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Quantum 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 Anderson model (Chap. 13), and many other models could be added. In fact, in Chap. 14 we prove that this type of behavior is universal among the field‑field interactions with conservation of momentum.

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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