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FEYNMAN-VERNON INFLUENCE FUNCTIONAL AND ITS APPLICATIONS. I

S. ALBEVERIO1),2),3, L. CATTANEO1), L. DI PERSIO2), S. MAZZUCCHI2),4)

Keywords: Oscillatory integrals, Feynman path integrals, Feynman-Vernon influence functional, Caldeira-Leggett model, open quantum systems.

AMS classification : 28, 35C15, 81C35, 81S40, 81Q05.

Abstract. A rigorous representation of the Feynman-Vernon in-fluence functional used to describe open quantum systems is given, based on the theory of infinite dimensional oscillatory integrals. An application to the case of the density matrices describing the Caldeira-Leggett model of two quantum systems with a quadratic interaction is treated.

1. Introduction

One of the crucial problems of modern physics consists in under-standing the behaviour of an open quantum system, i.e. of a quantum system coupled with a second system often called reservoir or enviro-ment. One is interested in the dynamics of the first system, taking into account the influence of the enviroment on it. A typical exam-ple is the study of a quantum particle submitted to the measurement of an observable. In fact, from a quantum mechanical point of view, the interaction with the measuring apparatus cannot be neglected and modifies the dynamics of the particle. On the other hand the evolution of the measuring instrument is not of primary interest.

A particularly intriguing approach to this problem was proposed in 1963 by Feynman and Vernon ( see [FH65, FV63]) within the path inte-gral formulation of quantum mechanics. In 1942 R.P. Feynman [Fey42], following a suggestion by Dirac (see [Dir33, Dir47], proposed an alter-native (Lagrangian) formulation of quantum mechanics (published in [Fey48]), that is an heuristic, but very suggestive representation for the

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solution of the Schr¨odinger equation  i~∂tψ = −~2 2M∆ψ + V ψ ψ(0, x) = ψ0(x) (1) describing the time evolution of the state ψ of a d−dimensional quan-tum particle. The parameter ~ is the reduced Planck constant, m > 0 is the mass of the particle and F = −∇V is an external force. Accord-ing to Feynman’s proposal the wave function of the system at time t evaluated at the point x ∈ Rdis heuristically given as an “integral over

histories”, or as an integral over all possible paths γ in the configura-tion space of the system with finite energy passing in the point x at time t: ψ(t, x) = “ Z {γ|γ(t)=x} e~iSt◦(γ)Dγ −1Z {γ|γ(t)=x} e~iSt(γ)ψ 0(γ(0))Dγ ” (2) where St(γ) is the classical action of the system evaluated along the

path γ, i.e. : St(γ) ≡ St◦(γ) − Z t 0 V (γ(s))ds, (3) St(γ) ≡ M 2 Z t 0 | ˙γ(s)| 2ds, (4)

Dγ is an heuristic Lebesgue “flat” measure on the space of paths and (R{γ|γ(t)=x}e~iSt◦(γ)Dγ)−1 is a normalization constant.

Feynman and Vernon (see [FH65, FV63]) generalized this idea to the study of the time evolution of the reduced density operator of a system in interaction with an enviroment. Let denote ρA, ρB, respectively, the

initial density matrices of the system and of the enviroment, SA, SB,

respectively, the action functionals of the system and of the enviroment and SI the contribution to the total action due to the interaction. Then

the kernel of the reduced density operator of the system ρR (obtained

by tracing over the environmental coordinates) is heuristically given by: ρR(t, x, y) =“ Z γ(t)=x γ′(t)=y e~i(SA(γ)−SA(γ′))F (γ, γ′)ρ A(γ(0), γ′(0))DγDγ′ ” (5) where F is the formal influence functional (IF):

F (γ, γ′) = “ Z Γ(t)=Q Γ′(t)=Q e~i(SB(Γ)−SB(Γ′))e i ~(SI(Γ,γ)−SI(Γ′,γ′))× × ρB(Γ(0), Γ′(0))DΓDΓ′dQ” (6)

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The number of spin-offs originated by the seminal work [FV63] is so large that it is nearly impossible to give here a complete list, and we limit ourselves to shortly mention some of them.

Probably the most influential contributions can be found in [CL83a, CL83b] where Caldeira and Leggett applied the heuristic IF method in order to study the quantum Brownian motion (QBM) (i.e. the analogue of Brownian motion of a quantum particle) and the tunneling phenom-enon in dissipative systems. Latter papers triggered a chain-reaction which is actually far from its end. In [Leg84] (see also [CL81, Cal83]) Leggett determined the imaginary-time functional which supplies the tunnelling rate form of a metastable state at zero temperature, in a formal WKB limit, in presence of an arbitrary linear dissipation mech-anism. In [CL85] an explicit calculation of the time-dependent den-sity matrix is given describing the damping on quantum interference between two gaussian wave packets in a harmonic potential and the obtained results are in agreement with the quantum theory of mea-surement, see e.g. [Zur82].

In [HA85] the decoupled particle-bath initial condition previously used, was compared with the initial off-diagonal coherence of the re-duced density matrix, constituting the thermal initial condition.

A wide-range use of the IF approach was given in [LCD+87] where

the authors mixed their previous experiences giving a deep view to the dynamics of a two-state system coupled to a dissipative environment.

In [CH87] an application of the IF formalism was given in order to study the reduced density operator of a particle coupled with a fermionic environment. Similar applications may be found in [Sch82, Gui84, Che87, Zwe87, BSZ92], where the fluctuations in the motion of a heavy particle interacting with a free fermion gas are studied providing various type of classical and semiclassical expansion either with and without weak-potential or linear response assumptions.

Chen’ s approach was extended in the case of a boson bath in [CLL89]. The heuristic IF approach was generalized in [SC87, SC90] to a non-factorizable initial system-plus-reservoir density operator without spe-cific symmetry assumptions.

Since heterogeneous problems related to macroscopic effects in quan-tum system require extensions to the QBM theory, following [CL83a] various attempts to derive a master-equation (ME) were made in or-der to include general initial conditions and nonlinear couplings. The ME for linear coupling and ohmic environment at high temperature found in [CL83a] was first extended to arbitrary temperature in [UZ89] and afterwards obtained for more general environments and nonlocal

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couplings, which produce colored noise and nonlocal dissipation, see [GSI88, HPZ92, Bru93, HPZ93, BG03] and references therein.

A complementary use of the IF approach to the description of Mar-kovian open quantum systems can be found in [Str97] where the IF method is used in order to develop the ME of general Lindblad positive-semigroup (see [Lin76]) and the propagator in a formal stationary phase approximation is calculated.

Actually the derivation of the ME for the reduced dynamics of quan-tum system have gained a lot of contributions by the use of mathe-matical respectively physical path integrals (PI) techniques (see e.g. [Exn85, JL02] respectively [Wei99, BP02, Kle04, GZ04] and references therein).

The IF formalism was also used in parametric random matrices ap-proach to the problem of dissipation in many-body systems, see e.g. [BDK95, BDK96, BDK97, BDK98] and reference therein, where the derived form of the IF differs from the one in [CL83a] and recovers the latter as the first term of its formal Taylor expansion.

The emerging theory of Quantum Computation is another field of ap-plication of the IF method since the implementation of real quantum processors is often hampered by the quantum decoherence phenome-non, see e.g. [Deu89, Unr95, BDE95, DS98, PZ99, GJZ+03, SH04] and

references therein.

Despite the broad range of its applications, a rigorous mathematical construction of the IF is still missing.

Our aim is to fill this gap following the ideas introduced in [AHK76, AHK77] in connection with the rigorous mathematical definition of Feynman path integrals (2) and in order to realize formulae (5) and (6) as well defined infinite dimensional oscillatory integrals on a suitable Hilbert space.

Before we go over to a short description of our present work we would like to outline that there are rigorous works on models of particles in interaction with heat bath not based on the IF approach, e.g. see [Dav73, CEFM00] and references therein.

