DISPERSION RELATIONS IN THE STOCHASTIC

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Proceedings Trim Size: 9in x 6in

”AcCu05 ˙26 QP Proc–”

DISPERSION RELATIONS IN THE STOCHASTIC LIMIT OF QUANTUM THEORY∗

L. ACCARDI Centro Vito Volterra. Universit` a degli Studi di Roma “Tor Vergata”. 00133, Rome, Italy. e-mail: [email protected] F.G. CUBILLO Departamento de An´ alisis Matem´ atico. Universidad de Valladolid. 47005, Valladolid, Spain. e-mail: [email protected].

We apply new techniques based on the distributional theory of Fourier transforms to study, in the stochastic limit of quantum theory, the convergence of the rescaled creation and annihilation densities, which lead to the master fields, and the form of the drift term of the stochastic Schr¨ odinger equation obtained in such limit. This approach permit us to dispense with the “analytical condition” and other restrictions usually considered and also to establish the dependence of the stochastic golden rules on certain properties of the dispersion function of the quantum field. [cf. articoli-cubi0330.da ]

1. Introduction The stochastic golden rules [?, ?], which arise in the stochastic limit of quantum theory as natural generalizations of the Fermi golden rule, provide a natural tool to associate a stochastic flow, driven by a white noise equation, to any discrete system interacting with a quantum field. The stochastic limit captures the dominating contributions to the dynamics arising from the cumulative effects, on a large time scale, of small interactions; the physical idea is that, looked from the slow time scale of the system, the field looks like a very chaotic object: a quantum white noise, i.e a δ-correlated (in time) quantum field also called master field. The new evolution is an appro∗ 26th

conference on quantum probability and infinite dimensional analysis. levico terme, 20-26 february, 2005. 1

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ximation of the original one which preserves much nontrivial information on the original complex system related to its decay and shift properties. In this work we study, from an analytical point of view, the convergence of the rescaled creation and annihilation densities, which lead to the master fields, and the form of the drift term of the stochastic Schrödinger equation obtained in such limit, which contains the quantum mechanical fluctuation-dissipation relations. This approach permit us to dispense with the analytical condition and other restrictions usually considered –see Section ??– and also to establish the dependence of the stochastic golden rules on certain properties of the dispersion function of the quantum field. To be precise, we shall see that, for the region Γ1 where the dispersion function is regular and not constant, every Bohr frequency of the system in its range gives rise to an independent master field, which is a quantum white noise concentrated over the corresponding resonant surface, whereas both the rest of Bohr frequencies and the open regions Γαj , where the dispersion function is constant, give rise to zero master fields, except for the resonant case, see Theorem ??. In a similar way we will show that the regions Γαj do not contribute to the drift term whenever the resonant case is not present, whereas for the region Γ1 we obtain the usual expression, see Theorem ??. The contribution of the singular regions of dispersion has not been completely determined yet. 2. Preliminaries In what follows we shall consider quantum systems describing the interaction of a discrete spectrum system S with free Hamiltonian X HS := ε r Pε r r

and Bohr frequencies ω = εr − εr0 , (εr , εr0 ∈ Spec HS ), and a bosonic quantum field as reservoir R with free Hamiltonian (on Fock space) Z HR := dk ω(k)a+ (k)a(k), where ω(k) is the dispersion function, a± (k) are the creation and annihilation densities, and the reference vector is mean zero Gaussian and gauge invariant, with covariance of the form + N (k) 0 a (k)a(k 0 ) 0 h i= δ(k − k 0 ). (1) 0 N (k) + 1 0 a(k 0 )a+ (k)

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We will assume that the total Hamiltonian has the form H (λ) := H0 + λHI = HS + HR + λHI , where l is a real coupling parameter and the interaction Hamiltonian HI is of dipole type, i.e.a X HI = Dj∗ ⊗ A(gj ) + Dj ⊗ A∗ (gj ) , j

where Dj are system operators and Z ∗ A (gj ) := dk gj (k)a+ (k),

Z A(gj ) :=

dk g ∗ (k)a(k),

being the functions gj the cutoff or form factors. Often we will simplify the notations by omitting the symbol ⊗. In the stochastic limit approach we consider the time rescaling t → (λ) (λ) of the Schrödinger equation in t/λ2 in the solution Ut = eitH0 e−itH interaction picture: ∂ (λ) (λ) U = −iλHI (t) Ut , ∂t t HI (t) = eitH0 HI e−itH0 , and study the limits, in a topology to be specified, of the rescaled interaction Hamiltonian and of the rescaled propagator: 1 t lim HI =: ht , λ→0 λ λ2 (λ)

lim Ut/λ2 =: Ut .

