Self-induced decoherence formalism and the corresponding classical limit are extended from quantum integrable systems to non-integrable ones.Fil: Castagnino, Mario Alberto G. J.. Consejo Nacional de Investigaciónes Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Astronomía y Física del Espacio. - Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Astronomía y Física del Espacio; Argentin

Castagnino, Mario Alberto G. J.

Brazilian Journal of Physics

CONICET Digital

Brazilian Journal of Physics, vol. 35, no. 2B, June, 2005 375Classical Limit of Non-Integrable SystemsMario CastagninoInstitutos de Fı´sica de Rosario y de Astronomı´a y Fı´sica del Espacio,Casilla de Correos 67, Sucursal 28, 1628 Buenos Aires, ArgentinaReceived on 6 January, 2005Self-induced decoherence formalism and the corresponding classical limit are extended from quantum inte-grable systems to non-integrable ones.1 IntroductionDecoherence was initially considered to be produced by de-structive interference [1]. Later the strategy changed and de-coherence was explained as caused by the interaction withan environment [2], but this approach is not conclusive be-cause:i.- The environment cannot always be defined, e. g. inclosed system like the universe.ii.-There is not a clear definition of the ”cut” betweenthe proper system and its environment.iii.- The definition of the pointer basis is not simple.So we need a new and complete theory: The self-inducedapproach [3], based in a new version of destructive interfer-ence, which will be explained in this talk in its version fornon-integrable systems. The essential idea is that this inter-ference is embodied in Riemann-Lebesgue theorem where itis proved that if f(ν)²L1 thenlimt→∞∫ a−af(ν)e−iνt~ dt = 0If we use this formula in the case when ν = ω − ω′, whereω, ω′ are the indices of the density operator ρ̂, in such a waythat ν = 0 corresponds to the diagonal, we obtain a catastro-phe, since all diagonal and not diagonal terms would disap-pear. But, if f(ν) = Aδ(ν) + f1(ν), where now f1(ν)²L1,we havelimt→∞∫ a−af(ν)e−iνt~ dt = Aand the diagonal terms ν = 0 remain while the off-diagonalones vanish. This is the trick we will use below.2 Weyl-Wigner-Moyal mappingLet M =M∈(N+∞) ≡ R∈(N+∞) be the phase space. Thefunctions over M will be called f(φ), where φ symbolizesthe coordinates of Mφa = (q1, ..., qN+1, p1q, ..., pN+1q )Then the Wigner transform readssymbf̂ $ f(φ) =∫〈q +∆|f̂ |q −∆〉ei p∆~ dN+1∆where f̂ ²Â and f(φ)²Awhere Â is the quantum algebra andthe classical one isA. We can also introduce the star productsymb(f̂ ĝ) = symbf̂ ∗ symbĝ = (f ∗ g)(φ),(f ∗ g)(φ) = f(φ) exp(− i~2←−∂ aωab−→∂ b)g(φ)and the Moyal bracket, which is the symbol correspondingto the commutator{f, g}mb = 1i~(f ∗ g − g ∗ f) = symb(1i~[f, g])so we have(f∗g)(φ) = f(φ)g(φ)+0(~) , {f, g}mb = {f, g}pb+0(~2)(1)To obtain the inverse symb−1 we will use the symmetricalor Weyl ordering prescription, namelysymb−1[qi(φ)pj(φ)] =12(q̂ip̂j + p̂j q̂i)Then we have an isomorphism between the quantum algebraÂ and the classical one Asymb−1 : A →Â , ∫†mb : Â→ AThe mapping so defined is the Weyl-Wigner-Moyal symbol.For the state we haveρ(φ) = symbρ̂ = (2pi~)−N−1symb(for operators)ρ̂and it turns out that(ρ̂|Ô) = (symbρ̂|symbÔ) =∫dφ2(N+1)ρ(φ)O(φ) (2)Namely the definition ρ̂² Â′, as afunctional on Â, is equalto the definition symbρ² A′, as afunctional on A.376 Mario Castagnino3 Decoherence in non integrable sys-tems3.1 Local CSCO.a.