Table Of ContentQuantum Theory
and Its Stochastic Limit
Springer-Verlag Berlin Heidelberg GmbH
ONLINE LlBRARY
Physics and Astronomy
http://www.springer.de/phys/
Luigi Accardi Yun Gang Lu
Igor Volovich
Quantum Theory
and
Its Stochastic limit
, Springer
Professor Luigi Accardi Professor Yun Gang Lu
Centro Vito Volterra Dipartirnento di Matematica
Facolta di Economia Universita degli Studi di Bari
Universita di Roma Via Orabona, 4
Via di Torvergata 70125 Bari, Italy
00133 Roma, Italy
Professor Igor Volovich
Steklov Mathematical Institute
Gubkina8
117966 Moscow, GSP-l, Russia
Library of Congress Cataloging-in-Publication Data
Accardi, L. (Luigi), 1947-
Quantum theory and its stochastic limit/Luigi Accardi, Yun Gang Lu, Igor Volovich. p.cm.
Includes bibliographical references.
1. Quantum theory. 2. Stochastic processes. L Lu, Yun Gang, 1955-II. Volovich, LV, III. Title.
QCI74.17.S76 A33 2001 530.12-dc21 2001049456
ISBN 978-3-642-07543-8 ISBN 978-3-662-04929-7 (eBook)
DOI 10.1007/978-3-662-04929-7
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Preface
Nowadays it is becoming clearer and clearer that, in the description of natural
phenomena, the triadic scheme - microseopie, mesoscopic, macroscopic - is
only a rough approximation and that there are many levels of description,
probably an infinite hierarchy, in which the specific properties of a given level
express some kind of cumulative or collective behaviour of properties or sys
tems corresponding to the lower levels. One of the most interesting challenges
for contemporary natural sciences is the comprehension of the connections
among these different levels of description of reality and the deduction of the
laws of higher levels in this hierarchy from basic laws corresponding to lower
levels.
Since these cumulative or collective phenomena are, typically, nonlin
ear effects, the transition from this general program to concrete scientific
achievements requires the developement of techniques which allow physical
information to be extracted from nonlinear quantum systems. Explicitly in
tegrable examples of such systems are rare, and the most interesting physical
phenomena are not captured by them. Even in the case of linear systems the
fact that an explicit solution is formally available is often useless, since it is
impossible to interpret interesting physical phenomena from it.
In the absence of generally applicable methods one intro duces approxima
tions whose role is to capture those phenomena which become dominant at
different orders of magnitude of some physical parameters (e.g., long times,
low densities, weak couplings, low or high energies, low or high temperatures,
large masses). The usual asymptotic methods of quantum theory (pertur
bation theory, scattering, semi-classical limit, etc.) provide useful tools to
approximate the values of individual quantities of physical interest; however,
in the past ten years a new idea has begun to emerge independently in
several fields of physics and mathematics: instead of approximating individual
quantities, let us look for an approximating theory from which the individ
ual approximations can be obtained by standard procedures. Such a theory
should work simultaneously as a magnifying glass and as a filter, in the sense
that all phenomena pertaining to the scales of magnitudes we are interested in
should be magnified while those pertaining to all the remaining scales should
be filtered away. One way to achieve this goal is through a new technique, the
stochastic limit: the purpose of the present book is to explain this technique
VI Preface
in a simple and self-contained way and to describe how the new ideas and
structures, which emerge from it, apply to concrete physical problems.
The main result of the stochastic limit is that, combining the basic ideas
of scattering (long times) and perturbation theory (small parameter), it au
tomatically selects from the dynamics the dominating terms (in this regime)
and shows that they can be resummed, giving rise to a new evolution operator
which is again unitary and such that, in many cases, the corresponding
evolution equations are explicitly integrable. This unitary evolution is an
approximation of the original one; however, although simpler, it preserves
much nontrivial information on the original complex system. A first indi
cation of this complexity is the fact that these equations are singular: the
limit Hamiltonian is a functional of some white noise whose explicit form is
uniquely determined by the original system. So, in order to deal with them,
one needs white noise ca1culus. This explains the name "stochastic limit".
