The Wigner Semi–circle Law in Quantum Electro Dynamics
L. Accardi Y.G. Lu
Centro V. Volterra
Universit`a degli Studi di Roma - Tor Vergata Via Orazio Raimondo, 00173 Roma – Italy
Contents
1 Introduction 3
2 The QED Model 6
3 The Collective Operators and Collective Vectors 10
4 Non Gaussianity of the quantum noise 13
5 The Limit of the Spaces of the Collective Vectors 16
6 Non crossing Pair Partitions 22
7 Limits of Matrix Elements of Arbitrary Products of
Collec-tive Operators 24
8 Why Hilbert Modules? 30
9 The Wigner Semicircle Law 34
ABSTRACT
In the present paper, the basic ideas of the stochastic limit of quantum theory are applied to quantum electro-dynamics. This naturally leads to the study of a a new type of quantum stochastic calculus on a Hilbert 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 called interacting Fock space. A kind of Fock space in which the n quanta, in the n-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 expo-nential 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 the Fermi golden rule.
1
Introduction
Quantum electro–dynamics (QED) studies the interaction between matter and radiation. Due to the nonlinearity of this interaction (c.f. (2) below), an explicit solution of the equations of motion is not known and, for their study, several types of approximations have been introduced.
Probably the best known of these approximations is the dipole approxima-tion in which the so called response term (eik·q in (2)) which couples matter,
represented by a electron in position q, to the k-th mode of the EM-field, is assumed to be 1. The dipole approximation has its physical motivations in the fact that, at optical frequencies and for atomic dimensions one estimates that k · q ≈ 10−3 (cf. [10] ) and therefore very small. Most of the concrete applications of QED (e.g. in quantum optics, laser theory, . . .) have been ob-tained under the dipole approximation or by replacing the response term by the first few (at most 3) terms in its series expansion (the so–called multipole terms).
The first step towards the elimination of the dipole approximation was done in [2], [3], where the quasi–dipole approximation was introduced. In the
present investigation this approximation is not assumed: we shall just keep the response term.
The most surprising consequence of this more precise analysis is that the limiting noise, which approximates the quantum electromagnetic field is no longer a quantum Brownian motion, but a nontrivial generalization of the free noises, introduced by K¨ummerer and Speicher [9], [16] who were inspired by Voiculescu’s free central limit theorems [17], and developed by Fagnola in [7]. This free noise has been generalized by Bozejko and Speicher [5], [6] and the quantum noise which, according to the present paper, arises canonically from QED is a further generalization in this direction.
The main difference between the usual gaussian and the so–called free gaussian fields is that the vacuum expectation values of products of creation and annihilation fields are obtained, in the free gaussian case, by summing the products of pair correlation functions not over all the pair partitions, as in the usual (boson or fermion gaussian) case, but only on the so–called non crossing pair partitions, defined in Section 5 of the present paper. In terms of graphs this means that the summation does not run over all graphs, but only over the so–called rainbow (or half–planar – cf. [20]) graphs. However it seems that the connection between these graphs and the Wigner semi–circle law was not realized in the previous physical literature.
Another novel feature, arising from the present analysis (and which al-ready emerged at the level of quasi–dipole approximation (cf. [3])) is that the noise does not live on a Hilbert space, but on a Hilbert module over the momentum algebra of the electron. This Hilbert module is described in Section §7 of the present article. The general notion of Hilbert module was introduced for purely mathematical reasons and up to now this notion had found its main applications in K–theory for operator algebras (we refer to [4], [8], [14], [15], for the general theory of Hilbert modules). This circumstance has required the developement of a theory of quantum stochastic calculus over a Hilbert module (see [11], [12], [13]).
A third result of the present analysis is the emergence of a new type of Fock space, which we call the interacting Fock space because the quanta in the n-particle space are not independent but interact in a highly nonlin-ear way (cf. Section 7 below and in particular Theorem 4). The vacuum distribution of the field operators in this space is not Gaussian but a non-linear modification of the Wigner semicircle law to which it reduces exactly when the nonlinear factor arising in the interacting Fock space is put equal to zero. The Wigner semicircle law was discovered by Wigner [18] starting from
a purely phenomenological model to mimic the behaviour of Hamiltonians of heavy nuclei. It is rather surprising that it arises naturally in QED and that its appearance is accompanied by the emergence of some new mathematical structures whose properties make them natural candidates for the description of those phenomena in which the self–interaction of quantum fields plays an essential role.
Since the dipole approximation is effective at low frequencies and small atomic dimensions, it is reasonable to hope that the results of the present paper might shed some light on a class of phenomena in which high fre-quencies or finite atomic dimensions play an essential role. The systematic investigation of this possibility seems to deserve further attention.
The outline of the present paper is as follows: in Section1 we describe the Hamiltonian model: it is essentially a generalization of the Fr¨olich polaron Hamiltonian, to which it reduces when one puts p = 1 and replaces the fac-tor 1/Sectionqrt| k | by 1/ | k | in formula (2) below, which describes the interaction Hamiltonian. In Section2 we describe the collective vectors and determine the 2–point function of the quantum noise, to which the initial quantum field converges when λ → 0. Section Section3 is devoted to prove the non Gaussianity of the limit noise. It describes, in the simplest possible case of the 4–point function, the mechanism through which only the non crossing pair partitions (rainbow graphs) survive in the limit. This essen-tially results from a combined effect of the CCR plus the Riemann–Lebesgue Lemma. The full limit space of the quantum noise is obtained in Section4. In Section5 we introduce some combinatorial properties of the non crossing pair partitions which shall be used in the following sections. The technical core of the paper is Section6, where we obtain the limit of the joint correla-tions of arbitrary products of collective creation and annihilation operators, i.e. we prove the convergence, in the sense of mixed vacuum moments, of the collective process to the quantum noise which shall be described in Section 7.
This allows to obtain our main goal, i.e. to compute the limit of the matrix elements of the wave operator in arbitrary collective vectors.
In Section7 we identify the limit noise space to a Hilbert module over the momentum C∗–algebra of the electron (the small system in our terminology). This is the interacting Fock space (more precisely – Fock module) mentioned above. In Section8 we compute the vacuum distribution of the noise field and we show that, if the interaction among the field modes is neglected, it reduces to a convex combination of Wigner semi–circle laws parametrized by
the momentum of the electron. Finally, in Section 9, we describe, without proof, the quantum stochastic differential equation (cf. (121)) satisfied by the weak coupling limit of the wave operator at time t (the unitary Markovian cocycle, in the language of quantum probability). This has to be meant in the sense of stochastic calculus on Hilbert modules, as developed in [11], [12], [13]. This Section has been included for completeness. We did not include the proof because, although long and elaborated from the technical point of view, it does not need new ideas and techniques, being based on a procedure which has now become standard in the stochastic limit of quantum theory, namely: one considers the matrix elements, in some collective vectors, of the wave operator at time t and show that, in the limit λ → 0, they satisfy the same ordinary differential equation, with the same initial condition, as the corresponding matrix elements of the solution of the stochastic equation, which is known to be unique from the general theory [11].
In the revised version of the present paper we have enlarged the introduc-tion, added several comments and corrected several notational and linguistic misprints. No statement or proof has been changed with respect to the orig-inal version of the paper (submitted for publication in November 1992).
2
The QED Model
We consider a free particle, called, the system, and characterized by: –- its Hilbert space L2(Rd) with d ≥ 3
–- its Hamiltonian
HS := −∆/2 = p2/2
where ∆ is the Laplacian and p = (p1, p2, ..., pd) the momentum operator.
This system interacts with a quantum field in Fock representation whose free Hamiltonian is informally written as:
HR:=
X
k∈Zd
| k | a+
kak (1)
where to each mode k ∈ Zd is associated a representation of the CCR with
The interaction between the system and the field is informally written as: λHI = λ X k eik·qp ⊗ a + k p| k | + h.c. = λ(A(q) · p + p · A(q)) (2) where A(q) is the vector potential, the tensor product p ⊗ ak means
p ⊗ ak = d
X
j=1
pj ⊗ aj,k (3)
and the factor λ is a small scalar (coupling constant). This interaction is obtained from the usual minimal coupling interaction by neglecting the term of order λ2.