In Section 2 we recall some known results, extend the definition of infi-nite dimensional oscillatory integrals and prove some important prop-erties (see [AGM03, AGM04, AM05b, AM05a, AM04a, AM04b, AM] and references therein). In Section 3 the new functional integral is used in the study of the time evolution of two linearly interacting quantum

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systems. A mathematical formalization of the Feynman-Vernon’s the-ory of the IF is given in Section 4 The main results of the paper are Theorems (6) and (7) where a conseguence of the appendix A is used in order to prove the integrability of certain function. The last section is devoted to the study of the Caldeira-Leggett model (see [CL83a]) in the case of a finite dimensional heat bath.

In the second part of this work we shall use our representation of the IF to study rigorously the limit of an infinite dimensional heat bath and of various physically relevant limits, partly discussed in above physical literature’s references.

2. Infinite dimensional oscillatory integrals

In this section we give the definitions of infinite dimensional oscilla-tory integrals and prove some important properties which will be used in the application to the study of the time evolution of a quantum system.

In the following we shall denote by H a (finite or infinite dimensional) real separable Hilbert space, whose elements will be denoted by x, y ∈ H and the scalar product with hx, yi. f : H → C will be a function on H and L : D(L) ⊆ H → H an invertible, densely defined and self-adjoint operator.

Let us denote by M(H) the Banach space of the complex bounded variation measures on H, endowed with the total variation norm, that is:

µ ∈ M(H), kµk = supX

i

|µ(Ei)|,

where the supremum is taken over all sequences {Ei} of pairwise

dis-joint Borel subsets of H, such that ∪iEi = H. M(H) is a Banach

algebra, where the product of two measures µ ∗ ν is by definition their convolution:

µ ∗ ν(E) = Z

Hµ(E − x)ν(dx), µ, ν ∈ M(H)

and the unit element is the vector δ0.

Let F(H) be the space of complex functions on H which are Fourier transforms of measures belonging to M(H), that is:

f : H → C f (x) = Z

H

eihx,βiµf(dβ) ≡ ˆµf(x).

F(H) is a Banach algebra of functions, where the product is the point-wise one; the unit element is the function 1, i.e. 1(x) = 1 ∀x ∈ H and the norm is given by kfk = kµfk.

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The study of oscillatory integrals on Rn with quadratic phase

func-tions, i.e. the ”Fresnel integrals”, Z

e2~ihx,xif (x)dx, ~ > 0, (7)

is a largely developed topic, and has strong connections with several problems in mathematics, e.g. in the theory of Fourier integral opera-tors, and physics, e.g. in optics. Following H¨ormander, the integral in (7) can be defined even if f (H) is not summable by exploiting the can-cellations due to the oscillatory behavior of the integrand, by means of a limiting procedure. More precisely the Fresnel integrals can be defined as the limit of a sequence of regularized, hence absolutely convergent, Lebesgue integrals.

Definition 1. A function f : Rn→ C is Fresnel integrable if and only

if for each φ ∈ S(Rn) such that φ(0) = 1 the limit

lim

ǫ→0(2πi~)

−n/2Z e2~ihx,xif (x)φ(ǫx)dx (8)

exists and is independent of φ. In this case the limit is called the Fresnel integral of f and denoted by

f Z

e2~i hx,xif (x)dx (9)

In [ET84] this definition was generalized to the case Rn is replaced

by an infinite dimensional real separable Hilbert space H. In fact an infinite dimensional Fresnel integral can be defined as the limit of a sequence of finite dimensional approximations:

Definition 2. Let (H, h , i) be a real separable (infinite dimensional) Hilbert space. A function f : H → C is Fresnel integrable if and only if for any sequence Pn of projectors onto n-dimensional subspaces of

H, such that Pn ≤ Pn+1 and Pn → 1 strongly as n → ∞ (1 being the

identity operator in H), the finite dimensional approximations (2πi~)−n/2

Z

PnH

e2~i hPnx,Pnxif (Pnx)d(Pnx),

are well defined (in the sense of definition 1) and the limit lim n→∞(2πi~) −n/2 Z PnH e2~i hPnx,Pnxif (Pnx)d(Pnx) (10)

exists and is independent of the sequence {Pn}.

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by:

f Z

e2~ihx,xif (x)dx.

A complete “ direct description ” of the largest class of Fresnel in-tegrable functions is still an open problem, even in finite dimension. However it is possible to find some interesting subsets of it, as the following result shows.

Theorem 1. Let L : H → H be a self adjoint trace-class operator, such that (I − L) is invertible. Let y ∈ H and let f : H → C be the Fourier transform of a complex bounded variation measure µf on

H. Then the function e−2~i hx,Lxieihx,yif (x) is Fresnel integrable and the

corresponding Fresnel integral can be explicitly computed in terms of a well defined absolutely convergent integral with respect to a σ−additive measure, by means of the following Parseval-type equality:

f Z e2~i hx,xie− i 2~hx,Lxieihx,yif (x)dx = = (det(I − L))−1/2 Z H e−i~2hα+y,(I−L)−1(α+y)iµ f(dα) (11)

where det(I−L) = | det(I−L)|e−πi Ind (I−L) is the Fredholm determinant

of the operator (I − L), | det(I − L)| its absolute value and Ind((I − L)) is the number of negative eigenvalues of the operator (I − L), counted with their multiplicity.

Proof: The result follows directly by theorem 2.1 in [AB93], see also [ET84], which states that for g ∈ F(H)

f Z e2~i hx,xie−2~ihx,Lxig(x)dx = p 1 det(I − L) Z H e−i~2hα,(I−L)−1(α)iµ g(dα)

By taking µg := δy∗ µf the conclusion follows.

By expression (11) the following result follows easily:

Corollary 1. Under the assumptions of theorem 1, the functional f ∈ F(H) 7→f

Z

e2~ihx,(I−L)xieihx,yif (x)dx

is continuous in the F(H)-norm.

Let us introduce now a new type of infinite dimensional oscillatory integrals on the product space H × H that will be applied in the next section to the time evolution of open quantum systems.

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Definition 3. Let f : H×H → C. If for any sequence Pn of projectors onto n-dimensional subspaces of H, such that Pn ≤ Pn+1 and Pn → 1

strongly as n → ∞ (1 being the identity operator in H), the finite dimensional oscillatory integrals

1 (2π~)n Z PnH Z PnH

e2~i hPnx,Pnxie−2~i hPny,Pnyif (Pnx, Pny)d(Pnx)d(Pny),

are well defined and the limit 1 (2π~)n Z PnH Z PnH e2~ihPnx,Pnxie− i 2~hPny,Pnyif (Pnx, Pny)d(Pnx)d(Pny) (12) exists and is independent of the sequence {Pn}, then it is denoted by:

f Z fZ

e2~ihx,xie− i

2~hy,yif (x, y)dxdy.

It is possible to prove a result analogous to theorem 1

Theorem 2. Let L : H → H be a trace class operator, such that I − L is invertible. Let f : H × H → C be the Fourier transform of a complex bounded variation measure µf on H × H. Then the integral

f Z fZ e2~i hx,xie− i 2~hy,yie− i 2~hx−y,L(x+y)if (x, y)dxdy

is well defined and is equal to: 1 det(I − L) Z H Z H e−i~2hα+β,(I−L)−1(α−β)idµf(α, β) (13)

where det(I − L) is the Fredholm determinant of the operator (I − L)

Proof: By definition, taking a sequence Pnof projectors onto n-dimensional subspaces of H, such that Pn ≤ Pn+1 and Pn→ 1 strongly as n → ∞

f Z fZ

e2~ihx,xie−2~i hy,yie−2~i hx−y,L(x+y)if (x, y)dxdy

= lim n→∞ 1 (2π~)n Z PnH Z PnH e2~ihxn−yn,(In−Ln)(xn+yn)if (xn, yn)dxndyn

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where xn := Pnx, x ∈ H, In− Ln := I|PnH− PnLPn. On the other

hand, the finite dimensional approximations are defined by the follow-ing sequence of regularized integrals:

1 (2π~)n Z PnH Z PnH e2~i hxn−yn,(In−Ln)(xn+yn)if (xn, yn)dxndyn = lim ǫ→0 1 (2π~)n Z PnH Z PnH

e2~i hxn−yn,(In−Ln)(xn+yn)iφ(ǫx, ǫy)f (xn, yn)dxndyn

with φ ∈ S(Rn× Rn), φ(0) = 1.