We develop a new technique to bring such singular equations to normal order
(which differs from the usual normal order because it involves not the usual
but the causal commutator); once this is done, most quantities of physical
interest can be ca1culated simply by solving a linear equation. Aremarkable
feature of this procedure is that a normally ordered white noise Hamiltonian
equation is a stochastic Schrödinger equation, Le. a stochastic differential
equation. In the past ten years this type of equation has been widely used
to build a multiplicity of phenomenological models in quantum optics, solid
state physics, quantum field theory, quantum measurement theory, etc. Thus
the stochastic limit allows these phenomenological models to be deduced
from the basic laws of physics. In fact it should be emphasized that, in the
stochastic limit, randomness is not postulated apriori but it is derived from
the microscopic quantum equations. In this sense we can say that the stochas
tic limit describes the microscopic structure of quantum noise by identifying
it with the fast degrees of freedom of the original system. The intuitions that
the fast degrees of freedom of a system can act as driving random forces
on the slow ones (a generalization of Haken' s "slaving princip Ie" ), that chaos
can be an infinite reservoir creating ordered structures and forms by means of
a stochastic resonance principle, etc., not only become exact in the stochastic
limit, but also find in it a precise quantititative formulation in the sense that
the distinction between fast and slow degrees of freedom is not determined
from the outside but realized by the dynamics itself; the fast degrees of free
dom are those which in the limit become quantum white noises (also called
"master fields"). Another advantage of the transition from phenomenological
models to models deduced from the basic laws is that the wealth, beauty and
variety of the new mathematical structures, hidden in these basic laws and
made explicit by the stochastic limit, by far exceed those used in the phe
nomenological models. The stochastic limit also provides a general approach
to the phenomenon of quantum decoherence which is important, in particular
for quantum computers.
Preface VII
The stochastic limit takes inspiration from the pioneering studies of quan
tum dynamical systems by Fermi, Bogoliubov, van Hove and Prigogine, and
its main goal is a detailed qualitative study of quantum dynamics, in analogy
to Poincare's qualitative study of classical dynamics.
Summing up: the basic philosophy of the stochastic limit can be for
mulated in a single sentence: if we look at the fast degrees of freedom of
a nonlinear system with a clock, adapted to the slow ones, then the for
mer look like an independent increment process, typically a white noise (the
highest degree in the chaos hierarchy!). The universality of this technique is
proved by the fact that it can be applied to a great variety of Hamiltonians.
Since in the simplest situations a quantum white noise is given by a pair
of noncommuting classical white noises, it follows that the stochastic limit
also provides a link between classical prob ability and quantum theory in
real time, without the need for imaginary times or analytical continuations.
In particular the quantum process obtained in the stochastic limit, when
restricted to some special sets of compatible observables (abelian algebras)
gives rise to interesting classical stochastic processes (e.g. birth and death
processes describing the population dynamics of atomic levels), thus explain
ing the efficiency of classical probabilistic techniques in the description of
several quantum phenomena such as stimulated emission in lasers, cosmic
ray cascades, branching in neutron diffusions, etc. Another example of this
phenomenon is Glauber's dynamics of the open Ising model which emerges
here as the restriction of a more interesting quantum Hamiltonian flow. The
symmetry of the resulting quantum Markov semigroup (which takes place if
the initial state of the field is an equilibium state, but not in the Fock case)
is related to the symmetry expressed by the Onsager relations.