Notice that, dropping the λ2–term, breaks the gauge invariance of the
theory. The realization of the present program without dropping the λ2– term is a nontrivial problem.
We conjecture that the limit should be the same. Even more difficult is the problem of realizing the present program for a non free particle.
There are indications that this program should be realizable for some classes of potentials. These topics will be discussed elsewhere. They are mentioned here only to indicate some possible lines of development. The total Hamiltonian we are going to consider is H = HS + HR+ λHI. The
most important object for such interacting model is the wave operator at time t. In the interacting picture it can be written as
Ut(λ) := eit(HS+HR)e−it(HS+HR+λHI) (4)
where Ut(λ) is the solution of the Schr¨odinger equation: d dtU (λ) t = −iλHI(t)U (λ) t , U (λ) 0 = 1 (5)
and the evolved interaction Hamiltonian HI(t) is given by
HI(t) := eit(HS+HR)HIe−it(HS+HR) (6)
Replacing in (1), (2) the sum by a continuous integral and the factor 1/p| k | by a cut-off function g(k), HI(t) is given by the expression:
HI(t) = i
hZ
Rd
dkeitp2/2eik·qe−itp2/2(−ip) ⊗ (Stg)(k)a+k − h.c.
i
where g is a good function (e.g. a Schwartz function) and {S0
t}t≥0 is the
unitary group on L2(Rd) given, in momentum representation, by
(St0g)(k) = e−it|k|g(k) (8) By the CCR, we have
e−itp2/2e−ik·qeitp2/2= e−ik·qe−itk·pe−it|k|2/2 (9) e−ik·qe−itk·p = e−itk·pe−ik·qeit|k|2 (10) Therefore, one can rewrite (7) in the form
HI(t) = i
hZ
Rd
dke−ik·qe−itk·pe−it|k|2/2(−ip) ⊗ (St0g)(k)a+k − h.c.i (11) Without loss of generality we can simply forget the factor e−it|k|2/2 by trans-fering it into S0
t. More precisely, form now on, the 1–particle free evolution
shall not be the original one (8), but the modified one (Stg)(k) := e−it(|k|+k
2/2)
g(k) (12)
The integrals (7), (11), should be meant in the weak topology on the subspace S(Rd) ⊗ E (of L2(Rd) ⊗ Γ(L2(Rd))), algebraic tensor product of the Schwartz functions and the (algebraic) linear span of the exponential and number vectors. This means that the matrix elements of HI(t) in these
vectors are well defined (and this is the only thing that shall be used in the following). More precisely, if f and h are in S(Rd) and t ∈ R, then the
functions
Tf,h,j,t : k ∈ Rd7→< f, eitp
2/2
eik·qe−itp2/2(−ipj)h >L2(Rd) ; j = 1, 2, ..., d
(13) are also in S(Rd) and the integrals
A+j(Tf,h,j,tStg) =
Z
Rd
dk < f, eitp2/2eik·qe−itp2/2(−ipj)h > (Stg)(k)a+j,k (14)
(j = 1, 2, ..., d) define independent copies of the Boson Fock creation field over L2(Rd) (notice that the Schwartz functions are an algebra, thus the
above and ψ, ψ0 ∈ E the matrix element < f ⊗ ψ, HI(t)h ⊗ ψ0 > can be interpreted as d X j=1 < ψ, A+j (Tf,h,j,tStg)ψ0 >
In conlusion we recall the basic strategy of the stochastic limit in quantum theory: the starting point is the formal solution of equation (5), given by the iterated series: Ut(λ) = ∞ X n=0 (−i)nλn Z t 0 dt1· · · Z tn−1 0 dtnHI(t1) · · · HI(tn) (15)
In usual perturbation theory one considers the first few terms of the series (15), in increasing powers of λ. For this procedure to make sense, it is required that the coupling constant be small in some sense. In the stochastic limit in quantum field theory one renormalizes time by the replacement t ,→ t/λ2 and studies the limit, as λ → 0, of matrix elements of the form:
< Φλ(ξ), U (λ) t/λ2Φ
0
λ(η) > (16)
where, ξ, η ∈ L2(Rd) and the Φ
λ are the so-called collective vectors. They are
chosen according to criteria which depend on the model and are suggested by the usual perturbation theory. The basic goal of the theory is to prove that, as λ −→ 0, the limit of (16) exists and has the form
< Φ(ξ), U (t)Φ0(η) > (17) where Φ(ξ), Φ0(η) are vectors in the tensor product space of the system space with a certain limiting space H, called the noise space, whose explicit form has to be determined, and the limit process {U (t)}t≥0is unitary on the tensor
product space for each t ≥ 0.
In the following, the subspace S(Rd) of the Schwartz functions shall be often denoted KSectionubsetL2(Rd). For any pair f, g ∈ K, the condition
Z
R
|< f, Stg >| dt < ∞ (18)
3
The Collective Operators and Collective
Vec-tors
One of the basic heuristic rules of the stochastic limit in quantum field theory is that the choice of the collective vectors is suggested by first order analysis of the usual perturbation theory. Following this rule, in this section we shall introduce some preliminary considerations which give an intuitive idea on how to define the collective vectors.
For each t ∈ R+, define
A+(Stg) :=
Z
Rd
dke−ik·qeitk·p(Stg)(k) ⊗ a+k (19)
as explained in formula (14) and define A(Stg) as the formal adjoint of
A+(S
tg) i.e., recalling (12)
A(Stg) :=
Z
Rd
dke−itk·peik·q(S−tg)(k) ⊗ a¯ k (20)
Both A+(S
tg), A(Stg) act on L2(Rd) ⊗ Γ(L2(Rd)), where Γ(L2(Rd)) denotes
the Boson Fock space over L2(Rd), and behave like creation and annihilation operators respectively, more precisely, for each ξ, f in the Schwartz space S(Rd): A(Stg) ξ ⊗ Φ(f )= Z Rd
dke−itk·peik·qξ ⊗ (S−tg)(k)f (k)Φ(f )¯ (21)
where Φ(f ) is the coherent or number vector with test function f and the integral in (21) is meant weakly on the domain S ⊗ E (S) where E (S) denotes the space algebraically generated by the coherent or number vectors with test functions in S. In these notations, we have that
H(t) = ihA+(Stg)(−ip) − h.c.
i
(22) and the first order term of the interated series for Ut(λ) is:
−iλ Z t 0 dt1H(t1) = λ Z t 0 dt1[A+(St1g)(−ip)−h.c.] = A + (λ Z t 0 dt1St1g)(−ip)−h.c. (23)
For simplicity, in (22), we write (−ip) instead of (−ip) ⊗ 1Γ, where 1Γ is the
identity operator on Γ(L2(Rd)).
In particular, in the iterated series for Ut/λ2 the first order term is
A+ (λ Z t/λ2 0 dt1St1g)(−ip) − h.c.
Following the above mentioned heuristic rule (see also [1]), we define the collective annihilator process by:
Aλ(0, t, g) := A(λ Z t/λ2 0 dt1St1g) = λ Z t/λ2 0 dt1 Z Rd
dke−itk·peik·q⊗Stg(k)ak; t ∈ R
(24) and its conjugate, the collective creator process, by:
A+λ(0, t, g) := A+(λ Z t/λ2 0 dt1St1g) = λ Z t/λ2 0 dt1 Z Rd
dke−ik·qeitk·p⊗Stg(k)a+k ; t ∈ R
(25) More generally, for any bounded interval [S, T ]SectionubsetR and Schwartz function f , we shall define the collective creators and annihilators by:
A+λ(S, T, f ) := λ Z T /λ2 S/λ2 dt Z Rd
dke−ik·qeitk·p⊗ Stf (k)a+k (26)
Notice that, with these notations, the first order term of the iterated series (1.14) can be written in the form:
λ Z t/λ2
0
dt1HI(t1) = i[pAλ(0, t, g) − A+λ(0, t, g)p] (27)
which has a formal similarity with the dipole approximation Hamiltonian (c.f. [2] ) except for the fact that in (27) the collective annihilators and creators also contain operators of the system space, hence they do not commute with p.