- When our quantum system is endowed with a CSCOof N + 1 observables, containing Ĥ, the underlying clas-sical system is integrable. In fact, let N + 1−CSCO be{Ĥ,Ô1, ..., Ô N} the Moyal brackets of these quantities are{OI(φ), OJ(φ)}mb = symb(1i~[ÔI , ÔJ ])= 0where I, J, ... = 0, 1, ..., N and Ĥ = Ô0. Then when ~→ 0from Eq.(1) we know that{OI(φ), OJ (φ)}pb = 0 (3)then as H(φ) = O0(φ) the set {OI(φ)} is a complete set ofN+1 constants of the motion in involution, globally definedover all M, and therefore the system is integrable. q. e. d.b.- If this is not the case N + 1 constants of the motionin involution {H,O1, ..., O N} always exist locally, as canbe shown integrating the system of equations (3). Then, ifφi²M there is maximal domain of integration Dφ〉 aroundφi²M where these constants are defined. In this case thesystem in non-integrable. Moreover we can repeat the pro-cedure with the system{OI(φ), OJ(φ)}mb = 0 (4)Then we can extend the definition of the constant{H,O1, ..., O N}, defined in each Dφ〉 , outside Dφ〉 as nullfunctions. Their Weyl transforms {Ĥ,Ô1, ..., Ô N} can beconsidered as a local N + 1-CSCOs related each one witha domain Dφ〉 that we will call {Ĥ,Ô1φi , ..., Ô Nφi} (weconsider that Ĥ is always globally defined).c.-We also can define an ad hoc positive partition of theidentity1 = I(φ) =∑iIφi(φ)where Iφi(φ) is the characteristic function or index func-tion, i.e.:Iφi(φ) ={1 if φ²Dφi0 if φ /∈ Dφiwhere the domains Dφi ⊂ Dφ〉 Dφi ∩ Dφj = ∅. Then∑i Iφi(φ) = 1. Then we can define Aφi(φ) = A(φ)Iφi(φ)andA(φ) =∑iAφi(φ)and using symb−1Â =∑iÂφiWe can further decomposeÂφi =∑jAjφi |j〉φi〈j|φi (5)where the |j〉φi are the corresponding eigenvectors of the lo-cal N+1−CSCO of Dφi ⊂ Dφ〉 where a local N+1-CSCOis defined.. SoÂ =∑ijAjφi |j〉φi〈j|φiall over M. It can be proved that for i 6= k it is〈j|φi |j〉φk = 0so the last decomposition is orthonormal, thus decomposi-tion (5) generalizes the usual eigen-decomposition of inte-grable system to the non-integrable case. We will use thisdecomposition below.3.2 Decoherence in the energy.a.- Let us define in each Dφi a local N + 1−CSCO {Ĥ,Ôφi} (as we have said we consider that Ĥ is always glob-ally defined) asĤ =∫ ∞0ω∑im|ω,m〉φi〈ω,m|φidω,ÔφiI =∫ ∞0∑mOmIφi |ω,m〉φi〈ω,m|φidωwhere we have used decomposition (5). The en-ergy spectrum is 0 ≤ ω < ∞ and mIφi ={m1φi , ...,mNφi},mIφi²N. ThereforeĤ|ω,m〉φi = ω|ω,m〉φi , ÔφiI |ω,m〉φi = OmIφi |ω,m〉φiwhere, from the orthonormality of the eigenvector andEq.(5), we have〈ω,m|φi |ω′,m′〉φj = δ(ω − ω′)δmm′δijb.