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Physics Today

Does Quantum Mechanics Have Nonlinear Terms?

Barbara Goss Levi

Published Oct 1, 1989• 2 citations
Publisher

Abstract

One hears about testing quantum electrodynamics or relativity theory, but the theoretical basis of quantum mechanics is rarely questioned. Nevertheless, Steven Weinberg (University of Texas) has recently called for high-precision tests of quantum mechanics that are independent of any particular quantum mechanical theory. To pave the way for an examination of quantum mechanics, he has suggested one possible way of generalizing quantum mechanics to make it nonlinear. As Eugene Wigner pointed out in a 1939 paper, the linearity of quantum mechanics is an important assumption and one that may not necessarily always prove true. Weinberg stressed to us that he does not really feel that quantum mechanics is in any immediate danger, but he does believe that we can learn by questioning it: If we find that quantum mechanics cannot be generalized any further, we may come to understand better why it works so well. If we find that the theory can be generalized in a plausible way, then we must ask why ordinary quantum mechanics is so nearly valid—and we may discover some hidden physics in the process.

DOI
10.1063/1.2811176

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One hears about testing quantum electrodynamics or relativity theory, but the theoretical basis of quantum mechanics is rarely questioned. Nevertheless, Steven Weinberg (University of Texas) has recently called for high-precision tests of quantum mechanics that are independent of any particular quantum mechanical theory. To pave the way for an examination of quantum mechanics, he has suggested one possible way of generalizing quantum mechanics to make it nonlinear. As Eugene Wigner pointed out in a 1939 paper, the linearity of quantum mechanics is an important assumption and one that may not necessarily always prove true. Weinberg stressed to us that he does not really feel that quantum mechanics is in any immediate danger, but he does believe that we can learn by questioning it: If we find that quantum mechanics cannot be generalized any further, we may come to understand better why it works so well. If we find that the theory can be generalized in a plausible way, then we must ask why ordinary quantum mechanics is so nearly valid—and we may discover some hidden physics in the process.
Atomic physics experiments which test a nonlinear generalization of quantum mechanics recently formulated by Weinberg are described. The experiments search for a dependence of hyperfine transition frequencies or nuclear spin precession frequencies on the relative populations of the hyperfine or nuclear spin states. set limits less than 10 pHz on the size of the possible nonlinear contributions to these frequencies. In some cases this can be interpreted as a limit of less than- on the fraction of binding energy per nucleon that could be due to a nonlinear correction to a nuclear Hamiltonian. nonlinear addition to quantum mechanics violates causality is discussed. The experiments The possibility that a
IntroductionPhysics has always had several different domains of application in on-going development, and physicists have always striven for unification among its different domains.Unification is usually achieved through development of so-called 'covering theories'.In the nineteenth century, the stunning example was Maxwell's Electrodynamics (MED), which unified electricity and magnetism as one domain of theory.Another major domain of theory then present was Newton's Mechanics (NM), which in the eighteenth century had really launched modern physics as a mathematical discipline.At the turn of the twentieth century, NM and MED were well in place, and were fulfilling many technologically important requirements.But there seemed to be an incompatibility between them.The problem concerned their invariance with respect to choice of reference frame: NM exhibited invariance if the allowed reference frames were all connected through Galilean transformations, whereas MED exhibited invariance if the allowed reference frames were all connected through Lorentz transformations.It looked as though one of these two theories must be more nearly correct than the other, but it was not clear which one was the better one.That problem seemed resolved with the advent of Einstein's Special Relativity Theory (SRT).SRT was believed to capture the true meaning of MED concerning the behavior of light signals, and SRT was certainly an endorsement of Lorentz transformation, so SRT was believed to offer the one possible revision of NM that could make mechanics fully consistent with MED.But meanwhile, new phenomena were being discovered at the micro scale of physics, and they often seemed inexplicable with any known theory, whether NM, SRT, or MED.These were phenomena suggesting quantization of light, quantized atomic states, atomic, molecular and crystal structures, radioactivity, etc.So at almost the same time as one problem seemed to be resolved, other problems were emerging.Since the earlier situation between NM and MED had demanded that Physics allow two seemingly discordant theories to co-exist until some good argument could replace one of them, the situation then presented by the new phenomena being discovered naturally invited the development of another potentially discordant theory: Quantum Mechanics (QM).The discovery of the photoelectric effect, and the introduction of the idea of the photon, initiated QM.Almost immediately, QM was developed to handle the Hydrogen atom, and the ground state thereof, the stability of which was thought to be impossible with MED.www.intechopen.
The usual interpretation of the quantum theory is self-consistent, but it involves an assumption that cannot be tested experimentally, viz., that the most complete possible specification of an individual system is in terms of a wave function that determines only probable results of actual measurement processes. The only way of investigating the truth of this assumption is by trying to find some other interpretation of the quantum theory in terms of at present "hidden" variables, which in principle determine the precise behavior of an individual system, but which are in practice averaged over in measurements of the types that can now be carried out. In this paper and in a subsequent paper, an interpretation of the quantum theory in terms of just such "hidden" variables is suggested. It is shown that as long as the mathematical theory retains its present general form, this suggested interpretation leads to precisely the same results for all physical processes as does the usual interpretation. Nevertheless, the suggested interpretation provides a broader conceptual framework than the usual interpretation, because it makes possible a precise and continuous description of all processes, even at the quantum level. This broader conceptual framework allows more general mathematical formulations of the theory than those allowed by the usual interpretation. Now, the usual mathematical formulation seems to lead to insoluble difficulties when it is extrapolated into the domain of distances of the order of ${10}^{\ensuremath{-}13}$ cm or less. It is therefore entirely possible that the interpretation suggested here may be needed for the resolution of these difficulties. In any case, the mere possibility of such an interpretation proves that it is not necessary for us to give up a precise, rational, and objective description of individual systems at a quantum level of accuracy.
All measurable predictions of classical mechanics can be reproduced from a quantum‐like interpretation of a nonlinear Schrödinger equation. The key observation leading to classical physics is the fact that a wave function that satisfies a linear equation is real and positive, rather than complex. This has profound implications on the role of the Bohmian classical‐like interpretation of linear quantum mechanics, as well as on the possibilities to find a consistent interpretation of arbitrary nonlinear generalizations of quantum mechanics.