The first step of the program of the stochastic limit in quantum theory is to show that in a certain sense, as λ −→ 0, i.e. the collective operators converge to some kind of creation and annihilation operators A+(S, T, f ) and A(S, T, f ) acting on some limit Hilbert space. Symbolically:
In order to determine the stucture of this space, the consideration of the two point function is naturally suggested by the fact that the vacuum distri-bution of the creation and annilhilation fields, before the limit, is gaussian.
The limit of the two point function of the collective creation and annihi-lation operators can be obtained as follows: for each s, t ≥ 0, one has
λ2 Z t/λ2 0 dt1 Z s/λ2 0 dt2 Z Rd dk1 Z Rd dk2 < 0 | ak1a + k2 | 0 > (28) < ξ, (−ip)e−it1k1·peik1·q(S t1g)(k1)e
−ik2·qeitk2·p(−ip)(S
t2g)(k2)η >= = λ2 Z t/λ2 0 dt1 Z s/λ2 0 dt2 Z Rd
dke−i(t2−t1)|k|2/2 < (ip)ξ, ei(t2−t1)k·p(ip)η > ¯g(k)(S
t2g)(k)
which, as λ → 0, tends to: t ∧ s Z +∞ −∞ dτ Z Rd
dk < (ip)ξ, eiτ k·p(ip)η > ¯g(k) (Sτg)(k) (29)
If, in (28), one replaces the zeros in the first two integrals by T /λ2, S/λ2
repectively, then the limit (29) is replaced by < χ[T ,t], χ[S,s] >L2(R) · Z +∞ −∞ dτ Z Rd
dk < (ip)ξ, eiτ k·p(ip)η > ¯g(k) (Sτg)(k)
(30) Keeping in mind the definitions (24) and (25) of collective annihilator and creator, the above result can be rephrased as follows: the approximate two point function
< ξ ⊗ Φ, Aλ(0, t, g)A+λ(0, s, g) η ⊗ Φ > (31)
tends to an object which for some aspects, is very similar to a two point function. In section (4.) we show how to substantiate this analogy.
Remark. Although the collective creation and, annihilation operators depend on the operator q, the limit of their 2-point function, i.e. (28), given by (29), is independent of q. However a remnant of the original q–dependence remains in the limit through the commutation relation (95).
The above considerations suggest to introduce the collective number vectors, which shall play a crucial role in the following.
Definition 1 A collective number vector is a vector of L2(Rd) ⊗ Γ(L2(Rd))
obtained by applying a (finite) product of collective creation operators to a vector of the form Φ ⊗ ξ where Φ is the vacuum in Γ(L2(Rd)) and ξ is an arbitrary vector in L2(Rd). Such a vector has the form
A+λ(S1, T1, f1) · ... · A+λ(Sn, Tn, fn)ξ ⊗ Φ (32)
In the following, when no confusion can arise, we shall use the simplified notation
n
Y
h=1
A+λ,hξ ⊗ Φ (33)
to denote a vector of the form (32). Notice, for future use, that the indices of the creators in (33) are increasing from left to right. The linear span of the collective vectors, for a given value of the coupling constant λ will be denoted Hλ and called the space of collective vectors.
It will be convenient to operate a change in notations with respect to the previous Sections and to write the state space of the composite system in the form Γ(L2(Rd)) ⊗ L2(Rd), rather than L2(Rd) ⊗ Γ(L2(Rd)).
4
Non Gaussianity of the quantum noise
From the previous two Sections we know that our original physical model is Boson Gaussian, i.e. the creation and annihilation fields satisfy the CCR and their vacuum distributions are Gaussian.
In this section we want to give an intuitive idea of the mechanism through which, by applying to the present model the procedure of the stochastic limit of quantum theory, in the limit λ → 0 of the vacuum correlation functions of the collective operators all the terms, corresponding to crossing partitions, vanish and the only nontrivial contributions come from the non crossing pair partitions (cf. Section (5) below for a quick review and references on this notion). This phenomen corresponds to the breaking of the gaussianity (be-cause in the gaussian case all pair partitions, and not only the non crossing ones, contribute to the correlations). The lowest order correlation function where the distinction between crossing and noncrossing pair partitions (and therefore between gaussian and non gaussian correlations) can become ap-parent corresponds to the four point function therefore in this Section we
shall study the limit of < A+λ(T1, f1)A+λ(T2, f2)Φ ⊗ ξ, A+λ(T10, f 0 1)A + λ(T 0 2, f 0 2)Φ ⊗ η > (34)
(the general case is dealt with is Sections (7) and (8)).
By the definition of collective creation and annihilation operators, (34) is equal to < λ2 Z T1/λ2 0 dt1 Z T2/λ2 0 dt2 Z Rd Z Rd dk1dk2e−ik1·qeit1k1·pe−ik2·qeit2k2·p (St1f1)(k1)(St2f2)(k2)a + k1a + k2 | 0 > ⊗ξ, λ2 Z T10/λ2 0 ds1 Z T20/λ2 0 ds2 Z Rd Z Rd e−ik01·qeis1k01·pe−ik02·qeis2k2·p (Ss1f 0 1)(k 0 1)(Ss2f 0 2)(k2)a+k0 1a + k0 2 | 0 ⊗ η >= = λ4 Z T1/λ2 0 dt1 Z T2/λ2 0 dt2 Z T10/λ2 0 ds1 Z T20/λ2 0 ds2 Z R4d dk1dk2dk10dk 0 2
< ξ, e−it2k2·peik2·qe−it1k1·peik1·qe−ik01·qeis1k01·pe−ik02·qeis2k20·pη >
(St1f1)(k1)(St2f2)(k2)(Ss1f 0 1)(k 0 1)(Ss2f 0 2)(k 0 2) < 0 | ak2ak1a + k0 1a + k0 2 | 0 > = λ4 Z T1/λ2 0 dt1 Z T2/λ2 0 dt2 Z T10/λ2 0 ds1 Z T20/λ2 0 ds2 Z R2d dk1dk2 (35)
[< ξ, e−it2k2·peik2·qe−it1k1·peis1k1·pe−ik2·qeis2k2·pη >
¯ f1(k1)(Ss1−t1f 0 1)(k1) ¯f2(k2)(Ss2−t2f 0 2)(k2)+
+ < ξ, e−it2k2·peik2·qe−it1k1·peik1·qe−ik2·qeis1k2·pe−ik1·qeis2k1·pη >
¯ f1(k1)(Ss2−t1f 0 2)(k1) ¯f2(k2)(Ss1−t2f 0 1)(k2)] (36)
By the CCR, one can move the q–factors together in the right hand side of (35). Thus one can rewrite (34) as
λ4 Z T1/λ2 0 dt1 Z T2/λ2 0 dt2 Z T1/λ2 0 ds1 Z T2/λ2 0 ds2 Z R2d dk1dk2
¯ f1(k1)(Ss1−t1f 0 1)(k1) ¯f2(k2)(Ss2−t2f 0 2)(k2)+
+ < ξ, e−it2k2·peis1k2·pe−it1k1·pe−is2k1·pei(s1−t1)k2·k1η >
¯ f1(k1)(Ss2−t1f 0 2)(k1) ¯f2(k2)(Ss1−t2f 0 1)(k2)] (37)
Notice that, in the first term in (37), f1 is paired with f10 and f2 with f20
(non crossing); while in the second one, f1 is paired with f20 and f2 with f10
(crossing).
With the change of variables
τ1 = λ2t1 , τ2 = λ2t2 , s1− τ1/λ2 = u , s2− τ2/λ2 = v (38)
the first term of (37) becomes Z T1 0 dτ1 Z T2 0 dτ2 Z (T10−τ1)/λ2 −τ1λ2 du Z (T20−τ2)/λ2 −τ2/λ2 dv Z R2d dk1dk2
< ξ, eivk2·peiuk1·pη > eiuk1·k2f¯
1(k1) ¯f2(k2)(Suf10)(k1)(Svf20)(k2) (39)
and as λ tends to zero, this goes to
< χ[0,T1], χ[0,T20]>L2(R) · +∞ Z −∞ du +∞ Z −∞ dv Z R2d dk1dk2f¯1(k1) ¯f2(k2)eiuk1·k2 (Suf10)(Svf20)(k2)· < ξ, ei(uk1+vk2)·pη > (40)
With the change of variables
τ1 = λ2t1, τ2 = λ2t2, u = s1− τ2/λ2, v = s2− τ /λ2 (41)
the second term of (37) becomes Z T1 0 dτ1 Z T2 0 dτ2 Z (T10−τ2)/λ2 −τ2/λ2 du Z (T20−τ1/λ2) −τ1/λ2 dv Z R2d dk1dk2 < ξ, ei(uk2+vk1)·pη > ·(S uf10)(k2)(Svf20)(k1) ¯f1(k1) ¯f2(k2) · eiuk1·k2·eik1·k2(τ2−τ1)/λ 2 (42) By the Riemann–Lebesgue lemma, (42) tends to zero as λ → 0.
