Nonlinear Quantum Mechanics Research Articles

Diffeomorphism groups and nonlinear quantum mechanics

This talk is dedicated to my friend and collaborator, Prof. Dr. Heinz-Dietrich Doebner, on the occasion of his 80th birthday. I shall review some highlights of the approach we have taken in deriving and interpreting an interesting class of nonlinear time-evolution equations for quantum-mechanical wave functions, with few equations; more detail may be found in the references. Then I shall comment on the corresponding hydrodynamical description.

Perfect State Distinguishability and Computational Speedups with Postselected Closed Timelike Curves

Bennett and Schumacher's postselected quantum teleportation is a model of closed timelike curves (CTCs) that leads to results physically different from Deutsch's model. We show that even a single qubit passing through a postselected CTC (P-CTC) is sufficient to do any postselected quantum measurement, and we discuss an important difference between "Deutschian" CTCs (D-CTCs) and P-CTCs in which the future existence of a P-CTC might affect the present outcome of an experiment. Then, based on a suggestion of Bennett and Smith, we explicitly show how a party assisted by P-CTCs can distinguish a set of linearly independent quantum states, and we prove that it is not possible for such a party to distinguish a set of linearly dependent states. The power of P-CTCs is thus weaker than that of D-CTCs because the Holevo bound still applies to circuits using them regardless of their ability to conspire in violating the uncertainty principle. We then discuss how different notions of a quantum mixture that are indistinguishable in linear quantum mechanics lead to dramatically differing conclusions in a nonlinear quantum mechanics involving P-CTCs. Finally, we give explicit circuit constructions that can efficiently factor integers, efficiently solve any decision problem in the intersection of NP and coNP, and probabilistically solve any decision problem in NP. These circuits accomplish these tasks with just one qubit traveling back in time, and they exploit the ability of postselected closed timelike curves to create grandfather paradoxes for invalid answers.

The wave-corpuscle properties of microscopic particlesin the nonlinear quantum-mechanical systems

We debate first the properties of quantum mechanics and its difficulties and the reasons resulting in these diffuculties and its direction of development. The fundamental principles of nonlinear quantum mechanics are proposed and established based on these shortcomings of quantum mechanics and real motions and interactions of microscopic particles and backgound field in physical systems. Subsequently, the motion laws and wave-corpuscle duality of microscopic particles described by nonlinear Schr?dinger equation are studied completely in detail using these elementary principles and theories. Concretely speaking, we investigate the wave-particle duality of the solution of the nonlinear Schr?dinger equation, the mechanism and rules of particle collision and the uncertainty relation of particle’s momentum and position, and so on. We obtained that the microscopic particles obey the classical rules of collision of motion and satisfy the minimum uncertainty relation of position and momentum, etc. From these studies we see clearly that the moved rules and features of microscopic particle in nonlinear quantum mechanics is different from those in linear quantum mechanics. Therefore, nolinear quantum mechanics is a necessary result of development of quantum mechanics and represents correctly the properties of microscopic particles in nonlinear systems, which can solve difficulties and problems disputed for about a century by scientists in linear quantum mechanics field.

非线性量子系统中由非线性Schrödinger方程描述的微观粒子的特性 The Properties of Microscopic Particles Described by Nonlinear Schrödinger Equation in Nonlinear Quantum Systems

