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DYNAMICS AND INFORMATION (Published by Uspekhi Fizicheskikh Nauk, Moscow, 1997) 400 p. ISBN 5-85504-008-9 B B Kadomtsev Russian Research Centre `Kurchatov Institute' Institute of Nuclear Fusion, pl. Kurchatova 46, 123182 Moscow Tel. (095) 196-98-14, Fax (095) 190-42-44 E-mail: [email protected] Even as a student, and for many years after, I have been feeling uncomfortable about the understanding of quantum mechanics. This discomfort stemmed probably from the fact that `understanding' for me was something more than is commonly expected. It was not enough for me to have a well-developed mathematical apparatus and to be able to use it for calculating any particular physical process. I always strived to see something hidden behind the formulas, something that could be put together to form a certain intuitive comprehension of physics as a whole. Let me dwell a little on what did not suit me in the traditional presentation of quantum mechanics. Any textbook on quantum theory starts with the SchroÈdinger equation and the wave function c that satisfies this equation. One can put up with the claim that there is no way of deriving SchroÈdinger's equation, and that it was the result of a purely intuitive insight into the weird microscopic world. It is hard, however, to accept the mystic properties of the wave function without giving it a more clear-cut physical interpretation. The wave function is known to be the key concept of quantum mechanics: all physical quantities in quantum theory are operators acting upon the wave function, while the numerical values of physical quantities are the eigenvalues of these operators. Accordingly, the time evolution of physical quantities is determined by the time evolution of the wave function. And it is the SchroÈdinger equation for many particles that describes this evolution. The idea seems to be simple. But that is not how it turns out. The numerical value of any physical quantity can only be found through measurements. SchroÈdinger's equation, however, has nothing to do with measurements. This circumstance was very clearly perceived in the early years of quantum mechanics. Without any reservations it is mentioned, for instance, in the paper of Fock [1]. In his book on mathematical foundations of quantum mechanics [2] von Neumann argues that there are two classes of quantum processes: the evolution according to the SchroÈdinger equation, and the measurements whose results can be obtained with the aid of projection operators. It remains unclear, however, what are the physical processes that lie behind these operators. Since the results of measurements are expressed in terms of probabilities, one may believe that the measurements reflect the properties of a certain random process. No tools for describing the physics of this random process, however, can be found in the textbooks. Unavailability of a clear-cut mathematical description of the processes of measurement give rise to a number of problems if not absurdities. First of all, it is the uncommon for an exact science a necessity to interpret the physical meaning of the wave function and the quantum mechanics itself. Moreover, such interpretations are not unique [3 ë 5], although they do not differ much from one another. Further on, since a measurement seems to require the presence of the observer, there are different views concerning the role of observer in the process. If we go from the microscopic object to the measuring instrument, and then on to the observer, it would only seem natural to use quantum physics at each step: both the instrument and the observer are physical systems, and there is nothing to prevent describing them in terms of SchroÈdinger equation for many particles. Then, however, we have to answer the question where exactly the wave function collapses to the one single eigenfunction, and simultaneouasly the physical quantity to its eigenvalue, and what is the mechanism of such collapse. One could, of course, assume in a purely formal way that the observer in his turn is being observed by someone else like `Dyson's friend'. But then someone must observe this second observer, and so on to infinity Ð a picture not too flattering for a physical science. 2 Information and dynamics Closely linked with the problem of measurement is the problem of transition to classical physics. In the textbooks on quantum mechanics one can often read that quantum mechanics needs classical mechanics for its justiécation. Classical mechanics, however, is not deéned in the quantum theory. Strictly speaking, the classical mechanics cannot be regarded as a special extreme case of quantum mechanics in the limit of physical bodies of large size and mass (although statements to this effect can be found in the academic publications). The point is that any classical particles or classical bodies have exact coordinates and dimensions. In quantum theory this situation corresponds to a wave packet with a very narrow localization which tends to zero as h ! 0. The orthodox quantum mechanics, however, gives no physical grounds for such localization. Any classical object may be identiéed with an arbitrarily absurd wave function, and there are no rules for selecting a more or less reasonable wave function. What is more, the superposition principle ought to apply to classical bodies as well, which leads to such a nonsense as the superposition of live cat and dead cat in the famous `SchroÈdinger's cat paradox' [6]. And still, there must be a relatively straightforward way to resolving these problems, since quantum mechanics with its probabilistic interpretation has been able after all to explain and describe practically all physical phenomena and processes. The approach used in this book is based on the concept of collapse of wave functions under the assumption that the wave function has a purely informational meaning. Let us explain this point. Assume that a beam of light strikes a black plate and is absorbed in the plate. The absorption can be characterized by the coefécient of absorption K, and so the light intensity I decreases with the depth of penetration x as I = I exp(ÿKx). The plate is black if its thickness L satisées the condition LK 4 1. Assume now that our beam of light contains one single photon. Obviously, it will be absorbed not across the entire surface of the plate, but rather within a small spot a few wavelengths across and about Kÿ deep. Now we argue that the absorption of photon is accompanied with its collapse. The most important thing for us here is that the wave function of collapsing photon is destroyed in the entire space with the exception of the small region of absorption. It is here that we meet with the collapse of wave