By introducing the new variables zn := xn− yn, wn := xn + yn, by

taking n ≥ ¯n and by Fubini theorem, the latter is equal to: lim ǫ→0 1 (4π~)n Z PnH Z PnH  Z PnH Z PnH eihα,z+w2 i+ihβ,w−z2 i e2~i hzn,(In−Ln)wniφ ǫz + w 2 , ǫ w − z 2  dzndwn  dµn(α, β) = lim ǫ→0 det(In− Ln)−1 (2π)2n Z PnH Z PnH  Z PnH Z PnH

e−i~2 hα+β−2ǫγ,(In−Ln)−1(α−β−2ǫδ)i

˜

φT(γn, δn)dγndδn



dµn(α, β)

where µn∈ F(PnH × PnH) is defined by:

R

PnHφ(xn, yn)dµn(xn, yn) :=

R

HχPnH(x, y)φ(Pnx, Pny)dµ(x, y) and φT ∈ S(PnH × PnH) is defined

by φT(zn, wn) := φ z+w2 ,w−z2



. In the third line we have used the fact that if I − L) is invertible, for any sequence {Pn}n∈N of projection

operators there exist an ¯n such that for any n ≥ ¯n the operator Pn(I −

L)PN is invertible, so that by taking n sufficiently large det(In− Ln) 6=

0. By applying Lebesgue’s dominated convergence theorem, and by the equality Z PnH Z PnH ˜ φT(γn, δn)dγndδn= (2π)2nφT(0, 0),

the latter is equal to: det(In− Ln)−1 Z PnH Z PnH e−i~2hα+β,(In−Ln)−1(α−β)idµn(α, β)

By taking the limit n → ∞ and by the convergence of det(In− Ln) to

det(I − L), we get the final result

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Corollary 2. Under the assumptions of theorem 2, the functional f ∈ F(H × H) 7→ fZ f

Z

e2~i hx,xie−2~i hy,yie−ihx−y,L(x+y)if (x, y)dxdy

is continuous in the F(H × H)-norm.

It is possible to prove the following Fubini type theorem on the change of order of integration between oscillatory integrals and Lebesgue integrals.

Let {µα : α ∈ Rd} be a family in M(H). We shall let

R

Rdµαdα denote

the measure defined by φ 7→ Z Rd Z H φ(x)dµα(x)dα whenever it exists.

Theorem 3. Let (H, h i) and L : H → H as in the assumptions of theorem 2. Let µ : Rd → M(H × H), α 7→ µ

α, be a continuous map

such that Z

Rd|µα|dα < ∞.

Let fα(x, y) = ˆµα(x, y), (x, y) ∈ H × H. Then RRdfαdα ∈ F(H × H)

and Z Rd f Z H f Z H

e2~ihx,xie−2~i hy,yie−2~i hx−y,L(x+y)if

α(x, y)dxdydα = f Z H f Z H

e2~ihx,xie−2~i hy,yie−2~i hx−y,L(x+y)i

Z Rd fα(x)dαdxdy (14) Proof: By definition of fα Z Rd fαdα = Z Rd Z H×H eihk,xi+ihh,yidµα(k, h)dα = Z H×H eihk,xi+ihh,yi Z Rd dµα(k, h)dα, so that RRdfαdα ∈ F(H).

By applying theorem 2 to the l.h.s. of (14), we have: Z Rd f Z H f Z H e2~ihx,xie− i 2~hy,yie− i 2~hx−y,L(x+y)ifα(x, y)dxdydα = det(I − L)−1 Z Rd Z H Z H e−i~2hk+h,(I−L)−1(k−h)idµ α(k, h)dα

By the usual Fubini theorem the latter is equal to: det(I − L)−1 Z H Z H e−i~2hk+h,(I−L)−1(k−h)i Z Rd dµα(k, h)dα

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that, by theorem 2 is equal to the r.h.s of of (14).

3. The Feynman-Vernon influence functional

The infinite dimensional oscillatory integrals of definition 2 provide a rigorous mathematical realization of the heuristic Feynman path in-tegral representation for the solution of the Schr¨odinger equation. The aim of the present section is the extension of these results to the Feyn-man path integral representation of the time evolution of an open quan-tum system.

Let Ut be the unitary evolution operator on L2(Rd) whose

genera-tor is the self-adjoint extension of the operagenera-tor defined on S(Rd) by

−2m∆ + 1

2xΩ2x + v(x), where m > 0, Ω is a positive symmetric

con-stant d × d matrix with eigenvalues Ωj, j = 1 . . . d, and v ∈ F(Rd),

v(x) = ˆµv(x).

The heuristic Feynman path integral representation (2) for the solution of the Schr¨odinger equation (1) is given by:

(U (t)ψ0)(x) = “ ^ Z γ(t)=x e2~i (m R t 0 ˙γ(s)2ds− R t 0γ(s)Ω2γ(s)ds)e− i ~ R t 0v(γ(s))dsφ0(γ(0))dγ”

Let us assume for notation simplicity that m = 1 (this condition will soon be relaxed) and let us introduce the Cameron-Martin space Ht,

i.e. the Hilbert space of absolutely continuous paths γ : [0, t] → R, such that γ(t) = 0, and square integrable weak derivative R0t| ˙γ(s)|2ds < ∞

endowed with the inner product hγ1, γ2i =

Rt

0 ˙γ1(s) · ˙γ2(s)ds. Let L :

Ht→ Ht be the trace class symmetric operator on Ht given by:

(Lγ)(s) = Z t s ds′ Z s′ 0 γ(s′′)ds′′, γ ∈ H t. (15) Let Hd

t := ⊕di=1Ht and let LΩ: Htd→ Hdt be the trace class symmetric

operator on Hd t given by: (LΩγ)(s) = Z t s ds′ Z s′ 0 (Ω2γ)(s′′)ds′′, γ ∈ Hdt.

One can easily verify that hγ1, LΩγ2i = R0tγ1(s)Ω2γ2(s)ds. Moreover

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invertible with: (I − LΩ)−1γ(s) = γ(s) − Ω Z t s sin[Ω(s′− s)]γ(s′)ds′+ + sin[Ω(t − s)] Z t 0 [cos Ωt]−1Ω cos(Ωs′)γ(s′)ds′, (16) and det(I − LΩ) = det(cos(Ωt))

see [ET84]. Thanks to these results and under suitable assumptions it is possible to realize the heuristic Feynman path integral representation for the solution of the Schr¨odinger equation as a well defined infinite dimensional oscillatory integral on the Hilbert space Hd

t.

Theorem 4. Let φ0 ∈ F(Rd). t 6= (n + 1/2)π/Ωj, n ∈ Z. Then the

vector φ(t) := Utφ0 is given by x 7→ φ(t)(x), with:

e−2~ixΩ2xtZg Hd t e2~i hγ,(I−L)γie−~i R t 0xΩ2γ(s)dse−~i R t 0v(γ(s)+x)dsφ 0(γ(0)+x)dγ (17) For a detailed proof see [ET84].

This result can be generalized to the Feynman path integral represen-tation of the time evolution of a mixed state:

Theorem 5. Let ρ be a density matrix operator on L2(Rd), such that

ρ admits a regular kernel ρ(x, y), x, y ∈ Rd. Let us assume moreover

that ρ admits a decomposition into pure states of the form ρ(x, y) = P

iλiei(x)e∗i(y), with λi > 0,Piλi = 1, hei, ejiL2(Rd)= δij, and ei(x) =

ˆ

µi(x), satisfying:

X

i

λi|µi|2 < ∞. (18)

Let t 6= (n + 1/2)π/Ωj, n ∈ Z. Then the density matrix operator at

time t admits a smooth kernel ρt(x, y) which is given by the infinite

dimensional oscillatory integral:

e−2~i(xΩ2x−yΩ2y)t ] Z Hm,dt ] Z Hm,dt e2~i hγ,(I−L)γie−2~ihγ′,(I−L)γ′i e−~i R t 0(xΩ2γ(s)−yΩ2γ′(s))dse−~i R t 0v(γ(s)+x)ds e~i R t 0v(γ′(s)+y)dsρ(γ(0) + x, γ′(0) + y)dγdγ′ (19)

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Proof: By decomposing ρ into pure states, by corollary 2 and condition (18) the integral (19) is equal to:

X i λi  e−2~i xΩ2xt ] Z Hm,dt e2~i hγ,(I−L)γe− i ~ R t 0xΩ2γ(s)dse−~i R t 0v(γ(s)+x)dse i(γ(0)+x)dγ   e2~i yΩ2yt ] Z Hm,dt e−2~i hγ′,(I−L)γ′e i ~ R t 0yΩ2γ′(s)dse i ~ R t 0v(γ′(s)+y)dse∗ i(γ(0) + y)dγ  =X i λi  e−2~i xΩ2xt ] Z Hm,dt e2~ihγ,(I−L)γe−~i R t 0xΩ2γ(s)dse−~i R t 0v(γ(s)+x)dsei(γ(0)+x)dγ   e−2~i yΩ2ytZ] Hm,dt e2~i hγ′,(I−L)γ′e−~i R t 0yΩ2γ′(s)dse−~i R t 0v(γ′(s)+y)dse i(γ(0)+y)dγ ∗ (20) By theorem 5 the latter line is equal toPiλiUtei(x)(Utei)∗(y) = ρt(x, y).