We shall now proceed to the proof of the fact that the vanishing, in the limit, of all the contributions coming from crossing partitions is a general feature of QED.
5
The Limit of the Spaces of the Collective
Vectors
Our next step will be to try to determine the limit lim
λ→0hΦλ, Φ 0
λi =: < Φ, Φ 0 >
of the scalar product of two collective number vectors. This will give the Hilbert space where the limits of the collective creation and annihilation operators act.
According to our choice of these vectors, this amounts to study the limit, as λ → 0, of scalar products of the form:
h n Y h=1 A+λ,hΦ ⊗ ξ, n Y h=1 A+λ,hΦ ⊗ ηi
The present section shall be devoted to the proof of the following result: Theorem 1 For any N, n ∈ N, the limit, as λ → 0, of the scalar product of two collective number vectors:
h N Y h=1 A+λ,hΦ ⊗ ξ, n Y h=1 A+λ,hΦ ⊗ ηi (43)
exists and is equal to: δN,n n Y h=1 hχ[Sh,Th], χ[Sh0,Th0]i · Z +∞ −∞ du1. . . Z +∞ −∞ dun Z Rnd dk1. . . dkn hξ, n Y h=1 eiuhkh·pηi n Y h=1 (Suhf 0 h)(kh)fh(kh) exp i n−1 X r=1 n−1 X h=r urkr· kh+1 (44) Remark. If the factor
hξ, n Y h=1 eiuhkh·pηi · exp i n−1 X r=1 n−1 X h=r urkh+1· kr (45)
were absent in (1), then the expression (1) would coincide with the scalar product
h⊗n
j=1(χ[Sj,Tj]⊗ fj), ⊗
n
defined on the tensor product of n copies of the space: L2(R) ⊗ K
where L2(R) has the usual scalar product, while KSectionubsetL2(Rd) has
the scalar product
(f | g) := Z
R
dthf, StgiL2(Rd) (47)
Notice that the scalar product (46) can also be written in the form h n Y j=1 A+(χ[Sj,Tj]⊗ fj)Ψ, n Y j=1 A+(χ[Sj0,Tj0]⊗ fj0)Ψi (48)
where A+(·) denotes the creation operator on the full Fock space over L2(R)⊗
K, with the scalar product (47), and Ψ is the vacuum vector in this space. Our goal in the following sections will be to write the limit (43) in the form (48) where the A+(·) are some sort of creation operators acting on some
sort of Fock space in which Ψ plays the role of vacuum vector.
The obstruction to the use of the usual Fock or full Fock space comes precisely from the scalar factor (45) which is itself the product of two fac-tors, each of which is related to the other two new features arising from our construction, namely:
–(i) the ξ − η–scalar product is related to the fact that the noise lives on a Hilbert module rather than a Hilbert space.
–(ii) The exp–factor is related to the fact that the space (more precisely, the module) over the test function space is not the usual Fock, or full Fock space, but the interacting Fock space.
These features will be better understood in the following. Without loss of generality, in the following we shall suppose that Sj = Sj0 = 0 and rewrite
Tj0 as Sj for all j = 1, · · · , n. Moreover, it is obvious that we have to prove
Theorem 43 only in the case n = N .
In order to prove Theorem 43 let us first notice that the explicit form of the scalar product (43) is:
λ2n Z T1/λ2 0 dt1 Z S1/λ2 0 ds1. . . Z Tn/λ2 0 dtn Z Sn/λ2 0 dsn Z R2nd dk1. . . dkndk10 . . . dk 0 n
hξ, e−itnkn·peikn·q. . . e−it1k1·peik1q· e−ik10qeis1k01·p. . . e−ik 0 n·qeisnk0n·pηi n Y h=1 (Sthfh)(kh) · (Sshf 0 h)(k 0 h)h0 | akn. . . ak1a + k0 1. . . a + k0 n | 0i (49)
By explicitly performing the vacuum expectation, (49) can be written in the form: X σ∈Sn λ2n Z T1/λ2 0 dt1 Z S1/λ2 0 ds1. . . Z Tn/λ2 0 dtn Z Sn/λ2 0 dsn Z Rnd dk1. . . dkn
hξ, e−itnkn·peikn·q. . . e−it1k1·peik1·qe−ikσ(1)·qeis1kσ(1)·p. . . e−ikσ(n)·qeisnkσ(n)·pηi
n Y h=1 (Ssh−tσ(h)f 0 h)(kσ(h))fh(kh) (50)
where Sn denotes the group of permutations over the symbols {1, 2, . . . , n}.
Lemma 1 In the limit λ → 0, only the identity permutation in the sum in (50), gives a non zero contribution.
Proof . The proof is an extension, to arbitrary n, of the argument used in Section (3.) for the case n = 2. Notice that the factors exp(±kjq) are
independent of the times sj, tj and each of them appears twice: once with
the “ + ” and another with the “ − ” sign. If σ(j) = h, by permuting the factor exp(ikhq) with all the factors
exp(itmkmp) , m = h − 1, h − 2, . . . , 1
and then with the factors
exp(isrkσ(r)p) , r = 1, . . . , σ−1(h) − 1 = j − 1
we shall erase this factor by multiplication with exp(−ikhq) = exp(−ikσ(j)q).
Because of the CCR, each of the first h − 1 exchanges gives rise to a scalar factor of the form
exp(−itmkm· kh) , m = 1, . . . , h − 1
and each of the other σ−1(h) − 1 = j − 1 exchanges gives rise to a scalar factor of the form
When all these exchanges have been performed, the scalar product in the expression (50) becomes equal to:
exp i 2 [Pσ−1(1)−1 r=1 srk1· kσ(r)− t1k1· k2 + Pσ−1(2)−1 r=1 srk2· kσ(r)− (t1k1· k3+ t2k2 · k3) +Pσ−1(3)−1 r=1 srk3· kσ(r)− . . . − (t1k1 · kn+ . . . + tn−1kn−1· kn) +Pσ−1(n)−1 r=1 srkn· kσ(r) ]ξ,Qn h=1e i(sh−tσ(h))kσ(h)·pηi = (51) = hξ, n Y h=1 ei(sh−tσ(h))kσ(h)·pηi·exp i n X h=1 σ−1(h)−1 X r=1 srkh· kσ(r)− n−1 X h=1 h X r=1 trkr· kh+1
We shall now rewrite the scalar exponential in (51) so to make more trans-parent the difference between negligible and non negligible terms. First of all notice that, for any permutation σ ∈ Sn:
n−1 X h=1 h X r=1 trkr· kh+1= t1k1· k2+ t1k1· k3+ t2k2· k3+ . . . + +t1k1· kn+ . . . + tn−1kn−1· kn = = n−1 X h=1 thkh· n X m=h+1 km = n X h=1 tσ(h)kσ(h)· n X m=σ(h)+1 km (52)
Notice that, to say that h1, . . . , hn ∈ {1, . . . , n} are such that the sequence
σ−1(h1), σ−1(h2), . . . , σ−1(hn) is ordered in increasing order, is equivalent to
say that
σ(m) = hm ; m = 1, . . . , n
With these notations we also have
n X h=1 σ−1(h)−1 X r=1 srkσ(r)· kh = n X m=1 σ−1(hm)−1 X r=1 srkσ(r)· khm = = n X m=1 m−1 X r=1 srkσ(r)· kσ(m) = n−1 X r=1 srkσ(r)· n X m=r+1 kσ(m) (53)
Thus the difference between (53) and (52), i.e. the factor appearing in the scalar exponential in (51), can be written in the form:
n−1 X h=1 (sh− tσ(h))kσ(h) n X m=h+1 kσ(m)+ n−1 X h=1 tσ(h)kσ(h) n X m=h+1 kσ(m)− n X m=σ(h)+1 km (54) Therefore, the expression (50) can be written as
X σ∈Sn λ2n Z T1/λ2 0 dt1 Z S1/λ2 0 ds1. . . Z Tn/λ2 0 dtn Z Sn/λ2 0 dsn Z Rnd dk1. . . dkn hξ, n Y h=1
ei(sh−tσ(h))kσ(h)·pηi exp
h i n−1 X h=1 (sh− tσ(h))kσ(h) n X m=h+1 kσ(h) i exphi n−1 X h=1 tσ(h)kσ(h)· Xn m=h+1 kσ(m)− n X m=σ(h)+1 km i (55) Now we replace the scalar product in (50) by the right hand side of (51) and make the following change of variables in (50):
τh = λ2th , uh = sh− τσ(h)/λ2 , h = 1, . . . , n
¿From the Riemann-Lebesgue Lemma it follows that, as λ → 0, the only terms (in (50)) that can give a nonzero contribution are those coming from the factors such that the identity
0 = n−1 X h=1 tσ(h)kσ(h)· n X m=h+1 kσ(m)− n X m=σ(h)+1 kh (56)
is satisfied for almost all (k1, . . . , kn) ∈ Rnd and almost all (t1, . . . , tn) ∈ Rn.