function and its informational content. The collapse of the wave function is similar to the collapse of probabilities. When we roll a die with six numbered sides, the probability of any one side being on top is 1/6 before the die stops. When the die stops, the probability for the top side being on top is unity, while the probabilities for all other sides simply vanish. As applied to the quantum theory, the principle of collapse of wave function of particles implies that, along with the time evolution of wave function according to SchroÈdinger equation, 0 1 one must also consider the process of collapse, whereby the wave function vanishes in a vast region of space where the particle in question is not present. Both types of processes have equal right to exist. It is important that the wave function in the presence of collapses must be regarded as a random function. Different methods can be used for describing random quantities and random processes. The most transparent one is the method of Langevin equation. Historically, the Langevin equation was proposed for describing the motion of Brownian particle. The force from the side of gas atoms acting upon the Brownian particle is represented as the sum of the regular force of friction proportional to the velocity of particle, and random impacts with a very short correlation time. A certain restriction (constraint) is imposed on the magnitude of these impacts so that the mean square velocity of Brownian particle is made equal to its thermal velocity. In case of quantum mechanics, the Langevin-type equation must be written directly for the wave function. Accordingly, the conventional SchroÈdinger equation must be supplemented by a ÿ c= t term with random operator, which actually is a sum of two parts. The regular term of the form i t 2 describes damping of the wave function with a certain characteristic time , while the term similar to random kicks in the Langevin equation takes care of the collapse itself. For a discrete set of eigenfunctions, the collapse corresponds to the random projection on one of the states with simultaneous elimination of all the remaining states. Collapses comply with the external condition Introduction (constraint): the probability of collapse into the state 3 cj must be proportional to jcj j 2 . Physically, this constraint amounts to the assumption that the collapse is caused by a very slight external perturbation which cannot change the diagonal elements of the density matrix, and only destroys its diagonal elements through decoherence Ð that is, through chaoti zation of phases. Leaving the mathematics aside, this approach to the description of collapses is very close to the ideas of the Copenhagen school in the early years of quantum theory. Some examples of collapses of wave functions are considered in the book. Simple models of scattering of quantum particles by macroscopic bodies with subsequent collapse of wave functions of the particles are used to demonstrate how the macroscopic bodies acquire classical properties. To wit, the collapses of wave functions of scattered particles cause collapses of wave functions of macroscopic bodies, so that the latter turn into narrow localized wave packets. Considered are also the collapses of wave functions in the case of radioactive decay. In the greatest detail, however, we study the collapse of wave functions of atoms or molecules of ordinary gas. We show that the quantum chaos of a gas appears as set of wave packets of gas atoms. The size of these packets is established and maintained through paired collisions of particles. This is associated with a very interesting effect of weak deviation from the universal law p jcj , where p is the probability of state, and c is the corresponding eigenfunction. In this connection it is worthwhile to recall the recommendation of R. Feynman [7]: `There are several problems of interpretation which call for further consideration... One is to prove that the probability interpretation of the cfunction is the only consistent interpretation of this quantity... It would be interesting to demonstrate that there is no other consistent interpretation of this quantity... Actually, no conventional distinction must be made between the observable and the observer, like we do now in the analysis of measurements in quantum mechanics; this question calls for thorough investigation.' As a matter of fact, the process of collapse occurs in a gas spontaneously, even without any observer. If this process is described in terms of the individual wave function collapses theh a small deviation from the universal law pj jcj j appears as a demand to satisfy the law of conservation of energy for gas atoms. (For this deviation it is important that the spectrum of states be continuous.) For each individual atom there is a slight uncertainty in the energy of the order of de h=t, where t is the mean time between collisions. It is to this accuracy that the law of conservation of energy holds for each individual atom. For the gas as a whole, however, the law of conservation of energy holds to a much higher accuracy. Because of this, the collapses exhibit a very slight asymmetry, a displacement of the wave packet about one wavelength along the direction of propagation of the wave packet at each act of scattering. Although the resulting effect is very small, it may give rise to macroscopically observable phenomena. In this book we give a rather detailed description of the Sokolov effect, which consists in the spontaneous polarization of excited hydrogen atoms passing by a metallic surface. This effect is attributed to the collapse of wave functions of free conduction electrons in the metal. The Sokolov effect is interesting in that it gives a new perspective to the feasibility of communication through quantum correlations. Indicated earlier for this purpose was the use of the so-called EPR pairs of correlated quantum particles (where EPR stands for the Einsteinë Podolsky ë Rosen paradox [8]). Individual pairs, however, are not suitable for that, since the law p jcj excludes the possibility of controlling the correlations of EPR pairs from a distance. As opposed to the individual correlation pairs of particles, the Sokolov effect results from the coherent superposition of EPR interactions, when one of the partners of EPR interaction (excited atom) has a huge number of counterparts (conduction electrons of the metal). The condition p jcj is not precisely valid in the Sokolov effect, and so the possibility of transmission of information to small distances by quantum correlations in this case, is in principle not excluded. We explain how to avoid violating the causality principle if the signal turns out to be superluminal one. Since we ascribe information