Towards a nonlinear quantum physics (original) (raw)
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Nonlinear quantum mechanics, the superposition principle, and the quantum measurement problem
Pramana, 2011
There are four reasons why our present knowledge and understanding of quantum mechanics could be regarded as incomplete. Firstly, the principle of linear superposition has not been experimentally tested for position eigenstates of objects having more than about a thousand atoms. Secondly, there is no universally agreed upon explanation for the process of quantum measurement. Thirdly, there is no universally agreed upon explanation for the observed fact that macroscopic objects are not found in superposition of position eigenstates. Fourthly, and perhaps most importantly, the concept of time is classical and hence external to quantum mechanics : there should exist an equivalent reformulation of the theory which does not refer to an external classical time. In this paper we argue that such a reformulation is the limiting case of a nonlinear quantum theory, with the nonlinearity becoming important at the Planck mass scale. Such a nonlinearity can provide insights into the problems mentioned above. We use a physically motivated model for a nonlinear Schrödinger equation to show that nonlinearity can help in understanding quantum measurement. We also show that while the principle of linear superposition holds to a very high accuracy for atomic systems, the lifetime of a quantum superposition becomes progressively smaller, as one goes from microscopic to macroscopic objects. This can explain the observed absence of position superpositions in macroscopic objects [lifetime is too small]. It also suggests that ongoing laboratory experiments maybe able to detect the finite superposition lifetime for mesoscopic objects, in the foreseeable future.
0 Perspectives on Nonlinearity in Quantum Mechanics
2000
Earlier H.-D. Doebner and I proposed a family of nonlinear time-evolution equations for quantum mechanics associated with certain unitary representations of the group of diffeomorphisms of physical space. Such nonlinear Schrödinger equations may describe irreversible, dissipative quantum systems. We subsequently introduced the group of nonlinear gauge transformations necessary to understand the resulting quantum theory, deriving and interpreting gauge-invariant parameters that characterize (at least partially) the physical content. Here I first review these and related results, including the coupled nonlinear Schrödinger-Maxwell theory, for which I also introduce the gauge-invariant (hy-drodynamical) equations of motion. Then I propose a further, radical generalization. An enlarged group G of nonlinear transformations, modeled on the general linear group GL(2, R), leads to a beautiful, apparently unremarked symmetry between the wave function's phase and the logarithm of its ampl...
2007
Einstein, de Broglie and others hoped that the schism between classical and quantum physics might one day be overcome by a theory taking into account the essential nonlinearity of elementary physical processes. However, neither their attempts, nor subsequent ones were able to supply a unifying principle that could serve as a startingpoint for a coherent understanding of both microphysical and macroscopic phenomena. In the late 1960s the phenomenon of amplitude quantization, or Macroscopic Quantum Effect (MQE), was discovered in a class of nonlinear oscillating systems in which two or more oscillating subsystems are coupled to each other by interactions having a specific phase-dependent character --so-called argumental interactions. Experimental and theoretical studies of the MQE, carried out up to the present time, suggest the possibility of a new conceptual framework for physics, which would provide a bridge between classical and quantum physics, replacing the Newtonian notion of "force" by a new conception of physical interaction. The present paper provides a brief introduction to the MQE and some ideas about its possible significance in the search for new approaches to the understanding of quantum phenomena.
A Solvable Model of a Nonlinear extension of Quantum Mechanics
arXiv (Cornell University), 2022
We introduce a particular nonlinear generalization of quantum mechanics which has the property that it is exactly solvable in terms of the eigenvalues and eigenfunctions of the Hamiltonian of the usual linear quantum mechanics problem. We hope that this simple example will elucidate some of the issues of interpreting nonlinear generalization of quantum mechanics that have been put forth to resolve questions about quantum measurement theory.
Perspectives on Nonlinearity in Quantum Theory
An enlarged group G of nonlinear transformations, modeled on the general linear group GL(2,R), leads to a beautiful, apparently unremarked symmetry between the wave function's phase and the logarithm of its amplitude. Equations Doebner and I earlier proposed are embedded in a wider, natural family of nonlinear time-evolution equations, on which G acts as a gauge group (leaving physical observations invariant). There exist G-invariant quantities that reduce to the usual probability density and flux for linearizable quantum theories in a particular gauge.
Linear quantum theory and its possible nonlinear generalizations
Annals of Physics, 1994
We show that the Schrödinger equation may be derived as a consequence of three postulates: 1) the hamiltonian formalism 2) a conformal structure 3) a projective structure. These suffice to deduce the geometrical structure of Hilbert space also. Furthermore, the quantum mechanical action principle, and unitary propagator, are reduced to special cases of classical results. Thereafter we explore the relaxation of these postulates. Of the possible generalizations only one appears physically fruitful. This is obtained from a simple gauge freedom of the standard theory. The result is a nonlinear quantum theory that has been the subject of recent interest. We consider the empirical status of this theory, and the problem of its interpretation. It is suggested that the only remaining option is to consider nonperturbative nonlinearities in the context of the quantum measurement problem. This is advanced as a principle of constraint.