Once the stochastic Schrödinger equation has been obtained, it is rela
tively easy to obtain the Langevin equation (stochastic limit ofthe Heisenberg
evolution). All the known types of master equations (and several new ones)
are obtained just by taking the partial expectation of Langevin equations with
respect to the reference state of the master fleld. This procedure corresponds
to the adiabatic elimination of the fastly relaxing variables, a technique also
called "coarse graining" , and establishes a connection between the stochastic
limit and the traditional "projection techniques" used to provide a micro
scopic Hamiltonian foundation to these equations and to derive the Kubo
Mori theory. This connection evidences how strongly irreversible and dissi
pative behaviours can be perfectly described by Hamiltonian dynamics, thus
providing a unified approach for reversible (Hamiltonian) and irreversible
(master equation) description of quantum systems in the spirit of Prigogine's
paradigm about the fundamental (as opposed to phenomenic) nature of ir
reversibility: if we think of decay phenomena as basic observable features
of irreversibility, we see that they are perfectly compatible with a reversible
dynamical evolution.
VIII Preface
It should be underlined, however, that the stochastic limit goes far beyond
the master equation because it does not eliminates the fast degrees of freedom.
This allows one to estimate the probabilities of some collective states, or more
generally the behaviour of a complex (nonlinear) system with many degrees of
freedom, in terms of relatively few functions of the microscopic characteristics
of the quickly relaxing degrees of freedom (according to the interpretation
these functions are called "order parameters", "kinetic" or "susceptibility"
or "transport coefficients", etc.).
The method is of very simple applicability in the sense that, for a large
dass of physically meaningful models, in addition to the energy shifts, broad
ening and lifetimes, which can also be obtained with other methods, it allows
one to guess the limit equations directly by inspection of the initial Hamil
tonian system and, from them, to deduce easily much information about
multipartide transitions, correlations, partide statistics, etc. We have tried to
condense the main results of the stochastic limit into the so-called stochastic
golden rules, which are a generalization of the Fermi golden rule and which
allow one to solve, just by inspection of the interaction Hamiltonian, the
following problem: given a quantum Hamiltonian system, write down im
mediately the associated stochastic Schrödinger equation (this, as explained
above, automatically gives also the Langevin and the master equation).
This rule is formulated in Chap. 4, and the reader already familiar with
the basic formalism of quantum field theory can begin reading this book
directly from this chapter. The examples covered there and in the following
chapter are variations of the spin-boson Hamiltonian. They are sufficient to
illustrate two of the basic principles (Le. general statements independent of
the specific model) which emerge from the stochastic limit, namely:
- the stochastic resonance principle (see Sect. 6.2);
- the time consecutive principle (see Sects. 8.3 and 8.4).
In Chap. 11, which condudes Part I, four other basic principles ofthe stochas
tic limit are formulated without proofs (these will be given in Part III):
- the stochastic universality dass principle;
- the block principle;
- the orthogonalization principle;
- the stochastic bosonization principle.
These principles are used to extend the stochastic golden rule to polynomial
interactions and to fermions. The block principle is particularly interesting
because it states that if the interaction is a monomial of degree n then some
special configurations in the n-particle space 01 the original field coalesce and
behave like a single pseudo-particle whose second quantization gives the mas
ter field. Stochastic bosonization (in dimensions strictly higher than two) is
a particular case of the block principle for fermions. Super-symmetric struc
tures arise when one starts from a fermi Hamiltonian induding polynomials
of both even and odd degree.
Preface IX
The fact that the stochastic golden rule allows one to obtain quite nontriv
ial results with practically no mathematical effort supports our expectation
that the stochastic limit could become an everyday tool for a multiplicity of
physicists. In view of this, one could even forgive the fact that the results are
also mathematically rigorous. In any case, to separate the new physical ideas
and effects deduced from this method from their mathematical proof, which
in some cases can be heavy, we have relegated the proofs of the main estimates
and principles of the stochastic limit to Part IH. In Parts I and H we empha
size the main ideas which allow one to obtain the correct answer quickly, at
a physicallevel of rigor. Since the Planck constant is one of the parameters
which can be involved in the time rescaling, the present method also provides
a natural second step after semiclassical approximation (for this reason it is
sometimes called semiquantum approximation) in the sense that, just as in the
semiclassicallimit one obtains a deterministic classical system (analogy with
the law of large numbers), one could say that the stochastic limit captures
a new typically quantum leading term in a different regime: analogy with the
central limit theorem. In the usual semiclassical approximation one obtains,
in the limit, classical trajectories. In the semiquantum approximation one
obtains Brownian motion trajectories in the simplest examples and new types
of classical or quantum white noises in the more sophisticated ones. Another
difference with the semiclassical approximation (as well as with relativity
theory) is that, while in these cases the old theory is recovered from the new
for a limiting value of a parameter (Planck constant, velo city of light), in
the stochastic limit it is the new theory which is recovered from the old for
a limiting value of some parameters (coupling constant, time, density, energy,
etc.).