原有量子力学的由Schrodinger方程不能很好地描述微观粒子的局域特性和波-粒二象性，因此，量子力学必须改进和发展。在非线性系统中我们考虑了粒子之间或它与背景场之间的非线性相互作用，于是必须用非线性Schrodinger方程去描述的微观粒子的状态。至此，微观粒子的状态和特征相对于线性系统发生了根本性的变化。在这种情况下不论外势场如何变化，但非线性Schrodinger方程都能给出具有波-粒二象特性的解，它由包络孤子和一个载波组成，有一个具有确定的位置的质心；同时，微观粒子此时也具有一个确定的质量，能量和动量，和它们遵守的质量，能量和动量守恒定律，在外场作用时粒子是稳定的，并满足经典运动方程和遵守哈密顿方程和拉格朗日方程。于是用非线性Schrodinger方程去描述的微观粒子具有一个明显的的局域特性和波-粒二象性， 从而彻底消除了原有量子力学存在近一个世纪的困难和问题。基于这些新的结果我们可建立起完整和系统的非线性量子力学理论体系，推动量子力学的发展。因此这一研究具有重要意义。 In original quantum mechanics the Schrodinger equation is, in essence, a wave equation, wherein the microscopic particles depicted have only a wave feature, and not a corpuscle nature. These descriptors do not agree only with the Broglie relation of wave-corpuscle duality, but also with experimental results and the traditional knowledge of particle concept. Meanwhile, the theory gives only some approximate solutions. Thus a series of contradictory representations and problems occur in quantum mechanics, which have re-sulted in durative disputations focused on the area of physics and have not led to any united conclusions until now. The only way to solve these problems and difficulties is to develop the quantum mechanics. We inves-tigate in detail wave-corpuscle duality of microscopic particles described by a nonlinear Schrodinger equa-tion in nonlinear quantum systems. Concretely speaking, we here study the properties of the solution of the nonlinear Schrodinger equation, the stability of microscopic particles, invariance and conservation laws of motion of particles, the Hamiltonian principle of particle motion and corresponding Lagrangian and Hamilton equations, the classical rule of microscopic particle motion, and so on. Studied results show that the solution of the nonlinear Schrodinger equation depicting microscopic particles is a soliton and have a wave-corpuscle duality, microscopic particles have always a mass center and possess determinant positions, sizes, mass, mo-mentum, energy and form, their mass, momentum and energy satisfy corresponding conservation laws, their dynamic states can be described by both nonlinear Schrodinger equation and classical Lagrangian and Ham-ilton equations, their motions obey the classical Newton-type law of motion. These properties indicate that the microscopic particles described by a nonlinear Schrodinger equation have a corpuscle nature. However, the solutions of dynamic equation are some solitary waves which are accompanied by a carrier wave to propagate in space-time, and possess certain frequency and wave speed. Thus microscopic particles described by nonlinear quantum mechanics have also wave feature. Thus we can affirm that the microscopic particles depicted by a nonlinear Schrodinger equation have a perfect wave-corpuscle duality, which are in essence different from those in linear quantum mechanics. Based on these results we can establish a new nonlinear quantum mechanics, which can solve these difficulties and problems disputed for about a century in linear quantum mechanics.

Nonlinear quantum mechanics, the superposition principle, and the quantum measurement problem

There are four reasons why our present knowledge and understanding of quantum mechanics can be regarded as incomplete. (1) The principle of linear superposition has not been experimentally tested for position eigenstates of objects having more than about a thousand atoms. (2) There is no universally agreed upon explanation for the process of quantum measurement. (3) There is no universally agreed upon explanation for the observed fact that macroscopic objects are not found in superposition of position eigenstates. (4) 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 aforesaid problems. We use a physically motivated model for a nonlinear Schrodinger 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 may be able to detect the finite superposition lifetime for mesoscopic objects in the near future.

On Nonlinear Quantum Mechanics, Noncommutative Phase Spaces, Fractal-Scale Calculus and Vacuum Energy

A (to our knowledge) novel Generalized Nonlinear Schrödinger equation based on the modifications of Nottale-Cresson’s fractal-scale calculus and resulting from the noncommutativity of the phase space coordinates is explicitly derived. The modifications to the ground state energy of a harmonic oscillator yields the observed value of the vacuum energy density. In the concluding remarks we discuss how nonlinear and nonlocal QM wave equations arise naturally from this fractal-scale calculus formalism which may have a key role in the final formulation of Quantum Gravity.

THE CHANGES OF STATES AND PROPERTIES OF MICROSCOPIC PARTICLES UNDER INFLUENCES OF DIFFERENT EXTERNAL POTENTIAL FIELDS

The properties of microscopic particles are studied using the linear Schrödinger equation in quantum mechanics and nonlinear Schrödinger equation, respectively. The results obtained show that the microscopic particles have only a wave nature in quantum mechanics, but a wave-corpuscle duality in nonlinear systems depicted by the nonlinear Schrödinger equation, no matter the form of external potentials. Thus we know that the kinetic energy term in dynamic equations determines the wave feature of the particles; the nonlinear interaction term determines the corpuscle feature; their combination makes the microscopic particles have a wave-corpuscle duality. However the external potential term can change the phase and group velocities of motion, phase, amplitude, frequency and form of wave for the particles in both quantum mechanics and the nonlinear quantum systems, although it cannot change these fundamental natures of particles, no matter the forms. Meanwhile, we find that the changes of positions of the microscopic particles by increasing the time under action of an external potential satisfy the Newton-type equation of motion in nonlinear quantum systems. Thus the investigations make us not only see the limits and approximations of quantum mechanics but also know the necessity and importance of developing nonlinear quantum mechanics on the basis of the nonlinear Schrödinger equation.