- A generic observable, in the orthonormal basis justdefined, reads:Ô =∑imm′∫ ∞0∫ ∞0dωdω′O˜(ω, ω′)φimm′ |ω,m〉φi〈ω′,m′|φiwhere O˜(ω, ω′)φimm′ is a generic kernel or distribution inω, ω′. As explained in the introduction, the simplest choiceto solve our problem is the van Hove choice [4].O˜(ω, ω′)φimm′ = O(ω)φimm′δ(ω − ω′) +O(ω, ω′)φimm′(6)where we have a singular and a regular term, so called be-cause the first one contains a Dirac delta and in the secondone the O(ω, ω′)φimm′ are ordinary functions of the realvariables ω and ω′. As we will see these two parts appear inevery formulae below. So our operators belong to an algebraÂ and they readÔ =∑imm′∫ ∞0dωO(ω)φimm′ |ω,m〉φi〈ω,m′|φi+Brazilian Journal of Physics, vol. 35, no. 2B, June, 2005 377∑imm′∫ ∞0∫ ∞0dωdω′O(ω, ω′)φimm′ |ω,m〉φi〈ω′,m′|φiThe observables are the self adjoint O† = O operators.These observables belong to a space Ô ⊂ Â . This spacehas the basis {|ω,m,m′)φi , |ω, ω′,m,m′)φi} defined as:|ω,m,m′)φi .= |ω,m〉φi〈ω,m′|φi ,|ω, ω′,m,m′)φi .= |ω,m〉φi〈ω′,m′|φic.- Let us define the quantum states ρ̂ ∈ Ŝ⊂Ô′ , where Ŝ is aconvex set. The basis of Ô′ is {(ω,mm′|φi , (ωω′,mm′|φi}and its vectors are defined as functionals by the equations:(ω,m,m′|φi |η, n, n′)φj = δ(ω − η)δmnδm′n′δij ,(ω, ω′,m,m′|φi |η, η′, n, n′)φj =δ(ω − η)δ(ω′ − η′)δmnδm′n′δij ,and all others (.|.) are zero. Then, a generic quantum statereads:ρ̂ =∑imm′∫ ∞0dωρ(ω)φimm′(ω,mm′|φi+∑imm′∫ ∞0dω∫ ∞0dω′ρ(ω, ω′)φimm′(ωω′,mm′|φiWe require that:ρ(ω, ω′)φimm′ = ρ(ω′, ω)φim′m,ρ(ω, ω)φimm ≥ 0,(ρ̂|Î) =∑im∫ ∞0dωρ(ω)φi = 1, (7)where Î =∫∞0dω∑im |ω,m〉φi〈ω,m|φi is the identity op-erator. Then, in fact, ρ̂ ∈ Ŝ , where Ŝ is a convex set, and wehave〈Ô〉ρ̂(t) = (ρ̂(t)|Ô) =∑imm′∫ ∞0dωρ(ω)φimm′O(ω)φimm′+∑imm′∫ ∞0dω∫ ∞0dω′ ρ(ω, ω′)φimm′× ei(ω−ω′)t/~O(ω, ω′)φimm′ (8)If we now take the limit t → ∞ and use the Riemann-Lebesgue theorem, being O(ω, ω′) and ρ(ω, ω′)φimm′ regu-lar (namely ′ρ(ω, ω′)φimm′O(ω, ω′)²L1 in the variable ν =ω − ω′), we arrive tolimt→∞〈Ô〉ρ̂(t) = (ρ̂∗|Ô) =∑imm′∫ ∞0dωρ(ω)φimm′O(ω)φimm′or to the weak limitW limt→∞ ρ̂(t) = ρ̂∗ =∑imm′∫ ∞0dωρ(ω)φimm′(ω,m,m′|φiwhere only the diagonal-singular terms remain showing thatthe system has decohered in the energy.Remarksi.- It looks like that decoherence takes place without acoarse-graining, or an environment. It is not so, the vanHove choice (6) and the mean value (8) are a restriction ofthe information as effective as the coarse-graining is to pro-duce decoherence.ii.-Theoretically decoherence takes place at t → ∞.Nevertheless, for atomic interactions, the characteristic de-coherence time is tD = 10−15s [5]. For macroscopic sys-tems this time is even smaller (e.g., 10−38s). Models withtwo characteristic times (decoherence and relaxation) canalso be considered [6].3.3 Decoherence in the other variables.By a change of basis we can diagonalize the ρ(ω)φimm′ inm and m′:ρ(ω)φimm′ → ρ(ω)φipp′ = ρφi(ω)p δpp′ .in a new basis orthonormal {|ω, p〉φi}. Thereforeρφi(ω)p δpp′ .is now diagonal in all its coordinates in a fi-nal local pointer basis in each Dφi , which, in the case ofthe observables is { |ω, p, p′)φi , |ω, ω′, p, p′)φi} (i. e. essen-tially {|ω′, p′〉φi}), so in this pointer basis we have obtaineda boolean quantum mechanics with no interference termsand we have the weak limit:W limt→∞ ρ̂(t) = ρ̂∗ =∑ip∫ ∞0dωρφi(ω)p(ω, p, p|φior in the case of P̂ with continuous spectra:W limt→∞ ρ̂(t) = ρ̂∗ =∑i∫ ∞0dω∫p²DφidpNρ(ω)φi(ω, p, p|φi (9)the only case that we will consider below.378 Mario Castagnino4 The classical statistical limita.