In Part II we deal with strongly nonlinear interactions and illustrate the
qualitatively new phenomena which arise in connection with them. The main
new feature here is the breaking oi the standard commutation relations and
consequently oi the usual statistics. This is a consequence of the domination of
the contribution of the noncrossing diagrams to the dynamics of these more
complex systems (the crossing diagrams are shown to tend to zero in the
stochastic limit). This leads to the emergence of a multiplicity of nonstandard
(Le. neither classical nor boson nor fermion) master fields (white noises) as
approximations of usual Hamiltonian systems: for example, in several cases
the master field, even if coming from a usual Boson field, exhibits a kind
of superlocalization phenomenon and becomes a bounded random variable
(such as a fermion). Among the prototype models in which these phenomena
appear we mention quantum electrodynamics without dipole approximation,
the polaron model (Chap. 12), the Anderson model (Chap. 13), and many
other models could be added. In fact, in Chap. 14 we prove that this type of
behaviour is universal among the field-field interactions with conservation of
moment um.
X Preface
The not ion of quantum entanglement acquires with the stochastic limit
a meaning which goes beyond the familiar not ion of superposition and leads
to the conclusion, supported by a large number of examples, that under
appropriate physical situations nonlinearly interacting quantum systems can
not be separated even at a kinematical level and behave as a single new
quantum object satisfying new types of commutation relations and therefore
new statistics. (Historically the first example of this phenomenon appeared
in nonrelativistic quantum electrodynamics without dipole approximation,
where the atomic degrees of freedom commute with the field operators be
fore the limit, but after they develop nontrivial commutation relations which
account for the nonlinearity of the interaction. ) This qualitative statement
has a mathematical counterpart in the two notions of the "Hilbert module"
and the interacting Fock space which have emerged, from the stochastic limit,
as natural candidates for the description of the state space of interacting
systems.
From the mathematical point of view, the stochastic limit has brought
a unification as weIl as a deep innovation in the theories of classical, quantum
stochastic and white noise calculus as developed respectively by Ito, Hudson
Parthasaraty and Hida, and also in the theory of generalized functions (the
developement of the theory of distributions on the standard simplex). More
precisely, it has motivated the developement of a white noise approach to
stochastic calculus which is completely new even at a classical level. The
introduction of the Ito formula in a white noise, i.e. generalized functions,
context leads to nontrivial generalizations of both classical and quantum
stochastic calculus which turn out to correspond to first-order (for the Wiener
process) or normally ordered second-order (for jump pro ces ses) powers of the
Fock white noise. Higher powers of white noise cannot be dealt with usual
(classical or quantum) probabilistic techniques and require renormalization.
Thus we can say that the present approach also reveals some unsuspected
probabilistic aspects of renormalization theory.
The stochastic limit is a generally applicable approximating theory which
captures the dominant phenomena in the weak coupling-Iong time regime.
Recently a general method for studying quantum dynamics has been devel
oped and corrections to the stochastic limit have been computed. Moreover
an exact general expression for matrix elements of the evolution operator
("ABC-formula") was obtained [ArVoOO].
The present book is addressed to researchers and students, physicists,
mathematicians, experts in quantum communication and information engi
neering and all those interested in nonlinear problems of quantum theory.
Much of the material presented here has been published in several articles in
the past ten years, but the general presentation and most of the proofs have
been simplified and appear for the first time in book form. Many of the re
sults obtained as applications of the stochastic limit technique have not been
included in this book: the most interesting applications of stochastic bosoniza-
