A Solvable Model of a Nonlinear extension of Quantum Mechanics (original) (raw)

Nonlinear Generalisation of Quantum Mechanics

2021

It is known since Madelung that the Schrödinger equation can be thought of as governing the evolution of an incompressible fluid, but the current theory fails to mathematically express this incompressibility in terms of the wavefunction without facing problem. In this paper after showing that the current definition of quantum-mechanical momentum as a linear operator is neither the most general nor a necessary result of the de Broglie hypothesis, a new definition is proposed that can yield both a meaningful mathematical condition for the incompressibility of the Madelung fluid, and nonlinear generalisations of Schrödinger and Klein-Gordon equations. The derived equations satisfy all conditions that are expected from a proper generalisation: simplification to their linear counterparts by a well-defined dynamical condition; Galilean and Lorentz invariance (respectively); and signifying only rays in the Hilbert space.

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.

Classical Mechanics as Nonlinear Quantum Mechanics

AIP Conference Proceedings, 2007

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.

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.

Towards a nonlinear quantum physics

2003

Introduction 1 2 Complementarity Principle and the Nonlocal Fourier Analysis 2.1 Introduction 2.2 Fourier nonlocal analysis 2.2.1 The quantum operators P x and X 2.3 The measurement and the collapse of the wave train 2.3.1 Measurement without physical interaction 2.4 Heisenberg uncertainty relations and the complementarity principle 2.4.1 Other derivations of Heisenberg's uncertainty relations 2.4.2 Meaning of the uncertainty relations 2.5 References 3 New Generation of Microscopes 39 3.1 Introduction 3.2 The common Fourier microscopes 3.3 Tunneling super-resolution microscope 3.3.1 Measurements of first and second kind 3.4 Apertureless optical microscope 3.5 References 4 Beyond Heisenberg's Uncertainty Relations 57 4.1 Introduction IX x Towards a Nonlinear Quantum Physics 4.2 Wavelet local analysis 4.3 Some elements of the causal theory of de Broglie 4.4 A nonlinear Schrodinger equation 71 4.4.1 Derivation of the nonlinear master Schrodinger equation 4.4.2 Some particular solutions of the nonlinear Schrodinger equation 80 4.4.3 A mathematical model for the quantum particle 4.4.4 An example of the application of the model 4.5 Local Paradigm in place of the nonlocal Fourier Paradigm 4.6 Derivation of a more general set of uncertainty relations 4.7 Experiments to test the general validity of the usual uncertainty relations 4.7.1 Photon Ring Experiment 4.7.2 Limitless Expansion of Matter Wave packets 4.8 An Example of a Concrete Measurement that goes beyond Heisenberg's Uncertainty Relations 4.9 References Contents xi 5.5 Experiments based on the complex interaction process of the causal quantum particles 5.5.1 Experimental test of the tired light model of de Broglie for the photon and its implications on the Cosmological expanding model of the Big Bang for the Universe 5.6 References 205 208 210

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...

Quantum Mechanics is Either Nonlinear or Non-Introspective

Modern Physics Letters A, 1998

The measurement conundrum seems to have plagued quantum mechanics for so long that impressions of an inconsistency amongst its axioms have spawned. A demonstration that such purported inconsistency is fictitious may then be in order and is presented here. An exclusion principle of sorts emerges, stating that quantum mechanics cannot be simultaneously linear and introspective (self-observing). The nonlinearity of this latter approach allows quantum mechanics to describe the entire measuring process, and also to be applied to the entire universe, for which there is no external observer.

The measurement problem in quantum mechanics

2019

In this paper, we discuss the importance of measurement in quantum mechanics and the so-called measurement problem. Any quantum system can be described as a linear combination of eigenstates of an operator representing a physical quantity; this means that the system can be in a superposition of states that corresponds to different eigenvalues, i.e., different physical outcomes, each one incompatible with the others. The measurement process converts a state of superposition (not macroscopically defined) in a well-defined state. We show that, if we describe the measurement by the standard laws of quantum mechanics, the system would preserve its state of superposition even on a macroscopic scale. Since this is not the case, we assume that a measurement does not obey to standard quantum mechanics, but to a new set of laws that form a “quantum measurement theory”.

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).

2 On the Physical Realizability of a Class of Nonlinear Quantum Systems

2016

In this paper, the physical realizability property is investigated for a class of nonlinear quantum systems. This property determines whether a given set of nonlinear quantum stochastic differential equations corresponds to a physical nonlinear quantum system satisfying the laws of quantum mechanics.