- Let us now take into account the Wigner transforms.There is no problem for regular operators which are con-sidered in the standard theory. Moreover these operators areirrelevant since they disappear after decoherence.b.- So we must only consider the singular ones asÔS =∑i∫p²DφidpN∫ ∞0Oφi(ω, p)|ω, p〉φi〈ω, p|φidωwhere now the P̂ have continuous spectra. SoÔS =∑iOφi(Ĥ, P̂φi) =∑iÔSφiBut Ĥ, P̂φi commute thussymbÔS = OS(φ) =∑iOφi(H(φ), Pφi(φ)) + 0(~2)and if Oφi(ω, p) = δ(ω − ω′)δ(p− p′) we havesymb|ω′, p′〉φi〈ω′, p′|φi = δ(H(φ)− ω′)(Pφi(φ)− p)(really up to 0(~2), but for the sake of simplicity we willeliminate these symbols from now on).Let us now consider the singular dual, the symbρ̂S asthe functional on M that must satisfy Eq.(2) that now reads(symbρ̂S |symbÔS) = (ρ̂S |ÔS)Then we define a density function ρS(φ) = symbρ̂S=∑i ρφiS(φ) such that∑i∫dφ2(N+1)ρφiS(φ)OφiS(φ) =∑i∫p²Dφi∫ ∞0ρφi(ω, p, )Oφi(ω, p)dωdpN (10)ρ̂S , is constant of the motion, so ρφi(φ) =f(H(φ), Pφi(φ)). Then we locally define at Dφi the lo-cal action-angle variables (θ0, θ1, ..., θN , J0φi , J1φi, ..., JNφi),where J0φi , J1φi, ..., JNφi would just be H,Pφi1, ..., PφiNand we make the canonical transformation φa →θ0φi , θ1φi, ..., θNφi , H, Pφi1, ..., PφiN so thatdφ2(N+1) = dq(N+1)dp(N+1) = dθ(N+1)φi dHdPNφiNow we will integrate of the functions f(H,Pφi) =f(H,Pφi , ..., Pφi) using the new variables.∫Dφidφ2N+2f(H,Pφi) =∫DφidθN+1φi dHdPNφi f(H,Pφi)=∫DφidHdPNφiCφi(H,Pφi)f(H,Pφi)where we have integrated the angular variablesθ0φi , θ1φi, ..., θNφi , obtaining the configuration volumeCφi(H,Pφi) of the portion of the hypersurface defined by(H = const., Pφi = const.) and contained in Dφi . SoEq.(10) reads∑i∫p²Dφi∫ ∞0ρφi(ω, p, )Oφi(ω, p)dωdpN =∑i∫dHdPNφiCφi(H,Pφi)ρφiS(H,Pφi)OφiS(H,Pφi)for any Oφi(ω, p) so ρSφi(H,P ) =1Cφiρφi(H,P ) forφ²Dφ〉 andρS(φ) = ρ∗(φ) =∑iρφi (H(φ), Pφi(φ))Cφi(H,Pφi)Putting ρφi(ω, p) = δ(ω−ω′)δN (p− p′) for some i and allother ρφj (ω, p) = 0 for j 6= i, we havesymb(ω′, p′, (φ)|φi =δ (H(φ)− ω′) δ(N)(P (φ)− p′φi)Cφi(H,Pφi)c.- Moreover the symb of Eq.(9) readsρS(φ) = ρ∗(φ) =∑i∫p²Dφidp×∫ ∞0dωρφi(ω, p)δ (H(φ)− ω) δ(N) (P (φ)− pφi)Cφi(H,Pφi)(11)So we have obtained a decomposition of ρ∗(φ) =ρS(φ) in classical hypersurfaces (H = ω, Pφi(φ) =pφi), containing chaotic trajectories (since the system isnot integrable), summed with different weight coefficientsρφi (ω, p) /Cφi(H,Pφi).d.- Finally only after decoherence the positive definitediagonal-singular part remains and from Eqs.(7) and (11)we see thatρφi(ω, p) ≥ 0⇒ ρ∗(φ) ≥ 0so the classical statistical limit is obtained.5 The classical limitThe classical limit can be decomposed into the followingprocessesQuantumMechanics− (decohence) −→Boolean QuantumMechanics−(symb and ~→ 0 ) −→Classical Statistical Mechanics−(choice of a trajectory)−→ Classical MechanicsBrazilian Journal of Physics, vol. 35, no. 2B, June, 2005 379where the first two have been explained. It only remains thelast one: For τ(φ) = θ0φi(φ) and at any fixed t we have∑i∫Dφiδ(τ(φ)−τ0−ωt)δ(θφi(φ)−θφi0−pφit)dτ0dθφi0 = 1then we can include this 1 in decomposition (11) and weobtainρ∗(φ) =∑i∫ρφi(ω, pφi)C(ω, pφi)δ(H(φ)− ω)δ(Pφi − pφi)×δ(τ(φ)−τ0−ωt)δ(θφi(φ)−θφi0−pφit)dωdNpφidτ0dθφi0namely a sum of classical chaotic trajectories satisfying:H(φ) = ω , τ(φ) = τ0 + ωt) ,Pφi = pφi , θφi(φ) = θφi0 + pφitweighted by ρφi (ω,pφi )C(ω,pφi ),where we can choose any one ofthem. In this way the classical limit is completed, in factwe have found the classical limit of a quantum system sincewe have obtained the classical trajectories, so the correspon-dence principle is also obtained as a theorem.6 Conclusioni.