of reality which is based on our intuition in a classical world. Nevertheless, it has often been tried to impose this understanding of reality on quantum theory - with limited success. Instead, it might be a better alternative to redefine the meaning of physical reality. This is the objective of the paper. A consideration of the quantum measurement process, conditional probabilities and some well-known typical quantum physical experiments provides the reasoning for the following three hypotheses: (1) Prior to a first measurement, a physical system is not in a quantum state. (2) Physical reality is all that and only that about which (classical) information is available in the universe. (3) Information creation is an independent process and is not covered by the Schroedinger equation. It is the first step of the quantum measurement process and does not have a classical counterpart. The first hypothesis makes sense only if the quantum measurement process can be described without presupposing an initial state for the system under consideration. This becomes possible by the objective conditional probabilities which represent the transition probabilities between the outcomes of successive quantum measurements and have been introduced by the author in some recent papers. The second hypothesis holds as well in the classical case, but a certain incompleteness of reality is typical of quantum mechanics and the origin of many quantum phenomena. Classically, the existence of a complete reality is presumed, and hypothesis 3 has no meaning then.
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.
We propose a new interpretation of Dempster-Shafer theory of evidence based on a stochastic model for a quantum system developed in terms of amplitude on entities. An entity provides an axiomatic description for the set of tests and states of a physical system and an amplitude density gives a mean for computing probability and pseudo-probability within this framework. The interference and independence of tests relative to an amplitude density are formulated. Various ways of combining entities and amplitudes are presented as superpositions of amplitude. We propose to extend the theory of evidence also to the physical observable experiment in quantum mechanics. So we can use, for quantum mechanics, all the knowledge in the evidence theory. On the contrary the knowledge in the evidence theory can give new insight in the quantum mechanics uncertainty. We give a modal logic interpretation of evidence theory and entity and tests definitions. We establish a bridge between mental approach to uncertainty by evidence theory and the physical phenomena represented by quantum mechanics. When one use the amplitude density in the evidence theory one can generate a new interpretation of joint and conditional definition of the basic probability assignment ( pseudo-probability).