Remark 1. Heuristically expression (19) can be written as f

Z fZ

e~i(St(γ+x)−St(γ′+y)ρ(γ(0) + x, γ′(0) + y)dγdγ′

where St(γ) is the classical action of the system evaluated along the

path defined in (3).

Let us consider now the time evolution of a quantum system made of two linearly interacting subsystems A and B. Let us assume that the state space of the system A is L2(Rd) while the state space of

the system B is L2(RN). Let the total Hamiltonian of the compound

systems be of the form HAB = HA+ HB+ HIN T, with HA= − ∆Rd 2M + 1 2xΩ2Ax + vA(x), x ∈ Rd, HB = − ∆RN 2m + 1 2RΩ2BR + vB(R) R ∈ RN,

HIN T = xCR, with C : RN → Rd is a linear operator and ΩA, resp.

ΩB, is a symmetric positive d × d (resp. N × N) matrix. Let us

assume that the quadratic part of the total potential, i.e. the function x, R 7→ 12xΩ

2

Ax + 12RΩ 2

BR + xCR is positive definite (so that the total

Hamiltonian is bounded from below). Let us assume moreover that the density matrix of the compound system factorizes ρAB = ρAρB and

has a smooth kernel ρAB(x, y, R, Q) = ρA(x, y)ρB(R, Q). We want to

prove an infinite dimensional oscillatory integral representation for the reduced density operator at time t, namely R(UtρABUt+)(x, y, R, R)dR

where the unitary operator Ut:= exp −1~Ht



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Z fZ γ(t)=x Γ(t)=R f Z γ′(t)=y Γ′(t)=R e~i(SA(γ)+SB(Γ)+SIN T(γ,Γ)−SA(γ′)−SB(Γ′)−SIN T(γ′,Γ′)× × ρA(γ(0), γ′(0))ρB(Γ(0), Γ′(0))dγdγ′dΓdΓ′dR (21) where γ and Γ represent the generic path in the configuration space of the system, respectively of the reservoir, and:

SA(γ) + SB(Γ) + SIN T(γ, Γ) := Z t 0 (M 2 ˙γ 2 (s) −12γ(s)Ω2Aγ(s) − vA(γ(s))ds + Z t 0 (m 2 ˙Γ 2 (s) −12Γ(s)Ω2BΓ(s) − vB(Γ(s))ds + Z t 0 γ(s)CΓ(s)ds (22) By the transformations in the path space, given by:

γ → γ/√M and Γ → Γ/√m (23) formula (21) becomes: Z ^Z γ(t)=x Γ(t)=R ^ Z γ′(t)=y Γ′(t)=R e2~i R t 0( ˙γ2(s)−γ(s) Ω2A M γ(s)−vA( γ(s) M )ds× × e2~i R t 0( ˙Γ2(s)−Γ(s) Ω2B mΓ(s)−vB( Γ(s) m )dse− i ~ R t 0γ(s)√mMC Γ(s)ds× ×e−2~i R t 0(˙(γ′)2(s)−γ′(s) Ω2A M γ′(s)−vA( γ′(s) M )dse− i 2~ R t 0(˙(Γ′)2(s)−Γ′(s) Ω2B mΓ′(s)−vB( Γ′(s) m )ds× × e~i R t 0γ′(s)√mMC Γ′(s)dsρA γ(0)√ M, γ′(0) √ M  ρB Γ(0) √ m, Γ′(0) √ m  dγdγ′dΓdΓ′dR, (24) By transformations in (23) it is possible to take unitary masses m and M in order to fulfill the hypotheses of theorems 4 and 5.

Let us consider the two Hilbert spaces: Hdt := H| t⊕ · · · ⊕ H{z }t

d−times

and HNt := H| t⊕ · · · ⊕ H{z }t N −times

We shall denote an element of Hd

t, respectively of HNt , by γ, respectively

Γ. Let L : Ht → Htbe the symmetric bounded operator on Ht, defined

by: Lγ(s) := Rstds′Rs′ 0 γ(s

′′)ds′′. Let L

A : Htd → Hdt, LB : HtN → HNt

and LAB : Hdt ⊕ HNt → Hdt ⊕ HNt be the self adjoint operators defined

by:

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LBΓ := LNΩ2Bm−1Γ (26) LAB(γ, Γ) := (LAγ + 1 √ mML dCΓ, L BΓ + 1 √ mML NCTγ) (27)

where, for all k ∈ N, Lk denotes the operator on Hk

t defined by: Lk := L(1)⊗ L(2)⊗ · · · ⊗ L(k) and: L(k) := 1 ⊗ 1 ⊗ · · · ⊗ 1 ⊗ |{z}L kthelement ⊗1 · · · ⊗ 1

Lemma 1. Let Ψ0 ∈ L2(Rn+d) ∩ F(Rn+d) and let vA ∈ F(Rd), vB ∈

F(RN). Let t 6= (n + 1/2)π/λ

j, where n ∈ Z and λ2j, j = 1, . . . d + N ,

are the eigenvalues of the matrix:  Ω′2 A C′ C′T ′2 B  Ω′A:= ΩA/ √ M , Ω′B := ΩB/√m, C′ := C/ √ M m (28) Then the solution of the Schr¨odinger equation evaluated at time t:

 i~∂

∂tΨ = HABψ

Ψ(0, x, R) = Ψ0(x, R), (x, R) ∈ Rd× RN (29)

is a smooth function and is represented by the infinite dimensional oscillatory integral: f Z Hd t⊕HNt e2~ih(γ,Γ),(Id+N−LAB)(γ,Γ)iG(γ, Γ, x, R)Ψ′ 0(γ(0)+x, Γ(0)+R)dγdΓ (30) where we have defined the functions:

Ψ′0(x, R) := Ψ0(x/ √ M , R/√m) and: G(γ, Γ, x, R) := e−2~itxΩ′2Ax−2~itRΩ′2BR−~ixC′Rt× × e−~i R t 0xΩ′2Aγ(s)ds−~i R t 0RΩ′2BΓ(s)ds−~i R t 0xC′Γ(s)ds−~i R t 0γ(s)C′Rds× × e−~i R t 0v′A(γ(s)+x)ds−~i R t 0v′B(Γ(s)+x)ds (31) while v′

A and vB′ are defined as follows:

vA′ (x) := vA(x/

M ) ; vB′ (R) := vB(R/√m)

Proof: Let ξ1, . . . ξd+N be a system of normal coordinates in Rd+N, with (x, R) = U (ξ1, . . . ξd+N), UT = U−1, the quadratic part of the action is

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by the invariance of the infinite dimensional oscillatory integrals under unitary transformation on paths space [AHK76], and by the infinite dimensional oscillatory integral representation for the solution of the Schr¨odinger equation with a potential of the type “ harmonic oscillator plus Fourier transform of measure” (see [ABHK82, ET84, AB93] for more details).