- We have defined the classical limit in the non-integrablecase.ii.- Essentially, we have presented a minimal formalismfor quantum chaos [7].iii.- We have deduced the correspondence principle.References[1] N.G. van Kampen, Physica XX, 603 (1954). A. Daneri, A.Loinger, and G.M. Prosperi, Nucl. Phys. 33, 297 (1962).[2] J.P. Paz and W. Zurek, “Environment-induced decoherenceand the transition from classical to quantum”, arXiv: quant-ph/0010011 (2000).[3] M. Castagnino and R. Laura, Phys. Rev., A62, 022107 (2000).M. Castagnino, Physica A 335, 511 (2004). M. Castagnino andO. Lombardi, Stud. Phil. and Hist. Mod. Phys. 35, 73 (2004).M. Castagnino and O. Lombardi, “Self-Induced decoherenceand the classical limit of quantum mechanics”, Phil. of Sci., inpress (2004).[4] L. van Hove, Physica 21, 517 (1955); 23, 441 (1957); 25, 268(1959).[5] M. Castagnino, M. Gadella, R. Liotta, and R. Id Betan, Journ.Phys. A34, 10067 (2001). M. Castagnino, R. Laura, R. Liotta,and R. Id Betan, Journ. Phys. A35, 6055 (2002).[6] M. Castagnino, “The master and Fokker-Plank equations ofthe self-induced approach”, in preparation (2004).[7] M. Castagnino, “Quantum chaos according to the self-inducedapproach”, in preparation (2004).

Classical limit of non-integrable systems

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Brazilian Journal of Physics, vol. 35, no. 2B, June, 2005 375Classical Limit of Non-Integrable SystemsMario CastagninoInstitutos de Fı́sica de Rosario y de Astronomı́a y Fı́sica del Espacio,Casilla de Correos 67, Sucursal 28, 1628 Buenos Aires, ArgentinaReceived on 6 January, 2005Self-induced decoherence formalism and the corresponding classical limit are extended from quantum inte-grable systems to non-integrable ones.1 IntroductionDecoherence was initially considered to be produced by de-structive interference [1]. Later the strategy changed and de-coherence was explained as caused by the interaction withan environment [2], but this approach is not conclusive be-cause:i.- The environment cannot always be defined, e. g. inclosed system like the universe.ii.-There is not a clear definition of the ”cut” betweenthe proper system and its environment.iii.- The definition of the pointer basis is not simple.So we need a new and complete theory: The self-inducedapproach [3], based in a new version of destructive interfer-ence, which will be explained in this talk in its version fornon-integrable systems. The essential idea is that this inter-ference is embodied in Riemann-Lebesgue theorem where itis proved that if f(ν)εL1 thenlimt→∞∫ a−af(ν)e−iνt~ dt = 0If we use this formula in the case when ν = ω − ω′, whereω, ω′ are the indices of the density operator ρ̂, in such a waythat ν = 0 corresponds to the diagonal, we obtain a catastro-phe, since all diagonal and not diagonal terms would disap-pear. But, if f(ν) = Aδ(ν) + f1(ν), where now