So for almost all (k1, . . . , kn) ∈ Rnd n X m=h+1 kσ(m) = n X m=σ(h)+1 km ; ∀ h = 1, . . . , n (57)
Letting h = n − 1 in (57), the fact that (57) is true for for almost all (k1, . . . , kn) ∈ Rnd implies that in the the sum
Pn
one term, i.e. the cardinality of the set {σ(h) + 1, · · · , n} is one. This is equivalent to say that
σ(n − 1) + 1 = n (58)
Letting h = n − 2 in (57) and using (58) one finds
kσ(n−1)+ kσ(n) = kn−1+ kn = kσ(n−2)+1+ kn
or equivalently
σ(n − 1) = σ(n − 2) + 1 = n − 1 (59) iterating one finds
σ = id (60)
i.e. σ must be the identity permutation.
Proof of Theorem 43. For σ = I, the last term of (51) is equal to exp i hPn h=1 Ph−1 r=1 srkr· kh− Pn−1 h=1 Ph r=1trkr· kh+1 i = expi Pn−1r=1Pn h=r+1srkr· kh− Pn−1 r=1 Pn−1 h=rtrkr· kh+1 = expi Pn−1 r=1 Pn h=r+1(sr− tr)kr· kh (61) Therefore (50) is equal to the limit of
λ2n RT1/λ2 0 dt1 RS1/λ2 0 ds1. . . RTn/λ2 0 dtn RSn/λ2 0 dsn R Rnddk1. . . dkn hξ,Qn h=1e i(sh−th)pηi ·Qn h=1(Ssh−thf 0 h)(kh)fh(kh) expi Pn−1 r=1 Pn−1 h=r(sr− tr)kr· kh+1 + o(1) (62) where o (1) denotes the sum over all the permutations δ ∈ Sn, different from
the identity, which, according to Lemma 1, tend to zero as λ −→ 0. With the change of variables:
τh := λ2th , uh = sh− τh/λ2 , h = 1, . . . , n
6
Non crossing Pair Partitions
Since the notion of non crossing pair partition will play a crucial role in the following, we devote the present section to introduce some basic properties of these partitions (for more complete information we refer to [5, 6, 16]).
Non crossing pair partitions play, for Wigner processes, the role played by the pair partitions for the usual Gaussian processes. More precisely: it is known that the mixed moments of order 2n (n is a natural integer) of a mean zero Gaussian process, are given by the sum, over all pair partitions of {1, . . . , 2n}, of the products of the 2-point functions (covariance) of the process computed over all the pairs of the partition.
If the sum over all pair partitions is replaced by the sum over all non crossing pair partitions, in the sense specified below, one obtains the notion of Wigner process.
To each pair of natural integers (mh, mh), we associate the closed interval:
[m0h, mh] := {x ∈ N : m0h ≤ x ≤ mh} (63)
and we shall say that the two pairs (m0h, mh), (m0k, mk) are non crossing if
[m0h, mh] ∩ [m0k, mk] = ∅ [m0h, mh] [m0k, mk] (64)
Condition (2) means that either the two intervals associated to the pairs do not intersect, or one of them is contained in the other one.
Definition 2 Let n be a natural integer. A non crossing pair of the set {1, 2, . . . , 2n} is a pair partition
(m01, m1), . . . , (m0n, mn) (65)
such that any two pairs (m0h, mh), (m0k, mk) of the partition are non crossing.
Lemma 2 Given n numbers
m1 < m2 < . . . < mn
in the set {1, . . . , 2n}. If there exists a non crossing pair partition (m01, m1), . . . , (m0n, mn)
of the set {1, . . . , 2n} such that
m0h < mh ; h = 1, . . . , n (66)
then it is unique. Moreover, for each h0 = 1, . . . , n, the number m0h0 is
uniquely determined by the condition:
m0h0 := max{x ∈ {1, . . . , 2n}\{mh}h=1n : x < mh0 and | {x + 1, . . . , mh0 − 1}∩
∩{m1, . . . , mn} |=| {x + 1, . . . , mh0 − 1} ∩ ({1, . . . , 2n}\{mh}
n h=1) |}
(67) where, | {· · ·} | denotes the cardinality of set {· · ·}
Remark. By construction {m0h}n
h=1 = {1, . . . , 2n}\{mh}nh=1 (68)
and the sequence {m0h}n
h=1 is not necessarily increasing in h.
Proof . The proof can be done by the induction using the following facts: – in the non crossing pair partition (65), m01 is surely equal to m1− 1;
– given (m01, m1) as above,
n
(m01, m1), · · · , (m0n, mn)
o
is a non crossing pair partition of {1, 2, · · · , 2n} if and only if
n
(m02, m2), · · · , (m0n, mn)
o is a non crossing pair partition of {1, 2, · · · , 2n} \ {m01, m1}
– since, by the previous two arguments, given the (mj), the pair (m01, m1)
is uniquely determined, the partition (65) is unique if and only if the corre-sponding partition of the set {1, 2, ..., 2n} \ {m01, m1} is unique.
– since, by the previous two arguments, given the (mj), the pair (m01, m1)
is uniquely determined, the partition (65) is unique if and only if the corre-sponding partition of the set {1, 2, ..., 2} \ {m01, m1} is unique.
In the following we shall use the notation
n.c.p.p.{1, . . . , 2n} (69)
7
Limits of Matrix Elements of Arbitrary
Prod-ucts of Collective Operators
In order to calculate the limit (16), the knowledge of the limits of scalar products of collective number vectors is not sufficient: one has to identify the limit of matrix elements of arbitrary products of collective creation and annihilation operators.
Goal of the present section is to describe these limits. More precisely, we shall prove the following theorem:
Theorem 2 For any natural integer n and with the notations: A0λ := Aλ ; A1λ := A + λ (70) the limit lim λ→0hΦ ⊗ ξ, X ε∈{0,1}2n Aε(1)λ (S1, T1, f1) . . . A ε(2n) λ (S2n, T2n, f2n)Φ ⊗ ηi (71)
exists and is equal to: X {m0 h,mh} n h=1∈n.c.p.p.{1,...,2n} n Y h=1 hχ[Smh,Tmh], χ[Sm0 h,Tm0h] iL2(R) Z +∞ −∞ du1. . . Z +∞ −∞ dun Z Rnd dk1. . . dkn n Y h=1 (Suhfmh)(kh)fm0h(kh) hξ, n Y h=1 eiuhkh·pηi · exp i n−1 X h=1 n X r=h+1 uhkh· krχ(m0 r,mr)(mh) (72)
In order to prove this theorem we must do some preparation. Without loss of generality we shall assume that Sh = 0 for all h = 1, · · · , 2n.
