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In the first part of the present paper, we have proved that, in order to find an explicit expression for the action, on the [Formula: see text]-orthogonal polynomials, of the 1-parameter unitary groups [Formula: see text] and [Formula: see text], the solution of the inverse normal order problem on the quantum algebra canonically associated to the classical semi-circle random variable is required. In this paper we solve this problem. The solution is obtained in two steps. First we determine the explicit form of normal order in the more general framework of [Formula: see text]-mode-type interacting Fock spaces. Then this result is applied to solve the inverse normal order problem in the semi-circle case.

In Ref. 3, the quantum Strassen’s theorem has been extended to the infinite-dimensional case. This theorem consists in the solution of the coupling problem for two states on the algebra of bounded operators on two Hilbert spaces [Formula: see text], [Formula: see text] with the additional constraint that the coupling state has support in a pre-assigned sub-space of [Formula: see text]. In this paper, we give an alternative proof of the main theorem in Ref. 3 that allows such extension.

The fact that any classical random variable with all moments has a quantum decomposition allows to associate to it a family of quantum moments. On the other hand, a classical random variable may have several inequivalent quantum decompositions, which lead to the same classical, but different quantum moments. Even in the simplest Central Limit Theorems (CLT), i.e. those of Bernoulli type, there are examples in which the corresponding quantum moments converge to the canonical quantum moments of the associated classical random variable, and examples in which this is not the case. This poses the problem to find a constructive criterium that characterizes the quantum moments associated to the canonical quantum decomposition (which is unique) with respect to the other ones. Theorem [Formula: see text] of the present paper provides such a criterium. Theorem [Formula: see text] deals with the case when one knows a priori that the quantum moments come from a central limit theorem (the motivation of the present paper arose in this context). It gives only a sufficient condition, but simpler to verify than the necessary and sufficient conditions of Theorem [Formula: see text]. Theorem [Formula: see text] naturally leads to a classification of Interacting Fock Spaces (IFS) into three types. We construct examples showing that all these possibilities can effectively take place. On the way, we prove that all the best known deformations of Heisenberg commutation relations can be obtained as special cases of a general construction within the algebraic approach to the theory of orthogonal polynomials.

We prove that, in the framework of generalized quantizations, orthogonality of $$n$$ -particle vectors corresponding to different multi-indexes, together with an additional condition on the norms of $$n$$ -particle vectors characterizes product states. Implications of this result for the non-linear quantization program are discussed.

We classify Bogolyubov endomorphisms of the CCR algebra over a general test function space. This classification requires a non-commutative extension of the usual hyperbolic functions. We consider semi-groups of such endomorphisms and characterize their generators under the assumption of norm continuity.

We propose a quantum extension of the Markov-Dobrushin inequality. As an application, we estimate the Markov-Dobrushin constant for some classes of quantum Markov channels, in particular for the Pauli channel, widely studied in quantum information theory.

After a short review of the quantum mechanics canonically associated with a classical real valued random variable with all moments, we begin to study the quantum mechanics canonically associated to the \textbf{standard semi--circle random variable} $X$, characterized by the fact that its probability distribution is the semi--circle law $\mu$ on $[-2,2]$. We prove that, in the identification of $L^2([-2,2],\mu)$ with the $1$--mode interacting Fock space $\Gamma_{\mu}$, defined by the orthogonal polynomial gradation of $\mu$, $X$ is mapped into position operator and its canonically associated momentum operator $P$ into $i$ times the $\mu$--Hilbert transform $H_{\mu}$ on $L^2([-2,2],\mu)$. In the first part of the present paper, after briefly describing the simpler case of the $\mu$--harmonic oscillator, we find an explicit expression for the action, on the $\mu$--orthogonal polynomials, of the semi--circle analogue of the translation group $e^{itP}$ and of the semi--circle analogue of the free evolution $e^{itP^2/2}$ respectively in terms of Bessel functions of the first kind and of confluent hyper--geometric series. These results require the solution of the \textit{inverse normal order problem} on the quantum algebra canonically associated to the classical semi--circle random variable and are derived in the second part of the present paper. Since the problem to determine, with purely analytic techniques, the explicit form of the action of $e^{-tH_{\mu}}$ and $e^{-itH_{\mu}^2/2}$ on the $\mu$--orthogonal polynomials is difficult, % aaa ask T if it is solved the above mentioned results show the power of the combination of these techniques with those developed within the algebraic approach to the theory of orthogonal polynomials., Comment: 28 pages