Lemma 2. Let f ∈ F(Hd

t ⊕ HNt ), f = ˆµ. Let t satisfy the following

inequalities t 6= (n + 1/2)π/ΩAj , n ∈ Z, j = 1 . . . d, (32) t 6= (n + 1/2)π/ΩBj , n ∈ Z, j = 1 . . . N, (33) t 6= (n + 1/2)π/λj, n ∈ Z, j = 1 . . . d + N, (34) where ΩA j, j = 1 . . . d, ΩBj, j = 1 . . . N , and λj, j = 1 . . . d + N are

respectively the eigenvalues of the matrices Ω′

A, Ω′B and of the matrix

given by (28). Let LA, LB, LAB be defined respectively by (25), (26)

and (27). Then the function: γ ∈ Hdt 7→ f Z HN t e2~ihΓ,(IN−LB)Γie−~ihΓ,L NC′Tγi f (γ, Γ)dΓ is Fresnel integrable and:

f Z Hd t⊕HNt e2~ih(γ,Γ),(Id+N−LAB)(γ,Γ)if (γ, Γ)dγdΓ = =f Z Hd t e2~i hγ,(Id−LA)γif Z HN t e2~i hΓ,(IN−LB)Γie−~ihΓ,L NC′Tγi f (γ, Γ)dΓdγ (35) Proof: By condition (33) the operator IN − LB is invertible and by theorem 1 we have: f Z HN t e2~ihΓ,(IN−LB)Γie− i ~hΓ,LNC′Tγif (γ, Γ)dΓ = det(IN − LB)−1/2 Z HN t e−i~2hΓ− LN C′T γ ~ ,(IN−LB)−1Γ− LN C′T γ ~ idµ γ(Γ) = det(IN − LB)−1/2e− i 2~hγ,C′LN(IN−LB)−1LNC′Tγi Z HN t

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where µγ is the measure on HNt defined by: Z HN t g(Γ)dµγ(Γ) := Z Hd t×HNt g(Γ)eihγ,γ′i dµ(γ′, Γ).

One can also easily verify that the operator on Hd

t defined by:

γ 7→ (LA+ C′LN(IN − LB)−1LNC′T)γ

is trace class and, if conditions (32),(32) and (34) are satisfied, the operator defined by:

γ 7→ (Id− LA+ C′LN(IN − LB)−1LNC′T)γ

is invertible. Moreover the function defined by γ 7→

Z

HN t

e−i~2hΓ,(IN−LB)−1Γieihγ,C′LN(IN−LB)−1Γidµγ(Γ)

is the Fourier transform of the bounded variation measure ν on Ht

defined by Z Hd t g(γ)dν(γ) := Z Hd t×HNt g(γ+C′LN(IN−LB)−1Γ)e− i~ 2hΓ,(IN−LB)−1Γidµ(γ, Γ)

By applying theorem 1 we have: f Z Hd t e2~ihγ,(Id−LA)γif Z HN t e2~ihΓ,(IN−LB)Γie−~ihΓ,L NC′Tγi f (γ, Γ)dΓdγ = det(Id− LA− C′LN(IN − LB)−1LNC′T)−1/2det(IN − LB)−1/2 Z Hd t⊕HNt e−i~2hγ+C′LN(IN−LB)−1Γ,(Id−LA−C′LN(IN−LB)−1LNC′T)−1(γ+C′LN(IN−LB)−1Γ)i e−i~2hΓ,(IN−LB)−1Γidµ(γ, Γ) (37)

On the other hand the oscillatory integral: f

Z

Hd t⊕HNt

e2~ih(γ,Γ),(Id+N−LAB)(γ,Γ)if (γ, Γ)dγdΓ

is equal, again by theorem 1, to: det(I − LAB)−1/2

Z

Hd t⊕HNt

e−i~2h(γ,Γ),(Id+N−LAB)−1(γ,Γ)idµ(γ, Γ) (38)

Where LAB is defined by (27), so that an element (γ′, Γ′) ∈ Hdt ⊕ HNt

is equal to (Id+N − LAB)−1(γ, Γ), (γ, Γ) ∈ Hdt ⊕ HNt if and only if



(Id− LA)γ′− LdC′Γ′ = γ

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and one can easily verify that the solution is: γ′ = (Id− LA− C′LN(IN − LB)−1LNC′T)−1γ+

+ (Id− LA)−1LdC′(IN − LB− LNC′(Id− LA)−1LdC′)−1Γ

Γ′ = (IN − LB)−1LNC′T(Id− LA− LdC′(IN − LB)−1LNC′T)−1γ+

+ (IN − LB− LNC′T(Id− LA)−1LdC′)−1Γ (40)

As a consequence the exponent in the integral (38) is equal to: h(γ, Γ), (Id+N − LAB)−1(γ, Γ)iHd t⊕HNt = = hγ, (Id− LA− C′LN(IN − LB)−1LNC′T)−1γiHd t + hγ, (Id− LA)−1LdC′(IN − LB− LNC′(Id− LA)−1LdC′)−1ΓiHd t+ + hΓ, (IN − LB− LNC′T(Id− LA)−1LdC′)−1ΓiHN t + hΓ, (IN− LB)−1LNC′T(Id− LA− LdC′(IN − LB)−1LNC′T)−1γiHN t (41) As one can easily verify that:

(IN − LB− LNC′T(Id− LA)−1LdC′)−1 = (IN − LB)−1

+(IN−LB)−1LNC′T(I−LA−C′LN(IN−LB)−1LNC′T)−1C′LN(I−LB)−1,

and analogously:

(Id− LA− C′LN(IN − LB)−1LNC′T)−1C′LN(IN − LB)−1

= (Id− LA)−1LdC′(IN − LB− LNC′T(Id− LA)−1LdC′)−1,

from which we conclude that the integral (38) is equal to the integral (37).

Equality (35) follows by the following relation:

det(I − LAB) = det(Id− LA− C′LN(IN− LB)−1LNC′T) det(IN − LB),

(42) that can be verified by writing the operator Id+N − LAB in block form,

i.e. Id+N − LAB =  IN − LB LNC′T LdCI d− LA 

, by taking the finite dimen-sional approximation of both sides of equation (42) and by the analo-gous equality valid for finite dimensional matrices.

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Lemma 3. Let ψA

0 ∈ L2(Rd) ∩ F(Rd), ψB0 ∈ L2(RN) ∩ F(RN). Let

t satisfy assumptions (32),(33) and (34). Then the solution of the Schr¨odinger equation (1) is equal to:

f Z Hd t e2~i hγ,(Id−LA)γi f Z HN t e2~i hΓ,(IN−LB)Γie− i ~hΓ,LNC′Tγi× × G(γ, Γ, x, R)ψ′0,A(γ(0) + x)ψ0,B′ (Γ(0) + R)dΓ ! dγ (43)

where G(γ, Γ, x, R) is given by (31) and ψ′

0,A(x) := ψA0(x/ √ M ), ψ′ 0,B(R) := ψB 0 (R/ √ m).

Proof: The result follows by lemma 1 and lemma 2 with ψ0 = ψA

0 ⊗ψB0.

Theorem 6. Let ρA

0 and ρB0 be two density matrix operators on L2(Rd)

and L2(RN) respectively. Let us assume that they have smooth kernels,

denoted by ρA

0(x, x′) and ρB0(R, R′). Let us assume moreover that they

decompose into sum of pure states ρA0 =X i wiAPψA i , ρ B 0 = X j wBj PψB j, ψ A i = ˆµAi , ψjB = ˆµBj , (44) with µA i ∈ F(Rd), µBj ∈ F(RN), and: X i,j wiAwjBAi |2Bj |2 < +∞. (45)

Let t satisfy assumptions (32), (33), (34).