Nonlinear gauge interactions: A possible solution to the "measurement problem" in quantum mechanics
2010
Two fundamental, and unsolved problems in physics are: i) the resolution of the "measurement problem" in quantum mechanics ii) the quantization of strongly nonlinear (nonabelian) gauge theories. The aim of this paper is to suggest that these two problems might be linked, and that a mutual, simultaneous solution to both might exist. We propose that the mechanism responsible for the "collapse of the wave function" in quantum mechanics is the nonlinearities already present in the theory via nonabelian gauge interactions. Unlike all other models of spontaneous collapse, our proposal is, to the best of our knowledge, the only one which does not introduce any new elements into the theory. A possible experimental test of the model would be to compare the coherence lengths-here defined as the distance over which quantum mechanical superposition is still valid-for, e.g., electrons and photons in a double-slit experiment. The electrons should have a finite coherence length, while photons should have a much longer coherence length (in principle infinite, if gravity-a very weak effect indeed unless we approach the Planck scale-is ignored).
Quantum Optical Mechanics (QOM): Abolishing 'Light'. [UET6]
This paper replaces the hypothetical 'object' called 'Light' (wave/photon). This sixth report on a new research programme that is investigating the electromagnetic (EM) interaction. This paper analyzes the effects of interactions arising from multiple, remote electrons on one or several, local 'target' electrons. These interactions are the result of the new quantized form of the EM impulse introduced in the previous paper. This model is used to re-interpret various optical effects that have previously required the existence of a fundamental object known as 'LIGHT': a basic entity, considered to be either a particle or a wave (or even both?-the 'photon') that travels across space. In contrast, this new EM model is constructed upon the key role of the 'light' emission processes, categorized as either oscillatory (as in antenna) or transitory (as within atoms). These real emission processes are now integrated into the asynchronous action-at-a-distance model of the EM interaction that is the basis of this new theory. Mathematically, this new model describes algebraically how variable or periodic phenomena (that have been assumed require the use of waves) can be explained by periodic, asynchronous, remote interactions between point particles without any use of differential equations (including the wave equation). This paper now extends the earlier pair-wise interaction between two electrons into the many-body world of macroscopic reality. The two key ideas of interaction saturation and selection are now introduced, which totally differentiate this theory from all other theories constructed around universal, continuous interaction (or 'force') models. By eliminating all the ray, wave and photon models of 'light' this paper now extends the original Newtonian mechanical philosophy of nature to the major domain of optics: both classical and quantum. The emphasis is on the electrons and on the relationship between electrons and not on some hypothetical 'carrier' that travels between them – this is the Newtonian action-at-a-distance particulate model extended to multiple times. The idea of selection leads to the introduction of information waves that identify the location and velocities of all other electrons that might participate in a ray-like exchange of momentum between pairs of electrons (saturation) that always act like particles (real trajectories across space). These supra-luminal waves do not carry momentum but ensure that the interaction minimizes the exchange of action across a non-local region of space. This new model resolves the long-time paradox of electrons as waves or particles: electrons are seen here as real point particles that interact periodically (rather than continuously) together; the focus is on the relationship between them that can be described by the discrete mathematics of particles or the periodic mathematics usually associated with waves. This paper includes the first analytical solution to the 3D scattering of two electrons – in the center-of-mass frame of reference both electrons are shown to go in quantized spiraling, conical motions: towards each other and then away from each other. The present theory provides an alternative to Feynman's mathematical approach to " the mysterious properties of light " while providing a physical explanation for some of the calculational diagrams introduced by Feynman in his approach to quantum electrodynamics (QED). This now replaces all field theories of 'light' without introducing the concept of the photon or virtual particles and so eliminates all QED infinities in the physical properties associated with the interactions of electrons arising from the false idea of vacuum polarization, returning the vacuum to its Newtonian role as the passive, empty space between real particles. This new EM theory establishes a firm foundation for a new quantum theory that covers all scales of nature from the macroscopic to the heart of the atomic nucleus, while covering the complete range of interaction sets from a pair of electrons to the myriads of electrons existing in macroscopic objects. The next (companion) paper will explain the wave-like properties of electrons while providing a new, comprehensive theory of quantum measurement. This next paper will finally establish the critical link between the realistic model of the micro-world introduced so far and the macroscopic world of scientific measurements.
A Local Interpretation of Quantum Mechanics
Foundations of Physics, 2015
It is shown that Quantum Mechanics is ambiguous when predicting relative frequencies for an entangled system if the measurements of both subsystems are performed in spatially separated events. This ambiguity gives way to unphysical consequences: the projection rule could be applied in one or the other temporal(?) order of measurements (being non local in any case), but symmetry of the roles of both subsystems would be broken. An alternative theory is presented in which this ambiguity does not exist. Observable relative frequencies differ from those of orthodox Quantum Mechanics, and a gendaken experiment is proposed to falsify one or the other theory. In the alternative theory, each subsystem has an individual state in its own Hilbert space, and the total system state is direct product (rank one) of both, so there is no entanglement. Correlation between subsystems appears through a hidden label that prescribes the output of arbitrary hypothetical measurements. Measurement is treated as a usual reversible interaction, and this postulate allows to determine relative frequencies when the value of a magnitude is known without in any way perturbing the system, by measurement of the correlated companion. It is predicted the existence of an accompanying system, the de Broglie wave, introduced in order to preserve the action reaction principle in indirect measurements, when there is no interaction of detector and particle. Some action on the detector, different from the one cause by a particle, should be observable.