GABA

A mathematical approach to the notion of complementarity in quantum physics is described and its historical development is shortly reviewed. After that, the notion of n-complementarity is introduced as a natural extension of complementarity and at the same time as weak form of stochastic independence. Several examples in which n-complementarity is realized but not independence are produced. The construction of these examples is based on the structure of Interacting Fock Space (IFS) that is strictly related to the classical theory of orthogonal polynomials. A brief description of both this notion and this connection is included to make the paper self-contained.

We investigate the spectrum for partial sums of m position (or gaussian) operators on monotone Fock space based on $\ell^2(\mathbb{N})$. In the basic case of the first consecutive operators, we prove it coincides with the support of the vacuum distribution. Thus, the right endpoint of the support gives their norm. In the general case, we get the last property for norm still holds. As the single position operator has the vacuum symmetric Bernoulli law, and the whole of them is a monotone independent family of random variables, the vacuum distribution for partial sums of $n$ operators can be seen as the monotone binomial with $n$ trials. It is a discrete measure supported on a finite set, and we exhibit recurrence formulas to compute its atoms and probability function as well. Moreover, lower and upper bounds for the right endpoints of the supports are given., Comment: 23 pages, 1 figure. Journal of Operator Theory, to appear

We investigate the existence of the limit of some high order weighted Cesaro averages., Comment: 11 pages, Results in Mathematics "online first", we inserted a "note added in proof"

We prove that, replacing the left Jordan–Wigner [Formula: see text]-embedding by the symmetric [Formula: see text]-embedding described in Sec. 2, the result of the corresponding central limit theorem changes drastically with respect to those obtained in Ref. 5. In fact, in the former case, for any [Formula: see text], the limit space is precisely the [Formula: see text]-mode Interacting Fock Space (IFS) that realizes the canonical quantum decomposition of the limit classical random variable. In the latter case, this happens if and only if [Formula: see text]. Furthermore, as shown in Sec. 4, the limit classical random variable turns out to coincide with the [Formula: see text]-mode version of the [Formula: see text]-deformed quantum Brownian introduced by Parthasarathy[Formula: see text], and extended to the general context of bi-algebras by Schürman[Formula: see text]. The last section of the paper (Appendix) describes this continuous version in white noise language, leading to a simplification of the original proofs, based on quantum stochastic calculus.

The [Formula: see text]-bit is the [Formula: see text]-deformation of the [Formula: see text]-bit. It arises canonically from the quantum decomposition of Bernoulli random variables and the [Formula: see text]-parameter has a natural probabilistic and physical interpretation as asymmetry index of the given random variable. The connection between a new type of [Formula: see text]-deformation (generalizing the Hudson–Parthasarathy bosonization technique and different from the usual one) and the Azema martingale was established by Parthasarathy. Inspired by this result, Schürmann first introduced left and right [Formula: see text]-JW-embeddings of [Formula: see text] ([Formula: see text] complex matrices) into the infinite tensor product [Formula: see text], proved central limit theorems (CLT) based on these embeddings in the context of ∗-bi-algebras and constructed a general theory of [Formula: see text]-Levy processes on ∗-bi-algebras. For [Formula: see text], left [Formula: see text]-JW-embeddings define the Jordan–Wigner transformation, used to construct a tensor representation of the Fermi anti-commutation relations (bosonization). For [Formula: see text], they reduce to the usual tensor embeddings that were at the basis of the first quantum CLT due to von Waldenfels. The present paper is the first of a series of four in which we study these theorems in the tensor product context. We prove convergence of the CLT for all [Formula: see text]. The moments of the limit random variable coincide with those found by Parthasarathy in the case [Formula: see text]. We prove that the space where the limit random variable is represented is not the Boson Fock space, as in Parthasarathy, but the monotone Fock space in the case [Formula: see text] and a non-trivial deformation of it for [Formula: see text]. The main analytical tool in the proof is a non-trivial extension of a recently proved multi-dimensional, higher order Cesaro-type theorem. The present paper deals with the standard CLT, i.e. the limit is a single random variable. Paper1 deals with the functional extension of this CLT, leading to a process. In paper2 the left [Formula: see text]-JW–embeddings are replaced by symmetric [Formula: see text]-embeddings. The radical differences between the results of the present paper and those of2 raise the problem to characterize those CLT for which the limit space provides the canonical decomposition of all the underlying classical random variables (see the Introduction, Lemma 4.5 and Sec. 5 of the present paper for the origin of this problem). This problem is solved in the paper3 for CLT associated to states satisfying a generalized Fock property. The states considered in this series have this property.