Can Closed Timelike Curves or Nonlinear Quantum Mechanics Improve Quantum State Discrimination or Help Solve Hard Problems?

We study the power of closed timelike curves (CTCs) and other nonlinear extensions of quantum mechanics for distinguishing nonorthogonal states and speeding up hard computations. If a CTC-assisted computer is presented with a labeled mixture of states to be distinguished--the most natural formulation--we show that the CTC is of no use. The apparent contradiction with recent claims that CTC-assisted computers can perfectly distinguish nonorthogonal states is resolved by noting that CTC-assisted evolution is nonlinear, so the output of such a computer on a mixture of inputs is not a convex combination of its output on the mixture's pure components. Similarly, it is not clear that CTC assistance or nonlinear evolution help solve hard problems if computation is defined as we recommend, as correctly evaluating a function on a labeled mixture of orthogonal inputs.

Two-Slit and Aharonov-Bohm Experiments in Magnetic Field

We discuss the two-slit experiment and the Aharonov-Bohm (AB) experiment in the magnetic field. In such a case the electron moving in the magnetic field produces so called synchrotron radiation. In other words the photons are emitted from the points of the electron trajectory and it means that the trajectory of electron is visible in the synchrotron radiation spectrum. The axiomatic system of quantum mechanics does not enable to define the trajectory of the elementary particle. The two-slit experiment and AB experiment in a magnetic field was never performed and it means that they are the missing experiments of quantum mechanics. The extension of the discussion to the cosmical rays moving in the magnetic field of the Saturn magnetosphere and its rings is mentioned. It is related to the probe CASSINI. The solution of the problem in the framework of the hydrodynamical model of quantum mechanics and the nonlinear quantum mechanics is also mentioned.

Features and states of microscopic particles in nonlinear quantum-mechanics systems

In this paper, we present the elementary principles of nonlinear quantum mechanics (NLQM), which is based on some problems in quantum mechanics. We investigate in detail the motion laws and some main properties of microscopic particles in nonlinear quantum systems using these elementary principles. Concretely speaking, we study in this paper the wave-particle duality of the solution of the nonlinear Schrodinger equation, the stability of microscopic particles described by NLQM, invariances and conservation laws of motion of particles, the Hamiltonian principle of particle motion and corresponding Lagrangian and Hamilton equations, the classical rule of microscopic particle motion, the mechanism and rules of particle collision, the features of reflection and the transmission of particles at interfaces, and the uncertainty relation of particle motion as well as the eigenvalue and eigenequations of particles, and so on. We obtained the invariance and conservation laws of mass, energy and momentum and angular momentum for the microscopic particles, which are also some elementary and universal laws of matter in the NLQM and give further the methods and ways of solving the above questions. We also find that the laws of motion of microscopic particles in such a case are completely different from that in the linear quantum mechanics (LQM). They have a lot of new properties; for example, the particles possess the real wave-corpuscle duality, obey the classical rule of motion and conservation laws of energy, momentum and mass, satisfy minimum uncertainty relation, can be localized due to the nonlinear interaction, and its position and momentum can also be determined, etc. From these studies, we see clearly that rules and features of microscopic particle motion in NLQM is different from that in LQM. Therefore, the NLQM is a new physical theory, and a necessary result of the development of quantum mechanics and has a correct representation of describing microscopic particles in nonlinear systems, which can solve problems disputed for about a century by scientists in the LQM field. Hence, the NLQM built is very necessary and correct. The NLQM established can promote the development of physics and can enhance and raise the knowledge and recognition levels to the essences of microscopic matter. We can predict that nonlinear quantum mechanics has extensive applications in physics, chemistry, biology and polymers, etc.