Let us consider the matrix elements of arbitrary products of collective creation and annihilation operators (in contrast with (50) where only anti-Wick-ordered products were considered):
hΦ ⊗ ξ, Aλ(0, T1, f1) . . . A+λ(0, T 0 1, f 0 1) . . . A + λ(0, T 0 n, f 0 n), . . . , Aλ(0, Tn, fn)Φ ⊗ ηi (73)
Counting from left to right, let
1 ≤ m1 < m2 < . . . < mn ≤ 2n (74)
denote the places where the creation operators appear. Since Φ is the vacuum vector, the expression (73) is clearly equal to zero if either mn< 2n or m1 = 1.
Remark. The scalar product (73) is 6= 0 only if for any r = 1, 2, . . . , n, the cardinality of {1, . . . , 2r}\{m1, . . . , mn} is less than or equal to max{h =
1, . . . , n; mh ≤ 2r}. That is, if it happens that, for any r = 1, · · · , n, in the
first 2r operators in the operator product in (73), there are more creators than annihilators. Using the definition (??) of Aλ and A+λ, (73) becomes
λ2n Z T1/λ2 0 dt1. . . Z Tm1/λ2 0 dtm1. . . Z T2n/λ2 0 dtmn Z R2nd d~k
hξ, e−itk1·peik1·q. . . e−ikm1qeism1km1·p. . . e−itn·kn·peikn·q. . . e−ikmn·qeitmnkmn·pηi
n Y h=1 (Stmhfm0 h)(kmh) · Y α∈{1,...,2n}\{m1,...,mn} (Stαfα)(kα) h0 | ak1. . . a + km1 . . . akn. . . a + k0 mn | 0i (75)
It is clear that the vacuum expectation value in (6.7) is of the form X (m1,...,mn)={1,...,2n}\{mh}nh=1 mh<mh,h=1,...,n n Y h=1 δkmh, kmh λ2n Z T1/λ2 0 dt1. . . Z T2n/λ2 0 dt2n Z Rnd d~k X (m1,...,mn)={1,...,2n}\{mn}nh=1 mh<mh,h=1,...,n
hξ, e−itm1k1·peik1·q. . . e−im1·qeitm1k1·pe−itmnkn·peikn·q. . . e−ikn·qeitmnkn·pηi
n
Y
h=1
Lemma 3 In the limit λ → 0 the only terms of the summation (76) which give a non zero contribution are those corresponding to those n–uples (m1, . . . , mn)
which satisfy the equation
χ[mr+1,2n−1](mh) − χ[mr+1,2n−1](mh) + χ[mr+1,2n−1](mh) − χ[mr,2n−1](mh) = 0
(77) Proof. Remember that the indices mj in (76) correspond to annihilators in
(73) and are associated to exponential factors of the form e−itmjkj·p
eikj·q (78)
in (76). While the indices mj correspond to creators and are associated to
exponential factors of the form
e−ikj·qeitmjkj·p (79)
Also in this case, as in the proof of Theorem 43, we shall permute the ex-ponentials in the scalar product (76) so to cancel the factors exp(±ikj · q).
However in this case we shall employ a different grouping strategy, namely, instead of permuting each factor exp(±ikj· q) until it is cancelled by its
con-jugate factor, we bring all the factors exp(±ikj · q) to the right of all the
factors exp(±tαjkj · p) (αj = mj or mj) with the exception of the factor
exp(itmnkn· p), which remains on the extreme right.
The scalar phase factor which arises when all the permutations have been performed has the form exp(iσ) where σ is a real number which can be expressed as the sum of four different types of terms:
− n−1 X r=1 n−1 X h=r tmhkr· kh (80) n−1 X h=1 n X r=1 χ[mh+1,2n−1](mr) tmrkh· kr (81) n−1 X r=1 n X h=1 χ[mh+1,2n−1](mr) tmrkh· kr (82) − n X r=1 n X h=1 χ[mh+1,2n−1](mr) tmrkr· kh (83)
The term (80) arises from the commutation of the factors exp(−ikr · q),
(r = 1, . . . , n − 1) with the factors exp(itmhkh· p), for h = r, . . . , n − 1.
The term (81) arises from the commutation of the factors exp(−ikh· q),
(h = 1, . . . , n) with the factors exp(itm¯rkr · p) for r = 1, . . . , n − 1. The
characteristic function in this term is motivated by the fact that the term exp(−ikh · q) first appears on the right of exp(−itm¯hkh · p) because of the
convention on the indices of the collective vectors (c.f. the end of Section 2) and the fact that ¯mh < mh this implies that among all the factors exp −itm¯·p,
it will have to commute only with those for which mh < ¯mr < 2n
(recall that the ¯mj are not increasing in general). Notice that, since mn= 2n
the n-th term of the second sum in (81) is zero.
The term (82) arises from the commutation of the factors exp(ikh·q), (h =
1, . . . , n) with the factors exp(itmrkr·p) for r = 1, . . . , n−1. The characteristic
function in this term is motivated by the fact that the term exp(ikh· q) first
appears on the right of exp(−itmhkh· p) for the above mentioned convention.
Therefore, among the factors exp(itmrkr · p), it will have to commute only
with those for which
mh < mr < 2n
Finally the term (83) arises from the commutation of the factors exp(ikh ·
q), (h = 1, . . . , n) with the factors exp(−itmrkr · p) for r = 1, . . . , n. The
characteristic function in this term has the same origin as in the previous one.
Now notice that the sum of the four expressions (80), (81, (82) (83) can be rewritten as
X
1≤h,r≤n
[χ[mr+1,2n−1](mh)tmh+ χ[mr+1,2n−1](mh)tmh
−χ[mr,2n−1](mh)tmh− χ[mr+1,2n−1](mh)tmh]kh· kr (84)
Where, in (84), the first term in square brackets corresponds to (81); the second one to (82); the third one to (80) (recall that the map j 7→ mj is
strictly increasing) and the fourth one to (83).
By adding and subtracting, into the square bracket of (84), the terms ±tmhχ[mr+1,2n−1](mh)kh· kr ; ±tmhχ[mr+1,2n−1](mh)kh· kr
the double summation (84) can be rewritten as P 1≤h,r≤nkh· kr(tmh− tmh)χ[mr+1,2n−1](mh) +P 1≤h,r≤ntmhkh· kr(χ[mr+1,2n−1](mh) − χ[mr+1,2n−1](mh)) −P 1≤h,r≤n(tmh− tmh)kh· krχ[mr,2n−1](mh) +P 1≤h,r≤ntmhkn· kr(χ[mr+1,2n−1](mh) − χ[mr,2n−1](mh))
By the same argument used in the proof of Theorem 43, we conclude that, in the limit λ → 0, only those terms of the summation (76) can survive, for which
X
1≤h,r≤n
tmhkh· kr[χ[mr+1,2n−1](mh) − χ[mr+1,2n−1](mh)
+χ[mr+1,2n−1](mh) − χ[mr,2n−1](mh)] = 0
for almost all (t1, . . . , t2n) ∈ R2n and almost all (k1, . . . , kn) ∈ Rnd and this
condition is equivalent to equation (77).
Lemma 4 For any natural number n and any choice of 1 < m1 < . . . <
mn= 2n in the set {1, . . . , 2n}, the equation (77), in the unknowns m1, . . . , mn
has a unique solution satisfying
mh < mh ; h = 1, . . . , n (85)
Moreover the solution (m1, . . . , mh) is characterized by the property that
(m1, m1) , (m2, m2), . . . , (mn, mn) (86)
is the unique non crossing partition of the set {1, . . . , 2n} associated to the set {mh}nh=1 in the sense specified by Lemma 2.
Proof. We distinguish two cases:
χ[mr+1,2n−1](mh) = 0 (87)
In this case, the non crossing property implies that one must also have: χ[mr,2n−1](mh) = 0
because mr< mr. Hence
χ[mr+1,2n−1](mh) = χ[mr+1,2n−1](mh)
and therefore
mh > mr ⇔ mh > mr
2) χ[mr+1,2n−1](mh) = 1
In this case, if χ[mr,2n−1](mh) = 1, by the same arguments as above, we have
mh > mr⇔ mh > mr ∀ h, r = 1, . . . , n
Notice that this is equivalent to saying that
mh > mr⇔ mh > mr ∀ h, r = 1, . . . , n
and clearly this can be reformulated as
mh > mr⇒ mh > mr ∀ h, r = 1, . . . , n
Now if χ[mr,2n−1](mh) = 0, one must have
χ[mr+1,2n−1](mh) = 1
That is, in any case, χ[mr+1,2n−1](mh) = 1 implies that
χ[mr+1,2n−1](mh) = 1
which is equivalent to
mh > mr ⇒ mh > mr
summing up, we have that either [mh, mh] ∩ [mr, mr] = ∅ or [mh, mh]
8
Why Hilbert Modules?