Common used Non Destruction Testing (NDT) technical is introduced. Combined with the bridge common damage problems and NDT present research situation and the current research trends, summed up the NDT detection contents used in bridge engineering. The NDT methods in theory study and engineering application development trends are forecasted. It will provide references for the research of NDT technology. Finally, modern sensor technology and wireless communication technology are suggested to be used in bridge NDT technology. Also the bridge health monitoring system is integrated into to provide the technical support for the bridge whole life design.

In the paper Accardi et al.: Identification of the theory of orthogonal polynomials in d–indeterminates with the theory of 3–diagonal symmetric interacting Fock spaces on \(\mathbb {C} ^d\), submitted to: IDA–QP (Infinite Dimensional Anal. Quantum Probab. Related Topics), [1], it has been shown that, with the natural definitions of morphisms and isomorphisms (that will not be recalled here) the category of orthogonal polynomials in a finite number of variables is isomorphic to the category of symmetric interacting fock spaces (IFS) with a 3–diagonal structure. Any IFS is canonically associated to a \(*\)–Lie algebra (commutation relations) and a \(*\)–Jordan algebra (anti–commutation relations). In this paper we continue the study of these algebras, initiated in Accardi et al. An Information Complexity index for Probability Measures on \(\mathbb {R}\) with all moments, submitted to: IDA–QP (Infinite Dimensional Anal. Quantum Probab. Related Topics), [2], in the case of polynomials in one variable, refine the definition of information complexity index of a probability measure on the real line, introduced there, and prove that the \(*\)–Lie algebra canonically associated to the probability measures of complexity index (0, K, 1), defining finite–dimensional approximations, in the sense of Jacobi sequences, of the Heisenberg algebra, coincides with the algebra of all \(K \times K\) complex matrices.

We prove in this paper that the vacuum expectation of any product of field operator of an interacting Fock space with the interactions λn's is driven by pair partitions if and only if each interaction λn is a linear combination of permutation operators.

This paper consists of two parts. Section 1 contains the main part of the talk given by the first named author at the 48th Toruń Symposium. Starting from Sect. 2 we illustrate the general program formulated at the end of Sect. 1 with an example of the simplest low density type interaction.

We prove that all finite joint distributions of creation and annihilation operators in Monotone and anti-Monotone Fock spaces can be realized as Quantum Central Limit of certain operators on a $C^*$-algebra, at least when the test functions are Riemann integrable. Namely, the approximation is given by weighted sequences of creators and annihilators in discrete monotone $C^*$-algebras, the weight being the above cited test functions. The construction is then generalized to processes by an invariance principle., Comment: 24 pages

In a recent paper, using a splitting formula for the multi-dimensional Heisenberg group, we derived a formula for the vacuum characteristic function (Fourier transform) of quantum random variables defined as self-adjoint sums of Fock space operators satisfying the multidimensional Heisenberg Lie algebra commutation relations. In this paper we use that formula to compute the characteristic function of quantum random variables defined as suitably truncated sums of the Virasoro algebra generators. By relating the structure of the Virasoro fields to the quadratic quantization program and using techniques developed in that context we prove that the vacuum distributions of the truncated Virasoro fields are products of independent, but not identically distributed, shifted Gamma-random variables.