Then the kernel ρt(x, x′, R, R′) of the density operator of the system

evaluated at time t is given by the following infinite dimensional oscil-latory integral (in the sense of definition 3):

f Z Hd t⊕HNt f Z Hd t⊕HNt

e2~i h(γ,Γ),(Id+N−LAB)(γ,Γ)ie−2~ih(γ′,Γ′),(Id+N−LAB)(γ′,Γ′)i

G(γ, Γ, x, R) ¯G(γ′, Γ′, x′, R′)ρ′0,A(γ(0) + x, γ′(0) + x′)

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where G(γ, Γ, x, R) is given by (31). It is also equal to: f Z Hd t f Z Hd t e2~ihγ,(Id−LA)γie− i 2~hγ′,(Id−LA)γ′if Z HN t f Z HN t e2~i hΓ,(IN−LB)Γi e−~ihΓ,L NCTγi e−2~i hΓ′,(IN−LB)Γ′ie~ihΓ′,L NCTγi G(γ, Γ, x, R) ¯G(γ′, Γ′, x′, R′) ρ′0,B(Γ(0) + R, Γ′(0) + R′)dΓdΓ′ρ′0,A(γ(0) + x, γ′(0) + x′)dγdγ′ (47) where ρ′ 0,A(x, y) := ρA0(x/ √ M , y/√M ) and ρ′ 0,B(R, Q) := ρB0(R/ √ m, Q/√m). Proof: If ρA

0 and ρB0 are pure states, the result is a direct consequence

of lemma 1 and lemma 3. For general ρA

0 and ρB0 satisfying assumptions (44) and (45) the result

follows by the continuity of the infinite dimensional oscillatory integral as a functional of F(RN +d) (corollary 2).

Theorem 7. Let ρA

0 and ρB0 be two density matrix operators on L2(Rd)

and L2(RN) respectively. Let us assume that they have regular kernels

as assumed in theorem 5, denoted by ρA

0(x, x′) and ρB0(R, R′). Let ρB0 ∈

S(RN × RN). Let us assume that t satisfies assumptions (32), (33),

(34) and that t is such that the determinant of the d × d left upper block of the n×n matrix cos(Ωt), Ω2 being the matrix (28), is non vanishing.

Then the kernel ρR(t, x, y) of the reduced density operator of the system

A evaluated at time t is given by: ρR(t, x, y) = e− it 2~xΩ′2Axe2~ityΩ′2Ayf Z Hd t f Z Hd t e2~i hγ,(Id−LA)γie− i 2~hγ′,(Id−LA)γ′i e−~i R t 0xΩ′2Aγ(s)dse~i R t 0yΩ′2Aγ′(s)dse−~i R t 0v′A(γ(s)+x)ds+~i R t 0v′A(γ′(s)+y)ds

F (γ, γ′, x, y)ρ′0,A(γ(0) + x, γ′(0) + y)dγdγ′ (48) where F (γ, γ′, x, y) is the influence functional is given by:

F (γ, γ′, x, y) := Z RN e−it~xC′Re+ it ~yC′Re− i ~ R t 0(γ(s)−γ′(s))C′Rds f Z HN t f Z HN t e2~i hΓ,(IN−LB)Γie− i 2~hΓ′,(IN−LB)Γ′ie− i ~hΓ,LNC′Tγie i ~hΓ′,LNC′Tγ′i e−~i R t 0RΩ2B(Γ(s)−Γ′(s))dse−~i R t 0(xC′Γ(s)+yC′Γ′(s))ds e−~i R t 0vB′ (Γ(s)+R)ds+~i R t 0v′B(Γ′(s)+R)dsρ′ 0,B(Γ(0) + R, Γ′(0) + R)dΓdΓ′dR (49)

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Proof:

Let us assume for notation simplicity that m = M = 1. The result in the general case can be obtained by replacing ΩA, ΩB, C, vA, vB, ρA0, ρB0

by Ω′

A, Ω′B, C′, vA′ , v′B, ρ′0,A, ρ′0,B.

First step: Let us prove first of all that the functional (γγ′) 7→ F (γ, γ, x, y)

is well defined for any γ, γ′ ∈ Hd

t, x, y ∈ Rd and it is Fresnel integrable

in the sense of definition 3.

By decomposing the mixed state ρB

0 into pure states according to the

formula (44), the influence functional can be written as: Z RN X j wjjB(x, γ; R)ψjB(y, γ′; R)dR where ψB

j (x, γ) is the solution of the Schr¨odinger equation with initial

datum ψB

j and Hamiltonian H = −12∆R+ 1 2RΩ

2BR + (x + γ(t))CR +

vB(R). In particular, by the unitarity of the evolution operator, kψBj (x, γ)kL2(RN)=

1 for any x ∈ Rd γ ∈ Hd

t. As, by Schwartz inequality:

X j wjB Z RN ψjB(x, γ; R)ψBj (y, γ′; R)dR ≤X j wBj kψBj (x, γ)kL2(RN)kψBj (y, γ′)kL2(RN) = 1

we can conclude that F (γ, γ′, x, y) is well defined for any x, y ∈ Rd

γ, γ′ ∈ Hd

t. Moreover, by Lebesgue’s dominated convergence theorem,

we have: F (γ, γ′, x, y) = lim ǫ→0+ Z RN e−ǫR2e−it~xCRe+ it ~yCRe− i ~ R t 0(γ(s)−γ′(s))CRds f Z HN t f Z HN t

e2~ihΓ,(IN−LB)Γie−2~ihΓ′,(IN−LB)Γ′ie−~ihΓ,LNCTγie i ~hΓ′,LNCTγ′i e−~i R t 0RΩ2B(Γ(s)−Γ′(s))dse−~i R t 0(xCΓ(s)+yCΓ′(s))ds e−~i R t 0vB(Γ(s)+R)ds+~i R t 0vB(Γ′(s)+R)dsρB 0(Γ(0) + R, Γ′(0) + R)dΓdΓ′dR

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By theorem 2 we have: F (γ, γ′, x, y) = | det(IN − LB)|−1 lim ǫ→0+ Z RN dRe−ǫR2e−it~xCR e+it~yCRe− i ~ R t 0(γ(s)−γ′(s))CRds ∞ X n=0 ∞ X m=0 1 n! 1 m! −i ~ n (~im Z t 0 . . . Z t 0 Z t 0 . . . Z t 0 n Y i=1 dsi m Y j=1 drj Z RN . . . Z RN Z RN . . . Z RN n Y i=1 dµv(ki) m Y j=1 dµv(hj) Z RN Z RN dk0dh0ρ˜b(k0, h0)eiR(k0−h0+ P n i=1ki+ P m j=1hj) e−i~2h(− LN CT γ ~ − vΩB ,R ~ −vCx~ +k0G0+ P n i=1kiGsi),(IN−LB)−1(−LN CT γ~ − vΩB ,R ~ −vCx~ +k0G0+ P n i=1kiGsi)i e+i~2h(− LN CT γ′ ~ − vΩB ,R ~ − vCy ~ +h0G0− P sj=1mhjGrj),(IN−LB)−1(−LN CT γ′~ − vΩB ,R ~ − vCy ~ +h0G0− P m j=1hjGrj)i where vB(R) = R RNeikRdµv(R), ρB(R, Q) = R RN R RNeik0R−ih0Qρ˜B(k0, h0)dk0dh0, vΩB,R, vCx, Gs ∈ H N t , s ∈ [0, t], are defined by hvΩB,R, Γi = Z t 0 RΩ2BΓ(s)ds, hvCX, Γi = Z t 0 xCΓ(s)ds, hGs, Γi = Γ(s).

By Fubini theorem we have:

F (γ, γ′, x, y) = | det(IN − LB)|−1 lim ǫ→0+ ∞ X n=0 ∞ X m=0 1 n! 1 m! −i ~ n (~im Z t 0 . . . Z t 0 Z t 0 . . . Z t 0 n Y i=1 dsi m Y j=1 drj Z RN . . . Z RN Z RN . . . Z RN n Y i=1 dµv(ki) m Y j=1 dµv(hj) Z RN Z RN dk0dh0˜ρb(k0, h0) g1(γ′)¯g1(γ)g2(γ′, h0, −h, r, y)¯g2(γ, k0, k, s, x) Z RN dRe−ǫR2e−~i R t 0(γ(s)+x−γ′(s)−y)CRdseiR(k0−h0+ P n i=1ki+ P m j=1hj) e−~iR R t 0Ω2B(I−LB)−1LNCT(γ(s)−γ′(s))ds)eiR R t 0Ω2B(I−LB)−1(−vC,x~ + vC,y ~ +(k0−h0)G0+ P n i=1kiGsi+ P m j=1hjGrj) (50) where, for every paths γ, γ′, x ∈ Rn, v