meaning to the wave function, we have to dwell on the concept of information. As usual, we deéne information according to Shannon; to énd out its linkage with 2 2 2 2 4 Information and dynamics the entropy we use Leo Szilard's `thermal micromachines' [9]. The special concept of perception is used for describing classical measurements in terms of information processes. Although the wave function is linked with information, it obviously differs from the latter in its content and physical meaning. By contrast to the irreversible processes, associated with the time evolution of probabilities, the wave function exhibits two kinds of evolution: the reversible time evolution according to SchroÈdinger's equation, and the irreversible `quantum jumps' or `quantum transitions' through collapses. To understand these two types of time evolution, it is convenient to use, after Yu. Orlov [10], the principles of `wave logic'. For this purpose we introduce the concept of `intention', which is very natural for the mental process of decision-making, and turns out to be quite compatible with the general principles of quantum theory. In particular, from the standpoint of the reversible processes of change of intents it becomes clear why it is the amplitudes that must be added together in the formalism of integration over paths proposed by Feynman [7]. The sequence of measurements and `decision-making' turns out to be characteristic not only of the mental process, but also for the evolution of quantum systems exchanging information with the environment. Let us focus our attention on this point. The analysis of quantum chaos in a gas reveals that the rigorous justiécation of irreversibility requires an assumption of weak interaction of gas with the irreversible environment. This interaction can be exceptionally weak, and this circumstance allows us to call it the `information link'. Closed systems, whether quantum or classical, only exhibit the reversible dynamic evolution. When, however, there is a weak link with the irreversible outside world, the pattern of dynamic behavior may change dramatically. In a classical gas this change occurs because of strong dynamic instability ì that it, because of the fast divergence of paths in the phase space. Because of this, even a very slight external disturbance transforms the Liouville equation into a parabolic equation with a small coefécient at second derivatives. We know that the solutions of such equations may be quite different from solutions of equations without the higher derivatives. Small external noises are reinforced by the dynamic chaos of the gas, and its behavior obeys the Boltzmann equation for the one-particle distribution function. In case of a quantum gas, the external triggering action, ampliéed by the quantum chaos, leads to `packetization' of the wave functions of gas atoms, endowing them with the features of classical particles. On the basis of arguments developed above we come to the following general picture. The world as a whole is irreversible, so that irreversible is any its part linked with the environment. Reversibility may only exist in the objects totally isolated from the external world. Our theories, however, have for the most part been developed from the analysis of closed systems. In particular, it is for the closed systems that the formalism of the orthodox quantum theory has been constructed. When there is even a weak link with the irreversible external world (this link may be referred to as informational), the behavior of complex quantum systems may differ radically from the behavior of closed systems. More precisely, we must take the collapse of wave functions into account. The more sophisticated the quantum system, the more important the processes of collapse. Ilya Prigogine [11] introduced the concept of open systems, which are systems through which the êows of energy and entropy may pass. When such êows are strong enough, the system may exhibit nonlinear processes of self-organization. Similar phenomena may take place in quantum systems. The linkage of quantum systems with the outside world may be quite weak; nevertheless, the existence of such links may give rise to radical changes in behavior and to quantum selforganization. Such systems may be referred to as informationally open systems. The strong impact of the environment on complex quantum system is associated with the possible decoherence ì that is, with the annihilation of phase correlations of different components of the wave function. When we are dealing with a single particle, this decoherence appears as a collapse with random destruction of the components of the wave function in extensive regions of space. In case of common macroscopic bodies, this `information exchange' with the environment results in that the wave functions (which depend on the coordinates of the center of mass) contract into very narrow wave packets, so that macroscopic bodies become conventional classical objects. Quantum Introduction 5 measurements bring together microscopic and macroscopic objects, which leads to collapse of their wave functions. As indicated above, it is convenient to describe the collapse of wave functions in terms of random functions satisfying the Langevin-type equation. The random impact of the environment, reinforced with the intrinsic quantum chaos, is accounted for in such equation by two terms: the regular damping, and the random creation of new wave packets. Metaphorically speaking, even at the microscopic level we encounter birth and gradual decay of wave packets and wave functions. In other words, life starts in the microcosm, and then multiplies and develops in open biological systems. Here we deal with the possible germination of new lines of evolution, which Kant [12] called `causality through freedom' (CausalitaÈt durch Freiheit). We all realize that such processes of birth of new things through development of very small initial disturbances are very important in the evolution of the Universe. Traditionally, however, the science of physics has been dealing with dynamic systems with precisely defined particles and strict causality. When we include nonlinear processes triggered by small instabilities, we must consider even the weakest couplings which may be called the information links. It is this theme ì the close connection between the deterministic behavior of dynamic systems and the éne information processes ì that constitutes the main topic of this book. A large part of it has been published earlier in Uspekhi Fizicheskix Nauk (Physics ë Uspekhi) [13]. Much of the material considered here has evolved from discussions and joint work with my son M.B. Kadomtsev. Bibliography ì146 PACS numbers: 03.65.Bz, 05.45.+b, 05.70.Ln, 89.70.+c