During the past 15 years a new technique, called the stochastic limit of quantum theory, has been applied to deduce new, unexpected results in a variety of traditional problems of quantum physics, such as quantum electrodynamics, bosonization in higher dimensions, the emergence of the noncrossing diagrams in the Anderson model, and in the large-N-limit in QCD, interacting commutation relations, new photon statistics in strong magnetic fields, etc. These achievements required the development of a new approach to classical and quantum stochastic calculus based on white noise which has suggested a natural nonlinear extension of this calculus. The natural theoretical framework of this new approach is the white-noise calculus initiated by T. Hida as a theory of infinite-dimensional generalized functions. In this paper, we describe the main ideas of the white-noise approach to stochastic calculus and we show that, even if we limit ourselves to the first-order case (i.e. neglecting the recent developments concerning higher powers of white noise and renormalization), some nontrivial extensions of known results in classical and quantum stochastic calculus can be obtained.

We prove that, in the stochastic limit of the Anderson model only the non-crossing diagrams survive for the transition amplitude from the first excited state of the free Hamiltonian to the first excited state of the interacting Hamiltonian. This confirms a conjecture of Migdal (1958) and Abrikosov, Gorkov, Dzyaloshinski (1975). From this we deduce a closed (nonlinear) Schwinger–Dyson type equation for the limit transition amplitude whose solution can be found and gives the explicit dependence of this amplitude on the momentum of the excited state.

The weak coupling limit for a quantum system, with discrete spectrum, in interaction with a quantum field reservoir is considered. Depending on the nature of the reservoir (i.e. bosonic or fermionic) and the degree of nonlinearity of the interaction, we discover that either a bosonization or fermionization of the collective multi-linear reservoir operators emerges. The stochastic evolution is determined after the weak coupling limit and is shown to be unitary: also we show that our calculations for the system-only dynamics coincide with those previously postulated by physicists.

In the present paper, the basic ideas of thestochastic limit of quantum theory are applied to quantum electro-dynamics. This naturally leads to the study of a new type of quantum stochastic calculus on aHilbert module. Our main result is that in the weak coupling limit of a system composed of a free particle (electron, atom,...) interacting, via the minimal coupling, with the quantum electromagnetic field, a new type of quantum noise arises, living on a Hilbert module rather than a Hilbert space. Moreover we prove that the vacuum distribution of the limiting field operator is not Gaussian, as usual, but a nonlinear deformation of the Wigner semi-circle law. A third new object arising from the present theory, is the so-calledinteracting Fock space. A kind of Fock space in which then quanta, in then-particle space, are not independent, but interact. The origin of all these new features is that we do not introduce the dipole approximation, but we keep the exponential response term, coupling the electron to the quantum electromagnetic field. This produces a nonlinear interaction among all the modes of the limit master field (quantum noise) whose explicit expression, that we find, can be considered as a nonlinear generalization of theFermi golden rule.

In the present paper we investigate the low density limit of a quantum “System + Reservoir” model in which the temperature is finite and the Hamiltonian of the system has a discrete spectrum. It is proved that the matrix elements of the time evolution operator, with a time rescaling and some proper choice of collective vector, tends to matrix elements of a solution of a quantum stochastic differential equation driven by a quantum Poisson process.

The basic ideas of the stochastic limit for a quantum system with discrete energy spectrum, coupled to a Bose reservoir are illustrated through a detailed analysis of a general linear interaction: under this limit we have quantum noise processes substituting for the field. We prove that the usual Schrodinger evolution in interaction representation converges to a limiting evolution unitary on the system and noise space which, when reduced to system's degrees of freedom, provides the master and Langevin equations that are postulated on heuristic grounds by physicists. In addition, we give a concrete application of our results by deriving the evolution of an atomic system interacting with the electrodynamic field without recourse to either rotating wave or dipole approximations.