0 ∈ R and vectors v = (v1, . . . , vn),

w= (w1, . . . , wn) we have defined the functions:

g1(γ) := e+

i

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and g2(γ, v0, v, w, x) := e+ i~ 2hv0G0+ Pn i=1viGwi−vC,x~ ,(IN−LB)−1(v0G0+ Pn i=1viGwi−vC,x~ )i× × e−ihLNCtγ,(IN−LB)−1(v0G0+ P n i=1viGwi−vC,x~ )i (52) By integrating with respect to R we have that (50) the latter is equal to: | det(IN − LB) |−1 lim ǫ→0+ ∞ X n=0 ∞ X m=0 1 n! 1 m!  −i ~ n i ~ m Z t 0 . . . Z t 0 Z t 0 . . . Z t 0 n Y i=1 dsi m Y j=1 drj Z RN . . . Z RN Z RN . . . Z RN n Y i=1 dµv(ki) m Y j=1 dµv(hj) Z RN Z RN π ǫ N/2 e−ω24ǫ  ˜ ρb(k0, h0)g1(γ′)¯g1(γ)g2(γ′, h0, −h, r, y)¯g2(γ, k0, k, s, x)dk0dh0 (53) were: ω := 1~ Z t 0 (I − L B)−1CT(γ(s) − γ′(s))ds − 1

~ (ΩBcos ΩBt)−1sin(ΩBt)CT(x − y)+

+ (cos ΩBt)−1(k0− h0+ n X i=1 cos(ΩBsi)ki+ m X j=1 cos(ΩBrj)hj) 2 (54) By introducing the new integration variables:

k′ 0 := 1 √ ǫ(k0− h0+ a), h ′ 0 := h0− 1 2a with: a := n X i=1 cos(ΩBsi)ki+ m X j=1 cos(ΩBrj)hj− cos(ΩBt) 1 ~ Z t 0 (I − L B)−1CT(γ(s)+ − γ′(s))ds) − 1~ (ΩB)−1sin(ΩBt)CT(x − y)) where: Z t 0 (I−L B)−1CT(γ(s)−γ′(s))ds = cos−1(ΩBt) Z t 0 cos(ΩBs)CT(γ(s)−γ′(s))ds

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the integral in (53), with k0 =√ǫk0′ + h′0− a2 and h0 = h′0+a2, can be written as: πN/2| det(IN − LB)|−1 lim ǫ→0+ ∞ X n=0 ∞ X m=0 1 n! 1 m! −i ~ n (~im Z t 0 . . . Z t 0 Z t 0 . . . Z t 0 n Y i=1 dsi m Y j=1 drj Z RN . . . Z RN Z RN . . . Z RN n Y i=1 dµv(ki) m Y j=1 dµv(hj) Z RN Z RN dk′ 0dh′0ρ˜b(√ǫk0′ + h′0− 1 2a, h ′ 0 + 1 2a)g1(γ ′g 1(γ) g2(γ′, h′0+ a 2, −h, r, y)¯g2(γ, √ ǫk0′ + h′0 a 2, k, s, x)e −1 4|(cos ΩBt)−1k0′|2

By letting ǫ → 0 and using dominated convergence, the integral reduces to the following form:

F (γ, γ′, x, y) = K(x, y, t)e−2~i h(γ−γ′),A(γ+γ′)ie− i ~hγ,CLN(IN−LB)−1vC,xi e~ihγ′,CLN(IN−LB)−1vC,yie i 2~CT(x−y) R t

0 sin(ΩBt) sin(ΩB(t−s))Ω2Bcos(ΩBt) C

T(γ(s)+γ(s))ds ∞ X n=0 ∞ X m=0 1 n! 1 m! −i ~ n (~im Z t 0 . . . Z t 0 Z t 0 . . . Z t 0 n Y i=1 dsi m Y j=1 drj Z RN . . . Z RN Z RN . . . Z RN n Y i=1 dµv(ki) m Y j=1 dµv(hj) Z RN dh′ 0ρ˜b(h′0− 1 2a, h ′ 0+ 1 2a) e−i~2 P n

i,j=1kisin(ΩB(t−si∨sj )) cos(ΩB(si∧sj ))ΩB cos(ΩBt) kj

ei~2

P

m

i,j=1hisin(ΩB(t−ri∨rj )) cos(ΩB(ri∧rj ))ΩB cos(ΩBt) hj

ei P n i=1kicos(ΩBsi)−cos(ΩBt)Ω2 Bcos(ΩBt) CTx ei P m j=1hjcos(ΩBrj )−cos(ΩBt)Ω2 Bcos(ΩBt) CTy ei(h0− a 2)Ω21−cos(ΩBt)Bcos(ΩBt)C Tx e−i(h0+ a 2)Ω21−cos(ΩBt)Bcos(ΩBt)C Ty e−2i R t 0 sin(ΩB(t−s))ΩB cos(ΩBt)CT(γ(s)+γ′(s))ds( P n i=1cos(ΩBsi)ki+ P m j=1cos(ΩBrj)hj)

ei~h′0ΩB cos ΩBtsin ΩBt ae−i~(h′0−a/2)

P

n

i=1sin(ΩB(t−si))ΩB cos(ΩBt) ki

e−i~(h′0+a/2) P m j=1sin(ΩB(t−rj ))ΩB cos(ΩBt) hj eihγ, P n i=1CLN(IN−LB)−1kiGsiieihγ′, P m j=1CLN(IN−LB)−1hjGrjieihγ−γ′,CLN(IN−LB)−1h′0G0i (55) where we have defined:

K(x, y, t) := πN2Ne− i 2~CT(x−y)  t Ω2B− sin(ΩBt) Ω3Bcos(ΩBt)  CT(x−y)

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and

e−2~ih(γ−γ′),A(γ+γ′)i := e− i

2~hCT(γ−γ′),LN(IN−LB)−1LNCT(γ+γ′)i×

× e+2~i cos(ΩBt)hCT(γ(s)−γ′(s)),(IN−LB)−1vihLN(IN−LB)−1G0,CT(γ(s)+γ′(s))i=

= e2~i R t 0CT(γ−γ′)(s)Ω−1 R s 0sin(ΩB(s−r))CT(γ+γ′)(r)drds (56) with v(s)i := t 2−s2 2 , i = 1 . . . N , s ∈ [0, t].

As we have assumed that the determinant of the d × d left upper block of the n × n matrix cos(Ωt) (Ω2 being the matrix (28)) is non

vanishing, it is possible to prove (see appendix A) that the operator I − LA − A is invertible. As F (γ, γ′, x, y) is of the form F (γ, γ′) =

e−2~ih(γ−γ′),A(γ+γ)i

f (γ, γ′), with f ∈ F(Hd

t ⊕ Hdt), we can conclude that

the influence functional is a Fresnel integrable function.

Second step Let us prove that the reduced density operator ρR(t, x, y)

is given by the infinite dimensional oscillatory integral (48).

Let ρ(t, x, y, R, Q) be the (smooth) kernel of the density operator of the compound system evaluated at time t. Then the integral giving the kernel of reduced density operator ρR(t, x, y) :=

R

ρ(t, x, y, R, R)dR is absolutely convergent and by Lebesgue’s dominated convergence theo-rem we have ρR(t, x, y) = limǫ→0R ρ(t, x, y, R, R)e−ǫR

2

dR.

On the other hand the influence functional can be written as F (γ, γ′) =

e− i 2~h(γ−γ′),A(γ+γ′)if (γ, γ′), with f : Hd t ⊕ Hdt → C defined as follows: f = lim ǫ→0fǫ, (57) and where: fǫ(γ, γ′) := πN/2| det(IN − LB)|−1 lim ǫ→0+ ∞ X n=0 ∞ X m=0 1 n! 1 m! −i ~ n (~im Z t 0 . . . Z t 0 Z t 0 . . . Z t 0 n Y i=1 dsi m Y j=1 drj Z RN . . . Z RN Z RN . . . Z RN n Y i=1 dµv(ki) m Y j=1 dµv(hj) Z RN Z RN dk′0dh′0ρ˜b(√ǫk0′ + h′0− 1 2a, h ′ 0+ 1 2a) g2(γ′, h′0+ a 2, −h, r, y)¯g2(γ, √ ǫk0′ + h′0 a 2, k, s, x)e −1 4|(cos ΩBt)−1k0′|2 with a′ := a + cos(Ω Bt)1~ Rt 0(I −LB)

−1CT(γ(s) −γ(s))ds) and the limit

(57) is meant in the F(Hdt ⊕ Hdt) sense.