In order to interpret the expression (4), obtained in Theorem 70, for the limit of matrix elements of arbitrary products of collective creators and annihila-tors, we describe in this section the Hilbert module on which the limits of the collective creation and annihilation processes live. By analogy with the Fock and the free Brownian motions, we shall refer to this process as the interacting free module Brownian motion. The first example of a quantum noise living on a (nontrivial) Hilbert module was considered in [3], the the-ory of stochastic integration and stochastic differential equations on Hilbert modules was developed in [11], [12], [13].
For each f in the Schwartz space, define ˜
f (t) := Z
Rd
(Stf )(k) e−ikqeikpdk (88)
then, ˜f is a map from R to the bounded operators on L2(Rd). Denote by
P the W∗− algebra generated by {eikp; k ∈ Rd} (i.e. the momentum W∗−
algebra of the system) and by F the P–right–linear span of { ˜f ; f ∈ K}. Then F is a P –right module and therefore the algebraic tensor product between L2(R) and F (denoted by L2(R) F ) is a P (in fact 1 P)–right
module. On the P–right module L2(R) F , we introduce a P–right bi-linear, P valued form: (· | ·) : L2(R) F × L2(R) F −→ P (89) by (α ⊗ ˜f | β ⊗ ˜g) :=< α, β >L2(R) · Z d R u Z d Rd ke−iukpf (k)(S¯ ug)(k) (90)
In the following we shall identify f with its eqivalence class with respect to the P–valued inner product defined by (4). Thus L2(R) F becomes a
P–pre–Hilbert module. The positivity of the right hand side of (4) is not obvious by inspection but it follows from Theorem 43. For each n ∈ N, we denote by (L2(R) F ) n the algebraic tensor product of n–copies of
L2(R) F and define the P–right sesqui-linear, P valued form: (· | ·) : (L2(R) F ) n× (L2(R) F ) n
by ((α1⊗ ˜f1) ⊗ · · · ⊗ (αn⊗ ˜fn) | (β1⊗ ˜g1) ⊗ · · · ⊗ (βn⊗ ˜gn)) := := n Y h=1 < αh, βh >L2(R)· Z d Rn u1· · · dun Z d Rnd k1· · · dkn n Y h=1 [e−iuhkhpf¯ h(kh)(Suhgh)(kh) exp(i X 1≤r≤h≤n−1 urkrkh+1) (92)
and we still identify the ˜fj’s with their equivalence classes with respect to the
equivalence relation stated before. Thus for each n ∈ N, with the P–right bi-linear, P valued form given by (6), (L2(R) F ) nbecomes a P–pre–Hilbert
module and the notion (L2(R) F )⊗n will be used to denote it.
Since for each n ∈ N, (L2(R) ⊗ F ) n is a P–pre–Hilbert module, (again
the positivity of (6) follows from Theorem 43), the direct sum C⊕L∞
n=1(L2(R)
F ) n makes sense and will be denoted by ΓL2(R) F and called the
Fock module over L2(R) F . In this pre–Hilbert module, the vector Ψ := 1 ⊕ 0 ⊕ 0 · · · is called the vacuum vector. One can easily show that
Lemma 5 The number vector subset
Γ := nA+(α1⊗ ˜f1) · · · A+(αn⊗ ˜fn)Φ; n ∈ N, αj ∈ L2(R), ˜fj ∈ F , j = 1, · · · , n
o (93) is a P–total subset of ΓL2(R) F.
Definition 3 For each element of L2(R) F , the creator with respect to
this element, denoted by A+(·), is defined on the P–right linear span of Γ by P–right linearity and
A+(α⊗ ˜f )h(α1⊗ ˜f1)⊗· · ·⊗(αn ˜fn)Ψ
i
:= (α⊗ ˜f )⊗(α1⊗ ˜f1)⊗· · ·⊗(αn⊗ ˜fn)Φ
(94) where n ∈ N, α, αj ∈ L2(R), ˜f , ˜fj ∈ F , j = 1, · · · , n. The formal adjoint
is called annihilator and denoted by A(·). Remark. In general, A+(α
1 ⊗ ˜f1)A+(α2 ⊗ ˜f2) is not necessarily equal to
Definition 4 For each k0 ∈ Rd, the left action of eik0p0 on Γ
L2(R) F⊗
H0 is defined by
eik0p0A+(α ⊗ ˜f ) := A+(α ⊗ ^eik0p1f )eik0p0 (95)
for all α ∈ L2(R), ˜f ∈ F , where p
0 is the momentum on H0 and p1 is the
momentum on the one–particle space of F .
It is easy to show that if * denotes the adjoint with respect to the P–valued scalar product given by (4), then eik0p0
∗
= e−ik0p0
Theorem 3 For each n ∈ N, α, αj ∈ L2(R), ˜f , ˜fj ∈ F (j = 1, · · · , n)
A(α ⊗ ˜f )A+(α1⊗ ˜f1) · · · A+(αn⊗ ˜fn)Ψ (96)
is equal to
(α ⊗ ˜f | α1⊗ ˜f1)A+(α2⊗ ˜f2) · · · A+(αn⊗ ˜fn)Ψ (97)
Proof. For each n ∈ N, α, αj ∈ L2(R), ˜f , ˜fj ∈ F (j = 1, · · · , n), βk ∈
L2(R), ˜gk ∈ F (k = 0, 1, · · · , n), since A = (A+)+, we have
< A+(α1⊗ ˜f1) · · · A+(αn⊗ ˜fn)Φ, A(α⊗ ˜f )A+(β0⊗˜g0)A+(β1⊗˜g1) · · · A+(βn⊗˜gn)Φ >=
=< A+(α⊗ ˜f )A+(α1⊗ ˜f1) · · · A+(αn⊗ ˜fn)Φ, A+(β0⊗˜g0)A+(β1⊗˜g1) · · · A+(βn⊗˜gn)Φ >
(98) Denote α := α0, ˜f := ˜f0. By definition (98) is equal to
n Y h=0 hαh, βhi · Z +∞ −∞ d u0. . . Z +∞ −∞ dun Z R(n+1)d dk0. . . dkn n Y h=0 eiuhkh·p· n Y h=0 (Suhgh)(kh)fh(kh) · exp i n−1 X r=0 n−1 X h=r urkr· kh+1 (99) Rewriting the expression (99) in the form:
n Y h=1 hαh, βhi · Z +∞ −∞ du1. . . Z +∞ −∞ dun Z Rnd dk1. . . dkn
n Y h=1 eiuhkh·p· n Y h=1 (Suhgh)(kh)fh(kh) · exp i n−1 X r=1 n−1 X h=r urkr· kh+1 hα, β0i · Z +∞ −∞ u.0 Z Rdk.0 eiu0k0·p· (S u0g0)(k0)f (k0) · exp i n X h=1 u0k0· kh (100) one finishes the proof.