Quantum Markov chain introduced by Accardi (cf. [1,2,3]) is one of natural generalization of classical Markov chain. It has many interesting applications in physics and the most important one is given by the paper of Fannes-Nachtergaele-Werner ([4]), where an application of quantum Markov chain’s technique enables us to understand the Valence bond states well.

In [2] a constructive quantum central limit is proved for any mean-zero real probability measure with moments of any order. The most important tool there used is interacting Fock space (IFS) (see and references therein for more details). More recently in the authors gave another proof of such result based on the realization that the convolution arising from addition of field operators in one-mode type IFS is the universal one of Accardi-Bozejko [l]. In this note we give a new proof of such a theorem for symmetric measures, where, even in the framework of IFS, a new approach is privileged. Namely, after the introduction of a creation-annihilation process on a suitable IFS, we prove that the central limits of its even moments satisfy a system of equations whose unique solution is given by the (even) moments of the measure. [5] a similar central limit result is obtained. That result has been successively generalized in [8]. But between that result and our central limit theorem proved in the present paper, there It is worth to mention that, in

Cabana-Duvillard and lonescu11have proved that any symmetric probability measure with moments of any order can be obtained as central limit theorem of self-adjoint, weakly independent and symmetrically distributed (in a quantum souse) random variables. Results of this type will be called "universal central limit theorem".Using Interacting Fock Space (IFS) techniques we extend this result in two directions: (i) we prove that the random variables can be taken to be generalized Gaussian in the sense of Accardi and Bożejko3and we give a realization of such random variables as sums of creation, annihilation and preservation operators acting on an appropriate IFS; (ii) we extend the above-mentioned result to the nonsymmetric case. The nontrivial difference between the symmetric and the nonsymmetric case is explained at the end of the introduction below.

The stochastic limit of a free particle coupled to the quantum electromagnetic field without dipole approximation leads to many new features such as: interacting Fock space, Hilbert module commutation relations, disappearance of the crossing diagrams, etc. In the present paper we begin to study how the situation is modified if a free particle is replaced by a particle in a potential which is the Fourier transform of a bounded measure. We prove that the stochastic limit procedure converges and that the overall picture is similar to the free case with the important difference that the structure of the limit Hilbert module is strongly dependent on the wave operator of the particle.

In the general scheme of interaction, described in Sect. 4.8, the techniques described in Part I do not apply to the case when not only the reservoir R but also the system S has a continuous spectrum. The usual way to handle these more complex interactions consists in replacing the response term by the first few terms in its series expansion. For example, in the case of nonrelativistic quantum electrodynamics (QED) (e.g. in nonlinear optics, laser theory, etc.) this amounts to replacing the dipole approximation by a multipole expansion. These approximations however break momentum conservation, and we shall see that by introducing them one loses information on some subtle cancella- tions which occur due to fast oscillations.

This chapter is added for completeness; to read it is not essential for an understanding the remainder of the book. We shall illustrate here how the functional integral approach can be used to derive the stochastic limit. In other words, we consider the basic formula for the stochastic limit of a scalar field, $$\mathop {\lim }\limits_{\lambda \to 1} \frac{1}{\lambda }\varphi \left( {\frac{t}{{{\lambda ^2}}},x} \right) = \Phi (t,x),$$ in the functional integral approach. We shall use freely the functional integral formalism, assuming that the reader is familiar with it. For some useful references on functional integrals see [AHK76] , [HKPS93] , [Hi01] , [SmSh91], [AcVo98].

