Mathematics

Quantum Theory and Its Stochastic Limit by Luigi Accardi,
Igor Volovich, Yun Gang Lu (Springer Verlag) The subject of this book is a new
mathematical technique, the stochastic limit developed for solving nonlinear
problems in quantum theory involving systems with infinitely many degrees of
freedom (typically quantum fields or gases in the thermodynamic limit). This
technique is condensed into some easily applied rules (called "stochastic golden
rule") which allow us to single out the dominating contributions to the
dynamical evolution of systems in regimes involving long times and small
effects. In the stochastic limit, the original Hamiltonian theory is
approximated using a new Hamiltonian theory that is singular. These singular
Hamiltonians still define a unitary evolution and the new equations give much
more insight into the relevant physical phenomena than the original Hamiltonian
equations. Especially, one can explicitly compute multitime correlations (e.g.
photon statistics) or coherent vectors, which are beyond the reach of typical
asymptotic techniques, as well as deduce in the Hamiltonian framework the widely
used stochastic Schrodinger equation and the master equation.

This monograph is well suited as a textbook in the emerging
field of stochastic limit techniques in quantum theory.

Nowadays it is becoming clearer and clearer that, in the
description of natural phenomena, the triadic scheme ‑ microscopic, 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 behavior of
properties or systems 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, nonlinear effects, the transition from this general program to
concrete scientific achievements requires the development of techniques that
allow physical information to be extracted from nonlinear quantum systems.
Explicitly integral 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
introduces approximations 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
(perturbation 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 individual
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 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 automatically selects from the dynamics the dominating terms (in
this regime) and shows that they carp 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
indication 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 calculus. 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 calculated simply by solving a linear equation. A remarkable feature of this
procedure is that a normally ordered white noise Hamiltonian equation is a
stochastic Schrodinger equation, i.e. 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, solidstate 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 a priori but it is derived from the microscopic
quantum equations. In this sense we can say that the stochastic 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 principle"), 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 freedom 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 phenomenological models. The
stochastic limit also provides a general approach to the phenomenon of quantum
decoherence which is important, in particular for quantum computers.

The stochastic limit takes inspiration from the pioneering
studies of quantum 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 formulated 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
former 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 probability 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 explaining 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 Schrodinger equation has been obtained,
it is relatively easy to obtain the Langevin equation (stochastic limit of the
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 field. 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 microscopic Hamiltonian foundation to these equations and to derive the
Kubo‑Mori theory. This connection evidences how strongly irreversible and
dissipative 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
irreversibility: if we think of decay phenomena as basic observable features of
irreversibility, we see that they are perfectly compatible with a reversible
dynamical evolution.

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 behavior 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 class of physically meaningful models, in addition to the
energy shifts, broadening and lifetimes, which can also be obtained with other
methods, it allows one to guess the limit equations directly by inspection of
the initial Hamiltonian system and, from them, to deduce easily much information
about multiparticle transitions, correlations, particle 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 immediately
the associated stochastic Schrodinger 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 (i.e. general statements independent of
the specific model) that 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 concludes Part 1, four other basic
principles of the stochastic limit are formulated without proofs (these will be
given in Part III):

‑ the stochastic universality class 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 of the
original field coalesce and behave like a single pseudo‑particle whose second
quantization gives the master field. Stochastic bosonization (in dimensions
strictly higher than two) is a particular case of the block principle for
fermions. Super‑symmetric structures arise when one starts from a fermi
Hamiltonian including polynomials of both even and odd degree.

The fact that the stochastic golden rule allows one to
obtain quite nontrivial 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 III. In Parts I
and II we emphasize the main ideas which allow one to obtain the correct answer
quickly, at a physical level 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 semiclassical limit 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 serniquantum 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, velocity 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 of the standard commutation relations
and consequently of 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 (i.e. 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

The notion of quantum entanglement acquires with the
stochastic limit a meaning which goes beyond the familiar notion of
superposition and leads to the conclusion, supported by a large number of
examples, that under appropriate physical situations nonlinearly interacting
quantum systems cannot 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 before 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 well 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 development of the theory of distributions on the standard
simplex). More precisely, it has motivated the development 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 processes) 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‑long time regime. Recently a general method for studying quantum
dynamics has been developed 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.

The present book is addressed to researchers and students,
physicists, mathematicians, experts in quantum communication and information
engineering 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 results
obtained as applications of the stochastic limit technique have not been
included in this book: the most interesting applications of stochastic
bosonization in higher dimensions and of quantum interacting particle systems
(barely mentioned in Sect. 5.19), many concrete applications of Belavkin's
theory of quantum filtering, the whole body of results pertaining to the
nonequilibrium and transport phenomena, in particular the temperature dependence
of the conductivity tensor, the fractional quantum Hall effect, the low‑density
limit and the deduction of the kinetic equations (Boltzmann, Vlasov, etc.) from
the stochastic limit. The basic idea of the low‑density limit is illustrated, in
the simplest (i.e. Fock) case, in Chap. 10. The really interesting case (finite
temperature) is much richer in structure and more difficult.

Some of the heaviest mathematical parts have been omitted.
In particular the proofs of the possibility of exchanging the summation of the
iterated series and the stochastic limit and taking the term‑by‑term limit of
the series (in those models where this is possible). The details of such proofs
are strongly model dependent and do not give much insight. We have preferred to
give, in Chaps. 15 and 16, detailed proofs of the general, model‑independent
estimates from which the estimates needed in each single model can be deduced by
rather standard arguments. The same can be said for the estimate of the error,
which, beyond its mathematical interest, is crucial for defining the range of
applicability of the whole theory. Since a discussion of all these topics would
have exceeded the limitations of this book, we have decided to publish these
results in a separate book.

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