Now let us compute the matrix elements of arbitrary products of creation and annihilation operators on the limit Hilbert module, which we interpret as limit noise space. For simplicity we shall not distinguish f from ˜f . What we must compute is the following object:
< Ψ, Aε(1)(α1⊗ f1) · · · Aε(2n)(α2n⊗ f2n)Ψ > (101)
where, n ∈ N, αj ∈ L2(R), fj ∈ F (j = 1, · · · , 2n), ε ∈ {0, 1}2n and
A0 := A, A1 := A+ It is clearly sufficient to compute (101) only in the case
ε(1) = 0 , ε(2n) = 1 (102)
Lemma 6 (101) is not equal to zero only if
2n
X
h=1
ε(h) = n (103)
Proof. The Lemma can be easily verified in the case n = 1. Suppose by induction thatP2n
h=1ε(h) 6= n implies that (7.14) is equal to zero and consider
< Ψ, Aε(1)(α1⊗ f1) · · · Aε(2(n+1))(αn+1⊗ f2(n+1))Ψ > (104)
Denote
h := min{x ∈ {1, · · · , 2n}; ε(x) = 1} (105) the position where the first creator is. By Theorem 3, (104) is equal to
Aε(h+1)(αh+1⊗ fh+1) · · · Aε(2(n+1))(αn+1⊗ f2(n+1))Φ > (106)
Now by applying Lemma 4, one can move the inner product (αh−1⊗ fh−1 |
αh⊗ fh) in (106) out from the inner product < Φ, · · · Φ >. Since in order to
produce this inner product we have used one creator and one annihilator, it follows that 2(n+1) X r=1 ε(r) 6= n + 1 ⇐⇒ X 1≤r≤2(n+1), r /∈{h−1,h} ε(r) 6= n (107)
By the induction assumption we complete the the proof. The same technique can be used to prove the following
Theorem 4 Denote
{mh}nh=1:= {r ∈ {1, · · · , 2n}; ε(r) = 1}, 1 < m1 < · · · < mn = 2n
(108) The inner product (101) is equal to zero if {mh}nh=1 does not define a non
crossing pair partition of {1, · · · , 2n}. If it does (and in this case we know from Lemma 2 that it is unique) (101) is equal to
n Y h=1 hαmh, αm0hiL2(R) Z +∞ −∞ du1. . . Z +∞ −∞ d un Z Rnd dk1. . . dkn n Y h=1 (Suhfmh)(kh)fm0h(kh) hξ, n Y h=1 eiuhkh·pηi · exp i n−1 X h=1 n X r=h+1 uhkh· krχ(m0 r,mr)(mh) (109) where, {m0h, mh}nh=1 is the unique pair partition of {1, . . . , 2n} determined by
{mh}nh=1.
9
The Wigner Semicircle Law
This section is devoted to describe how the Wigner semicircle law arises from the above considerations.
It is well known that the distinctive characteristic of the Wigner semicircle law is the role of the non crossing pair partition, in the expression of its momenta. But in our case, the situation becomes more complicated since the inner product of the n–particle space is not the product of n copies of the inner product of the one–particle space. Thus even if we obtain only non crossing pair partitions, in the sum each of them is weighted by a factor depending on the partition. We shall see that the Wigner semicircle law corresponds to the case in which these weighting factors are put equal to zero.
In order to evidentiate the above mentioned connection between the vac-uum distribution of the field operator and the Wigner semi–circle law, we introduce the probability spaces (Ωn, An, Pn) , n ∈ N:
–Ωn := {{m0h, mh}nh=1: noncrossing pair partition of {1, · · · , 2n}};
–ωm0,m:= {m0
h, mh}nh=1 ∈ Ωn;
–An is the discrete σ–algebra on Ωn;
–Pn : Ωn−→ [0, 1] is the probability defined by
Pn(ωm0,m) :=
1 | Ωn|
(110) For each 1 ≤ h, r ≤ n, define a random variable:
Xh,r(ωm0,m) := 0, if h ≥ r;
χ(m0
r,mr)(mh), if h < r
(111) Then
Lemma 7 For each non crossing pair partition {m0h, mh}nh=1
X {m0 h,mh} n h=1∈Ωn {1, · · · , 2n} expi 2 n−1 X h=1 n X r=h+1 uhkh· krχ(m0 r,mr)(mh) = =| Ωn|n ·Enexp i n−1 X h=1 n X r=h+1 uhkh· krXh,r (112) where En denotes expectation with respect to the random variables Xh,r,
defined by (8), with respect to the probability measure Pn and | Ωn |n
de-notes the cardinality of the set Ωn of all the non crossing pair partitions of
For each n ∈ N, α ∈ L2(R), f ∈ K, define the field operator by B α,f =
A(α ⊗ ˜f ) + A+(α ⊗ ˜f ). Then:
< Φ, Bα,f2n+1Φ >= 0 (113) Moreover, Theorem 4 shows that for each ξ ∈ L2(Rd)
< ξ, < Φ, Bα,f2nΦ > ξ >= X {m0 h,mh}nh=1∈Ωn kαk2n L2(R) Z +∞ −∞ du1. . . Z +∞ −∞ dun Z Rnd dk1. . . dkn Z Rd dµξ(k) n Y h=1 (Suhf )(kh)f (kh) · e iuhkh·k expi n−1 X h=1 n X r=h+1 uhkh· krχ(m0 r,mr)(mh) (114) where, µξ is the spectral measure of p with respect to the fixed vector ξ.
By applying Lemma 119, we have that < Φ ⊗ ξ, B2nα,fΦ ⊗ ξ >=| Ωn |nkαk2nL2(R) Z +∞ −∞ du1. . . Z +∞ −∞ dun Z Rnd dk1. . . dkn Z Rd dµξ(k) n Y h=1 (Suhf )(kh)f (kh) · e iuhkh·k Enexp i n−1 X h=1 n X r=h+1 uhkh· krXh,r (115) and in the following the right hand side of (115) will be denoted by
| Ωn |nkαk2nL2(R)
Z
Rd
dµξ(k) Mnf(k) (116)
Lemma 8 For each n ∈ N, the function Mnf(·) defined by (115), (116), has the following properties:
i) Mf
n(·) ≥ 0 and continuous;
ii) Mnf(·) satisfies the bound, uniform in k: Mnf(k) ≤h Z R du Z Rd dx | ¯f (x)(Suf )(x) | in ; ∀k ∈ Rd (117)
Notice that the vacuum odd moments of Bα,f are zero and, if each factor
Mnf(k) were the 2n–th power of some function cf(k), independent on n, then
the expression (115) would be the moment of order 2n of a random variable with distribution given by a convex combination of Wigner semi–circle laws with parameter
cf(k)kαkL2(R)
and mixing measure given by the spectral measure of the momentum operator in the state ξ. This is not the case because of the interaction term in (115), i.e. the factor under En–expectation. Neglecting this term, i.e. putting
Xh,r = 0 , ∀h, r, the right hand side of (115) reduces to
| Ωn|n kαk2nL2(R) Z Rd dµξ(k) [(f | f )(k)]n (118) where (f | f )(k) := Z R du Z Rd
dy(Suf )(y) ¯f (y)eiuy·k (119)
which is precisely of the type discussed above with cf(k) = (f | f )(k)
In this sense we have claimed in the introduction that the vacuum distribution of the limit field operator is a nonlinear modification of (a convex combination of) Wigner semi–circle laws.
10
The limit stochastic process
Up to now we have discussed the convergence, in the sense of mixed moments, of the collective creation and annihilation processes to a new type of quantum noise. The results proved in the previous Section, combined with techniques now standad in the stochastic limit of quantum theory, allow to deduce the explicit form of the stochastic equation for the limit of the time–rescaled wave operator. The full proof, which is unfortunately rather long, shall be published elsewhere [19]. We state her however the final result because the explicit form of the equation is particularly simple and easy to use.
Theorem 5 For each t ≥ 0, {Sh, Th}Nh=1, {S 0 h, T 0 h}N 0 h=1⊂ R, {fh}Nh=1, {f 0 h}N 0 h=1⊂
K and ξ, η ∈ L2(Rd), the limit < N Y h=1 A+λ(Sh, Th, fh)Φ ⊗ ξ, Ut/λ2 N0 Y h=1 A+λ(Sh0, Th0, fh0)Φ ⊗ η > (120) exsits and is equal to the solution of the quantum stochastic differential equa-tion with respect to the free module Brownian Moequa-tion:
U (t) = 1 + Z t
0
dA+s(g)(−ip) − (−ip)+dAs(g) − (−ip)+(g | g)−(−ip)ds
U (s) (121) on the full Fock P–module described in Section Section 7. Where, the half– inner product (· | ·)− is defined by
(f | g)− := Z 0 −∞ dt Z Rd dke−itkpf (k)(S¯ tg)(k) (122)
The proof that the solution of the quantum stochastic differential equation (121) is effectively a unitary operator and in fact the very meaning of this equation, depends on the theory of the free stochastic calculus over a Hilbert module, which has been recently developed in [13].
Aknowledgements
L.A. acknowledges partial support from the Human Capital and Mobility programme, contract number: erbchrxct930094.
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