In this chapter the initial state of the reservoir will be the vacuum state and we study the limits, as λ → 0, of matrix elements of the evolution operator in the collective vectors, $$\left\langle {\prod\limits_{h = 1}^N \lambda \int_{{{{S_h}} \mathord{\left/ {\vphantom {{{S_h}} {{\lambda ^2}}}} \right. \kern-\nulldelimiterspace} {{\lambda ^2}}}}^{{{{T_h}} \mathord{\left/ {\vphantom {{{T_h}} {{\lambda ^2}}}} \right. \kern-\nulldelimiterspace} {{\lambda ^2}}}} {{C_h}\left( {{s_h}} \right)d{s_h}\Phi ,{U_{{t \mathord{\left/ {\vphantom {t {{\lambda ^2}}}} \right. \kern-\nulldelimiterspace} {{\lambda ^2}}}}}} \prod\limits_{h = 1}^{N'} \lambda \int_{{{{{S'}_h}} \mathord{\left/ {\vphantom {{{{S'}_h}} {{\lambda ^2}}}} \right. \kern-\nulldelimiterspace} {{\lambda ^2}}}}^{{{{{T'}_h}} \mathord{\left/ {\vphantom {{{{T'}_h}} {{\lambda ^2}}}} \right. \kern-\nulldelimiterspace} {{\lambda ^2}}}} {{{C'}_h}\left( {{{s'}_h}} \right)d{{s'}_h}\Phi } } \right\rangle $$ (16.0.1) and of the corresponding matrix elements for the Heisenberg evolution $$\left\langle {\prod\limits_{h = 1}^N \lambda \int_{{{{S_h}} \mathord{\left/ {\vphantom {{{S_h}} {{\lambda ^2}}}} \right. \kern-\nulldelimiterspace} {{\lambda ^2}}}}^{{{{T_h}} \mathord{\left/ {\vphantom {{{T_h}} {{\lambda ^2}}}} \right. \kern-\nulldelimiterspace} {{\lambda ^2}}}} {{C_h}\left( {{s_h}} \right)d{s_h}\Phi ,{U_{{t \mathord{\left/ {\vphantom {t {{\lambda ^2}}}} \right. \kern-\nulldelimiterspace} {{\lambda ^2}}}}}\left( {1 \otimes X} \right)U_{{t \mathord{\left/ {\vphantom {t {{\lambda ^2}}}} \right. \kern-\nulldelimiterspace} {{\lambda ^2}}}}^*} \prod\limits_{h = 1}^{N'} \lambda \int_{{{{{S'}_h}} \mathord{\left/ {\vphantom {{{{S'}_h}} {{\lambda ^2}}}} \right. \kern-\nulldelimiterspace} {{\lambda ^2}}}}^{{{{{T'}_h}} \mathord{\left/ {\vphantom {{{{T'}_h}} {{\lambda ^2}}}} \right. \kern-\nulldelimiterspace} {{\lambda ^2}}}} {{{C'}_h}\left( {{{s'}_h}} \right)d{{s'}_h}\Phi } } \right\rangle $$ (16.0.2a) and $$\left\langle {\prod\limits_{h = 1}^N \lambda \int_{{{{S_h}} \mathord{\left/ {\vphantom {{{S_h}} {{\lambda ^2}}}} \right. \kern-\nulldelimiterspace} {{\lambda ^2}}}}^{{{{T_h}} \mathord{\left/ {\vphantom {{{T_h}} {{\lambda ^2}}}} \right. \kern-\nulldelimiterspace} {{\lambda ^2}}}} {{C_h}\left( {{s_h}} \right)d{s_h}\Phi ,U_{{t \mathord{\left/ {\vphantom {t {{\lambda ^2}}}} \right. \kern-\nulldelimiterspace} {{\lambda ^2}}}}^*\left( {1 \otimes X} \right){U_{{t \mathord{\left/ {\vphantom {t {{\lambda ^2}}}} \right. \kern-\nulldelimiterspace} {{\lambda ^2}}}}}} \prod\limits_{h = 1}^{N'} \lambda \int_{{{{{S'}_h}} \mathord{\left/ {\vphantom {{{{S'}_h}} {{\lambda ^2}}}} \right. \kern-\nulldelimiterspace} {{\lambda ^2}}}}^{{{{{T'}_h}} \mathord{\left/ {\vphantom {{{{T'}_h}} {{\lambda ^2}}}} \right. \kern-\nulldelimiterspace} {{\lambda ^2}}}} {{{C'}_h}\left( {{{s'}_h}} \right)d{{s'}_h}\Phi } } \right\rangle $$ (16.0.2b)