λ→0

In canonical form this reduces to find the limit of the rescaled creation and annihilation densities 1 ∓i t2 (ω(k)−ω) ± a± e λ a (k), (2) λ,ω (t, k) := λ ∂ Ut = −iht Ut , whose obtaining the white noise Schrödinger equation ∂t normally ordered form is the quantum stochastic differential equation

dUt = (−idH(t) − Gdt)Ut ,

(3)

asterisk ∗ denotes the Hermitian conjugate for operators and the complex conjugate for scalars. For distributional densities we use the symbol + instead ∗ . a The

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where Z dH(t) := lim

λ→0

t+dt

dτ t

τ 1 HI λ λ2

is called the martingale term and Z t+dt Z t1 t1 t2 1 dt1 dt2 hHI H i Gdt := lim 2 I 2 λ→0 λ λ λ2 t t

(4)

is known as the drift term. Among the usual assumptions to achieve this program we have the following: • the cut–off functions gj are Schwartz functions; • the dispersion function ω(k) and the cut–off functions gj are related by the following analytical condition: Z Z Z dk eitω(k) gi∗ (k)gj (k) < +∞; dt|hgi , eitω(p) gj i| = dt R

R

Rd

• the (d−1)–dimensional Lebesgue measure of the surface {k : ω(k) = 0} is equal to zero (this implies, in particular δ(ω(k)) = 0). In this work we apply new techniques, based on the distributional theory of Fourier transforms [?, ?, ?, ?], which permit us to dispense with the above conditions and to establish the dependence of the stochastic golden rules on certain properties of the dispersion function ω(k). 3. The Dispersion Function In what follows we shall assume that the dispersion function Rd 3 k 7→ ω(k) ∈ R is such that ω(k) ≥ 0 for all k ∈ Rd and we can write Rd = Γ1 ∪ Γ2 ∪ Γ3 , where: (i) Γ1 is an open set of Rd in which ω(k) is a C ∞ -function and ∇ω(k) 6= 0 for every k ∈ Γ1 . We shall denote by Γ11 the range of the restiction of ω(k) to Γ1 , i.e. Γ11 := Rang(ω|Γ1 ), and assume that the boundary ∂Γ11 of Γ11 has Lebesgue measure zero.

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(ii) Γ2 = ∪Γαj , being Γαj an open subset of Rd where the dispersion function ω(k) is constant and equal to αj , i.e. ω(k) = αj ,

∀ k ∈ Γαj .

d

(iii) Γ3 = R \(Γ1 ∪ Γ2 ), that is Γ3 contains the boundaries of Γ1 and Γ2 and other possible regions of singular points of the dispersion function ω(k). 4. Convergence of the Rescaled Densities Let us study the convergence, in the sense of correlators, of the rescaled creation and annihilation densities given in Eq.(??). To simplify the notation we restrict our attention to the vacuum reference vector, so that N (k) = 0 (see Eq.(??)). The extension of the results to the general case is immediate. Moreover, because the mean zero Gaussianity, we have only to prove the convergence, in the sense of Schwartz distributions [?], of the covariance 1 −i t−t2 0 (ω(k)−ω)+i t02 (ω−ω0 ) 0 0 λ λ haλ,ω (t, k)a+ δ(k − k 0 ), λ,ω 0 (t , k )i = 2 e λ i.e. we must calculate, for any Schwartz test functions φ, ϕ, f and g, Z 0 0 lim dt dt0 dk dk 0 φ(t)ϕ(t0 )f (k)g(k 0 )haλ,ω (t, k)a+ λ,ω 0 (t , k )i. λ→0

The following theorem shows that, on Γ1 , every Bohr frequency ω in the open range Γ11 of the dispersion function gives rise to an independent master field, which is a quantum white noise concentrated over the resonant surface ω(k)−ω = 0, and the rest of Bohr frequencies give rise to zero master fields, while, on the open regions Γαj where the dispersion function is constant, the limit does not exist in the resonant case αj = ω = ω 0 and again gives rise to zero master fields otherwise. Theorem 4.1. Under the conditions for ω(k) given above, in the sense of Schwartz distributions, i.e. in S 0 (R2d+2 ): (a) Over Γ1 , if ω doesn’t belong to the boundary ∂ Γ11 of Γ11 , 0 0 lim haλ,ω (t, k)a+ (t , k )i 0 λ,ω λ→0

Γ1

= δω,ω0 2πδ(t − t0 )δ(k − k 0 )δ(ω(k) − ω)χΓ11 (ω). (b) Over each Γαj ,

0 0 lim haλ,ω (t, k)a+ λ,ω 0 (t , k )i λ→0 Γαj

( =

doesn’t exist, if αj = ω = ω 0 , 0,

[∗]

if αj 6= ω or αj 6= ω 0 .