Nonlinear quantum mechanics: Results and open questions

About 15 years ago, we (Heinz-Dietrich Doebner and I) proposed a special type of nonlinear modification of the usual Schrodinger time-evolution equation in quantum mechanics. Our equation was motivated by certain unitary representations of the group of diffeomorphisms of physical space, in the framework of either nonrelativistic local current algebra or quantum Borel kinematics. Subsequently, we developed this and related approaches to nonlinearity in quantum mechanics considerably further, to incorporate theories of measurement, groups of nonlinear gauge transformations, symmetry and invariance properties, unification of a large family of nonlinear perturbations, and possible physical contexts for quantum nonlinearity. Some of our results and highlights of some open questions are summarized.

A Relativistic Gauge Theory of Nonlinear Quantum Mechanics and Newtonian Gravity

A new kind of gauge theory is introduced, where the minimal coupling and corresponding covariant derivatives are defined in the space of functions pertaining to the functional Schroedinger picture of a given field theory. While, for simplicity, we study the example of a U(1) symmetry, this kind of gauge theory can accommodate other symmetries as well. We consider the resulting relativistic nonlinear extension of quantum mechanics and show that it incorporates gravity in the (0+1)-dimensional limit, similar to recently studied Schroedinger-Newton equations. Gravity is encoded here into a universal nonlinear extension of quantum theory. A probabilistic interpretation (Born's rule) holds, provided the underlying model is scale free. Keywords: nonlinear functional Schroedinger equation, gauge symmetry, Newtonian gravity.

Quantum-classical dynamics of wave fields

An approach to the quantum-classical mechanics of phase space dependent operators, which has been proposed recently, is remodeled as a formalism for wave fields. Such wave fields obey a system of coupled nonlinear equations that can be written by means of a suitable non-Hamiltonian bracket. As an example, the theory is applied to the relaxation dynamics of the spin-boson model. In the adiabatic limit, a good agreement with calculations performed by the operator approach is obtained. Moreover, the theory proposed in this paper can take nonadiabatic effects into account without resorting to surface-hopping approximations. Hence, the results obtained follow qualitatively those of previous surface-hopping calculations and increase by a factor of (at least) 2, the time length over which nonadiabatic dynamics can be propagated with small statistical errors. Moreover, it is worth to note that the dynamics of quantum-classical wave fields proposed here is a straightforward non-Hamiltonian generalization of the formalism for nonlinear quantum mechanics that Weinberg introduced recently.

Nonlinear Quantum Mechanics at the Planck Scale

I argue that the linearity of quantum mechanics is an emergent feature at the Planck scale, along with the manifold structure of space-time. In this regime the usual causality violation objections to nonlinearity do not apply, and nonlinear effects can be of comparable magnitude to the linear ones and still be highly suppressed at low energies. This can offer alternative approaches to quantum gravity and to the evolution of the early universe.

An action principle for the quantization of parametric theories and nonlinear quantum cosmology

By parametrizing the action integral for the standard Schrödinger equation we present a derivation of the recently proposed method for quantizing a parametrized theory. The reformulation suggests a natural extension from conventional to nonlinear quantum mechanics. This generalization enables a unitary description of the quantum evolution for a broad class of constrained Hamiltonian systems with a nonlinear kinematic structure. In particular, the new theory is applicable to the quantization of cosmological models where a chosen gravitational degree of freedom acts as geometric time. This is demonstrated explicitly using three cosmological models: the Friedmann universe with a massless scalar field and Bianchi type I and IX models. Based on these investigations, the prospect of further developing the proposed quantization scheme in the context of quantum gravity is discussed.

Non-Linear Quantum Mechanics and High Energy Cosmic Rays

We point out that the presence of energetic cosmic rays above the GZK cutoff may be explained by fundamental non-linearities in quantum mechanics at the Planck level.

Some considerations on topologies of infinite dimensional unitary coadjoint orbits

The topology of the embedding of the coadjoint orbits of the unitary group U of an infinite dimensional complex Hilbert space H , as canonically determined subsets of the space T s of symmetric trace-class operators, is investigated. The space T s is identified with the B -space predual of the Lie-algebra L ( H ) s of the Lie group U . It is proved, that the orbits consisting of symmetric operators with finite range are (regularly embedded) closed submanifolds of T s . Such orbits play a role of “generalized phase spaces” of (also nonlinear) quantum mechanics. An alternative method of proving the regularity of the embedding is also given for the “one-dimensional” orbit, i.e. for the projective Hilbert space P( H ) . Closeness of all the orbits lying in T s is also proved.