We consider the following problem: a field interacts with a system S and after this interaction no direct measurement is made on S, but one extracts information on S by measuring the field before (input) and after (output) the interaction. For example, in the case of an atom which decays emitting radiation, if one knows its initial state and detects the radiated photons, then one can deduce some information on its new state. The idea is to deduce information on the system emitting the radiation from the measured radiation. The emitted radiation is a typical example of an output process. Other typical examples of output fields are the field operators (quadratures) evolved at time t. More generally, one is interested in the statistics of the output field with respect to a given initial state. Often, e.g. in quantum optics, by signal one means the mean value of the output process and by noise its variance. Typical choices of input fields are (Math) or linear combinations thereof. The corresponding output fields are $$A_{j,out}^ \in (t) = U_j^ + A_j^ \in (t){U_t}\;,\;{N_{jk,out}}(t) = U_t^ + {N_{jk}}(t){U_t},$$ where U t is an evolution operator involving the interaction between the system and the field.

In this chapter we begin the proof of the main results discussed in Chap. 4 by illustrating the first two new features which appear as a consequence of the interaction, with respect to the convergence of free fields, discussed in Chap. 3. These are: (i) the causal δ-function; (ii) the time consecutive principle.

Part III is devoted to some analytical results useful in the consideration of the stochastic limit. As explained in the Preface, we will not include the detailed proofs of all the equations deduced in the text. Instead of this we have tried to condense into a few mathematical theorems, the basic estimates which can be applied to a multiplicity of models. We have organized the material as follows: In Chap. 15, we give some basic analytical results. These essentially amount to an analytical representation and estimates of the Feynman diagrams. These estimates allow one to take the term-by-term limit of the iterated series expansion of the evolution operator.

In this chapter we first introduce some general notations that shall be used throughout the book and then explain the main ideas of the stochastic limit in a general framework (i.e. independent of any specific model). Additional notations shall be introduced when needed.

The generalized rotating-wave approximation condition (4.10.2) is too restrictive. In the present chapter we begin to realize the second step of the program described in Sect. 4.9, i.e. we shall generalize the multiplicative Hamiltonian (4.8.1) by dropping this condition and allowing the system operators to be arbitrary [modulo the analytical condition (4.9.3)]. We shall see that this leads to a new phenomenon, namely: even if we start with a single scalar boson Fock field, in the limit we shall have not a single, but an infinity of independent quantum noises — one for each Bohr frequency [see (4.8.4) for this notion] of the system; this is the stochastic resonance principle. In this chapter the stochastic golden rule is applied to the investigation of the stochastic limit for the general spin—boson Hamiltonian, describing a discrete system coupled with a boson field. The spin—boson Hamiltonian is widely used in physics [4], in studying quantum computing [Vol99], in studying stochastic resonance(1) [AcKoVo97, Gam98, Gri98, ImYuOh99] , etc.

In this chapter we discuss one of the main theses of the stochastic limit approach, namely the fact that in the stochastic limit the quantum fields become white noises, in the simplest possible situation, i.e. for mean zero Gaussian (free) fields. In these cases one has only to prove the convergence of the covariance (2-point) function and therefore the problem is reduced to the convergence of a d-dimensional integral.

In this chapter we consider the low-density limit and the associated equations, including the Boltzmann equation. Here we limit ourselves to a quick outline of the main ideas underlying the emergence of the number process in the stochastic limit and its connection with the 2-particle scattering operator. A full discussion of the low-density limit and of the associated transport equations (see [AcLu91a-b-c], [AcLu92]) will be given in a separate book, entirely devoted to this problem.