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The proof of this result cast some light on the resonant case αj = ω = ω 0 of item (b): Over each Γαj the final expression in our calculations isb Z 2π αj − ω αj − ω 0 ∧ lim 2 φ∨ ϕ dk f (k)g(k) λ→0 λ λ2 λ2 Γαj 2π ∨ φ (0) ϕ∧ (0) λ→0 λ2

Z dk f (k)g(k),

= lim

Γαj

which is equal to zero when φ∨ (0) = 0 or ϕ∧ (0) = 0, or ±∞ otherwise. Thus the limit also exists in this case, and is equal to zero, if we restrict our attention to test functions with zero mean in time. What happens over Γ3 or when ω ∈ ∂Γ11 ? For example, for dispersion functions of the form ω(k) = |k|µ ,

µ > 0,

we have Γ1 = Rd \{0}, Γ2 = ∅, Γ3 = {0}, Γ11 = (0, ∞) and ∂Γ11 = {0}, so that the frequency of interest is ω = 0. We obtain in this case 0 0 lim haλ,ω (t, k)a+ (t , k )i λ,ω 0 λ→0 ω=0

=

 

0, if d − µ > 0, 2π d/2+1  δω,ω0 δ(t − t0 )δ(k − k 0 )δ(k), if d − µ = 0. Γ(d/2)

When d − µ < 0, our techniques do not give an answer. 5. The Drift As Eq.(??) shows, the drift term Gdt in the stochastic Schrödinger equation given in Eq.(??) is the limit of the expectation value in the reservoir state use the following conventions: The Fourier transform f ∧ and the inverse Fourier transform f ∨ of a test function f ∈ S(Rd ) are given by Z Z 1 1 f ∧ (s) := dx f (x)eix·s , f ∨ (s) := dx f (x)e−ix·s , d/2 d/2 (2π) (2π) b We

so that f ∧∨ = f ∨∧ = f . The Fourier transform F ∧ and the inverse Fourier transform F ∨ of a distribution F ∈ S 0 (Rd ) are defined by the relations hF ∧ , f ∧ i = hF, f i,

hF ∨ , f ∨ i = hF, f i,

being dual pair h·, ·i antilinear on the left and linear on the right.

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of the second term in the iterated series solution for the rescaled Shrödinger equation in interaction picture. In the following theorem we show that the open region Γ2 does not contribute to the drift term whenever the resonant case αk = ω is not present, whereas for the region Γ1 we obtain the usual expression for the drift. The contribution of the singular region Γ3 to the drift has not been determined yet. Theorem 5.1. Under the conditions for ω(k) given above we have: (i) If Γ2 is not empty and no Bohr frequency ω of the system coincides with one of the values αk , then the contribution of the region Γ2 to the drift term is zero, whereas if any of the Borh frequencies ω of the system coincides with one of the values αk , then G does not exist. (ii) Otherwise XX ∗ +∗ ∗ G= (gi |gj )− ω Eω (Di ) Eω (Dj ) + (gi |gj )ω Eω (Di ) Eω (Dj ) ω ij +The part corresponding to the singular region Γ3 , where, for each Bohr frequency ω, the Eω (Dj ) are system operators defined by X Eω (Dj ) := Pεr −ω Dj Pεr , εr ∈Fω

Fω := {εr ∈ Spec HS : εr − ω ∈ Spec HS }, and the explicit form of the constants (gi |gj )± ω is Z (gi |gj )− dk gi∗ (k)gj (k)(N (k) + 1)δ(ω(k) − ω) ω = πχΓ11 (ω) Γ 1 Z (N (k) + 1) , −i P.P. dk gi∗ (k)gj (k) ω(k) − ω Γ1 Z + (gi |gj )ω = πχΓ11 (ω) dk gi∗ (k)gj (k)N (k)δ(ω(k) − ω) Z Γ1 N (k) −i P.P. dk gi∗ (k)gj (k) . ω(k) − ω Γ1 The constants (gi |gj )± ω are called generalized susceptivities and have an important physical interpretation. In some sense they contain all the physical information on the original Hamiltonian system and can be considered as the prototype of quantum mechanical fluctuation-dissipation relations, cf. [?].

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Acknowledgements F.G. Cubillo is grateful to L. Accardi and Centro Vito Volterra for support and kind hospitality. Bibliografia 1. L. Accardi, Y.G. Lu, I. Volovich, Quantum Theory and Its Stochastic Limit, Springer-Verlag, Berlin, 2002. 2. L. Accardi, S.V. Kozyrev, Quantum Interacting Particle Systems. In Quantum Interacting Particle Systems, World Scientific, Singapore, 2002, pp. 1–193. 3. I.M. Gelfand, G.E. Shilov, Les Distributions, Dunod, Paris, 1962. 4. V.G. Maz’ja, Sobolev Spaces, Springer-Verlag, Berlin, 1985. 5. L. Schwartz, Méthodes Mathématiques pour les Sciences Physiques, Hermann, Paris, 1966. 6. E.C. Titchmarsh, Introduction to the Theory of Fourier Integrals, Oxford University Press, Oxford, 1948.