By the continuity of the infinite dimensional oscillatory integral as a functional on F(Hd

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equation (48) is equal to: e−2~itxΩ2Axe2~ityΩ2Aylim ǫ→0 f Z Hd t f Z Hd t

e2~i hγ,(Id−LA)γie−2~ihγ′,(Id−LA)γ′i

e−~i R t 0xΩ2Aγ(s)dse~i R t 0yΩ2Aγ′(s)dse−~i R t 0vA(γ(s)+x)ds+~i R t 0vA(γ′(s)+y)ds e−2~i h(γ−γ′),A(γ+γ′)ifǫ(γ, γ′)ρA 0(γ(0) + x, γ′(0) + y)dγdγ′ (58)

On the other hand the latter is equal to:

e−2~itxΩ2Axe2~ityΩ2Aylim ǫ→0 f Z Hd t f Z Hd t e2~i hγ,(Id−LA)γie− i 2~hγ′,(Id−LA)γ′i e−~i R t 0xΩ2Aγ(s)dse~i R t 0yΩ2Aγ′(s)dse−~i R t 0vA(γ(s)+x)ds+ i ~ R t 0vA(γ′(s)+y)ds ( Z RN dRe−ǫR2e−it~xCRe+ it ~yCRe− i ~ R t 0(γ(s)−γ′(s))CRds f Z HN t f Z HN t

e2~ihΓ,(IN−LB)Γie−2~ihΓ′,(IN−LB)Γ′ie−~ihΓ,L NCTγi e~ihΓ′,L NCTγi e−~i R t 0RΩ2B(Γ(s)−Γ′(s))dse−~i R t 0(xCΓ(s)+yCΓ′(s))ds e−~i R t 0vB(Γ(s)+R)ds+ i ~ R t 0vB(Γ′(s)+R)dsρB 0(Γ(0)+R, Γ′(0)+R)dΓdΓ′dR)ρA0(γ(0)+x, γ′(0)+y)dγdγ′

By Fubini theorem (see theorem 3) and by the infinite dimensional oscillatory integral representation or the kernel of the density operator it is equal toR dRe−ǫR2

ρ(t, x, y, R, R). By letting ǫ → 0 the conclusion follows.

Remark 2. It is typical of the difficulties in handling rigorously Feyn-man path integrals (as infinite dimensional oscillatory integrals) that the passages to the limit cause mathematical problems, because of the lack of the dominated convergence and limited avalability of Fubini-type theorems. Our ǫ-cut-off trick was instrumental to perform such type of computation.

4. Application to the Caldeira-Leggett model

Let us compute now the influence functional F (γ, γ′, x, y) in the case

vB≡ 0, ρB0(R, Q) := QN j=1ρ (j) B (Rj, Qj, 0), where: ρ(j)B (Rj, Qj, 0) := r j π~ coth(~ωj/2kT )e− mωj 2~ sinh(~ωj /kT ) (R2j+Q2j) cosh ~ωj kT−2RjQj 

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ωj, j = 1 . . . n being the eigenvalues of the matrix ΩB. By notation

simplicity we put m = 1, the general case can be handled by replacing ΩA, ΩB, C, vA, vB, ρA0, ρB0 with Ω′A, ΩB′ , C′, v′A, vB′ , ρ′0,A, ρ′0,B.

By inserting this into the general formula (55) the influence func-tional becomes: F (γ, γ′, x, y) = K(x, y, t)e−2~i h(γ−γ′),A(γ+γ′)ie− i ~hγ,CLN(IN−LB)−1vC,xi e~ihγ′,CL N(I N−LB)−1vC,yie i 2~CT(x−y) R t

0 sin(ΩBt) sin(ΩB(t−s))Ω2Bcos(ΩBt) C

T(γ(s)+γ(s))ds Z RN dh′0ρ˜b(h′0− 1 2a, h ′ 0+ 1 2a)e i(h0−a2)Ω21−cos(ΩBt) Bcos(ΩBt) CTx e−i(h0+ a 2)Ω21−cos(ΩBt)Bcos(ΩBt)C Ty

ei~h′0ΩB cos ΩBtsin ΩBt aeihγ−γ′,CLN(IN−LB)−1h′0G0i (59)

where K(x, y, t) := πN2Ne i 2~CT(x−y)  t Ω2B− sin(ΩBt) Ω3B  CT(x+y) e−2~ih(γ−γ′),A(γ+γ′)i:= e− i 2~hCT(γ−γ′),LN(IN−LB)−1LNCT(γ+γ′)i

e+2~i cos(ΩBt)hCT(γ(s)−γ′(s)),(IN−LB)−1vihLN(IN−LB)−1G0,CT(γ(s)+γ′(s))i

= e2~i R t 0CT(γ−γ′)(s)Ω−1 R s 0sin(ΩB(s−r))CT(γ+γ′)(r)drds (60) a = − cos(ΩBt) 1 ~ Z t 0 (I−L B)−1CT(γ(s)−γ′(s))ds)− 1 ~ (ΩB)−1sin(ΩBt)CT(x−y))

By direct computation, we obtain: F (γ, γ′, x, y) = e2~i R t 0CT(γ(s)+x−γ′(s)−y)Ω−1B R s 0sin(ΩB(s−r))CT(γ(r)+x+γ′(r)+y)drds e−2~1 R t 0CT(γ(s)+x−γ′(s)−y)Ω−1B coth( ~ΩB 2kT) R s 0 cos(ΩB(s−r))CT(γ(r)+x−γ′(r)−y)drds (61) which yelds the result heuristically derived in [FV63].

Appendix A. Kernel of the operator I − LA− A

A vector γ ∈ Hd

t belongs to the kernel of the operator I − LA− A,

if it satisfies the following equation: γ(s) + Z t s ds′ Z s′ 0 ds′′ Z s′′ 0 CΩ−1B sin(ΩB(s′′− r))CTγ(r)dr+ − Z t s ds′ Z s′ 0 Ω2Aγ(s′′)ds′′ = 0 s ∈ [0, t] (62)

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with γ(t) = 0. Equation (62) is equivalent to: ¨ γ(s) + Ω2Aγ(s) − Z s 0 CΩ−1B sin(ΩB(s − r))CTγ(r)dr = 0 (63)

with the conditions: γ(t) = 0, ˙γ(0) = 0.

By differentiating equation (62), it is easy to see that its solution, if it exists, is a C∞ function and its odd derivatives, evaluated for s = 0,

vanish, while the even derivatives satisfy the following relation γ2(N +2)(0) + Ω2Aγ2(N +1)(0) −

N

X

k=0

(−1)kCΩ2kBCTγ2(N −k)(0) = 0 (64) By induction it is possible to prove that γ2N(0) = (−1)N[Ω2N]

d×dγ(0),

where [Ω2N]

d×d denotes the d × d left upper block of the N-th power

of the n × n matrix Ω2 (where Ω2 is given by equation (28)). One

can conclude that the solution of equation (62) is of the form γ(s) = [cos(Ωs)]d×dγ(0). By imposing the condition γ(t) = 0, one concludes

that if det([cos(Ωs)]d×d) 6= 0 then equation (62) cannot admit nontrivial

solutions and the operator I − LA− A is invertible.

Aknowledgements

We would like to thank the hospitality of the mathematics depart-ments in Bonn and Trento. L. Cattaneo gratefully acknowledges the financial support by the Lise Meitner Fellowship of the Land Nordrhein-Westfalen. S. Mazzucchi gratefully acknowledges the financial support by P.A.T. (Trento) and MIUR (Italy); this research is part of the I.D.A. project of INdAM.

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1)Institut f¨ur Angewandte Mathematik, Wegelerstr. 6, 53115 Bonn

(D)

2) Dip. Matematica, Universit`a di Trento, 38050 Povo (I)

3) BiBoS; IZKS; SFB611; CERFIM (Locarno); Acc. Arch.

(Mendri-sio)

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