Download DYNAMICS AND INFORMATION (Published by Uspekhi

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