Some considerations on topologies of infinite dimensional unitary coadjoint orbits

The topology of the embedding of the coadjoint orbits of the unitary group U of an infinite dimensional complex Hilbert space H , as canonically determined subsets of the space T s of symmetric trace-class operators, is investigated. The space T s is identified with the B-space predual of the Lie-algebra L( H) s of the Lie group U . It is proved, that the orbits consisting of symmetric operators with finite range are (regularly embedded) closed submanifolds of T s . Such orbits play a role of “generalized phase spaces” of (also nonlinear) quantum mechanics. An alternative method of proving the regularity of the embedding is also given for the “one-dimensional” orbit, i.e. for the projective Hilbert space P( H) . Closeness of all the orbits lying in T s is also proved.

Self-tunnelling oscillations in non-linear quantum mechanics and the electron-transfer problem

A small quantum system with a non-linear self-consistent potential may display a spontaneous symmetry breaking of the ground state. This simple result, which can be interpreted also as a spontaneous self-localization of the wavefunction, results in the presence of several equivalent ground states, which do not exist simultaneously. There is a virtual degeneracy in the system. The addition of a stochastic perturbation (noise) to such a system provides the possibility of tunnelling among the different equivalent ground states. The tunnelling process can lock itself in into a characteristic frequency related to the dynamics of the system and not contained in the input noise. A self-pulsating quantum system is then generated based on the intrinsic non-linearity of the dynamics and the presence of noise, which is used as a seed. We inquiry into the relevance of these phenomena for electron transfer processes in large biomolecules. By combining non-linearity and noise a controlled transfer rate can be achieved in what is effectively a quantum-switch regulated by external perturbations; a mechanism that could be at work in many electron-transfer processes in proteins.

Dynamics of Synchronising Systems

11R5. Dynamics of Synchronising Systems. - RF Nagaev (Dachny pr 9-2-66, St Petersburg, 198255, Russia). Springer-Verlag, Berlin. 2003. 326 pp. ISBN 3-540-44195-6. $149.00.Reviewed by I Andrianov (Inst fur Allgemeine Mechanik, RWTH, Templergraben 64, Aachen, D-52056, Germany).This book is monograph on the use of the averaging technique for solving the problems of theoretical, quantum and applied mechanics. The book is aimed at both graduate and postgraduate students as well as researchers in mechanics and physics with university or equivalent education. It is assumed that the reader has a basic knowledge of analytical mechanics, theory of nonlinear oscillations, rigid-body dynamics, quantum mechanics and electrical engineering, as well as of mathematics in the framework of a basic course taught at a technical university. The book is divided into 11 chapters. Chapter 1 is devoted to the concept of local integrability and consideration of possibilities of the choice of small parameters. Chapter 2 contains a brief description of mechanical and electromechanical systems that can be considered as being conservative. In Chapter 3 and 4 the single-and multivariable systems in “action-angle” variables are analyzed. In Chapter 5 multifrequency averaging of the system with a multidimensional rapidly rotating phase is carried out by a modified averaging procedure. Usage of canonical averaging of the equations of quantum mechanics, in particular the Schro¨dinger’s equation with various potentials, made it possible to obtain a solution in a more appropriate form than on the basis of the spectral analysis (Chapter 6). In Chapters 7, 8, and 10 the problems of weak interactions of quasiconservative dynamic systems are studied. Interesting effects of synchronization are discovered and studied. Periodic solutions in problems of excitation of mechanical oscillations form the subject of Chapter 11. It is noteworthy that all solutions obtained are valid for finite, but relatively extended time intervals. The book is very interesting from the standpoint of practically usable important results as well as for further development of averaging procedure. The subject index is informative, but somewhat short. The quality of the figures is high, but, unfortunately, no captions are included. References contain mainly papers and books written in Russian, just as Russian books now have been translated into English. Moreover, practically all-Russian journals referred to are cover-to-cover translated into English. For example, MTT is translated as Mechanics of Solids, PMM as PMM J Appl Math Mech, etc. It may therefore be more natural and convenient for readers to refer to English translations. This skillfully written book is a reader-friendly and well-organized textbook in the field of mathematical mechanics. As a rule, each section includes a pedagogical introduction. Dynamics of Synchronising Systems is highly recommended for purchase by both individuals and libraries.