f1(ν)εL1,we havelimt→∞∫ a−af(ν)e−iνt~ dt = Aand the diagonal terms ν = 0 remain while the off-diagonalones vanish. This is the trick we will use below.2 Weyl-Wigner-Moyal mappingLet M = M∈(N+∞) ≡ R∈(N+∞) be the phase space. Thefunctions over M will be called f(φ), where φ symbolizesthe coordinates of Mφa = (q1, ..., qN+1, p1q, ..., pN+1q )Then the Wigner transform readssymbf̂ $ f(φ) =∫〈q + ∆|f̂ |q −∆〉ei p∆~ dN+1∆where f̂ εÂ and f(φ)εAwhere Â is the quantum algebra andthe classical one isA. We can also introduce the star productsymb(f̂ ĝ) = symbf̂ ∗ symbĝ = (f ∗ g)(φ),(f ∗ g)(φ) = f(φ) exp(− i~2←−∂ aωab−→∂ b)g(φ)and the Moyal bracket, which is the symbol correspondingto the commutator{f, g}mb = 1i~(f ∗ g − g ∗ f) = symb(1i~[f, g])so we have(f∗g)(φ) = f(φ)g(φ)+0(~) , {f, g}mb = {f, g}pb+0(~2)(1)To obtain the inverse symb−1 we will use the symmetricalor Weyl ordering prescription, namelysymb−1[qi(φ)pj(φ)] =12(q̂ip̂j + p̂j q̂i)Then we have an isomorphism between the quantum algebraÂ and the classical one Asymb−1 : A →Â , ∫†mb : Â→ AThe mapping so defined is the Weyl-Wigner-Moyal symbol.For the state we haveρ(φ) = symbρ̂ = (2π~)−N−1symb(for operators)ρ̂and it turns out that(ρ̂|Ô) = (symbρ̂|symbÔ) =∫dφ2(N+1)ρ(φ)O(φ) (2)Namely the definition ρ̂ε Â′, as afunctional on Â, is equalto the definition symbρε A′, as afunctional on A.376 Mario Castagnino3 Decoherence in non integrable sys-tems3.1 Local CSCO.a.- When our quantum system is endowed with a CSCOof N + 1 observables, containing Ĥ, the underlying clas-sical system is integrable. In fact, let N + 1−CSCO be{Ĥ,Ô1, ..., Ô N} the Moyal brackets of these quantities are{OI(φ), OJ(φ)}mb = symb(1i~[ÔI , ÔJ ])= 0where I, J, ... = 0, 1, ..., N and Ĥ = Ô0. Then when ~→ 0from Eq.(1) we know that{OI(φ), OJ (φ)}pb = 0 (3)then as H(φ) = O0(φ) the set {OI(φ)} is a complete set ofN +1 constants of the motion in involution, globally definedover all M, and therefore the system is integrable. q. e. d.b.- If this is not the case N + 1 constants of the motionin involution {H, O1, ..., O N} always exist locally, as canbe shown integrating the system of equations (3). Then, ifφiεM there is maximal domain of integration Dφ〉 aroundφiεM where these constants are defined. In this case thesystem in non-integrable. Moreover we can repeat the pro-cedure with the system{OI(φ), OJ(φ)}mb = 0 (4)Then we can extend the definition of the constant{H, O1, ..., O N}, defined in each Dφ〉 , outside Dφ〉 as nullfunctions. Their Weyl transforms {Ĥ,Ô1, ..., Ô N} can beconsidered as a local N + 1-CSCOs related each one witha domain Dφ〉 that we will call {Ĥ,Ô1φi , ..., Ô Nφi} (weconsider that Ĥ is always globally defined).c.-We also can define an ad hoc positive partition of theidentity1 = I(φ) =∑iIφi(φ)where Iφi(φ) is the characteristic function or index func-tion, i.e.:Iφi(φ) ={1 if φεDφi0 if φ /∈ Dφiwhere the domains Dφi ⊂ Dφ〉 Dφi ∩ Dφj = ∅. Then∑i Iφi(φ) = 1. Then we can define Aφi(φ) = A(φ)Iφi(φ)andA(φ) =∑iAφi(φ)and using symb−1Â =∑iÂφiWe can further decomposeÂφi =∑jAjφi |j〉φi〈j|φi (5)where the |j〉φi are the corresponding eigenvectors of the lo-cal N +1−CSCO of Dφi ⊂ Dφ〉 where a local N +1-CSCOis defined.. SoÂ =∑ijAjφi |j〉φi〈j|φiall over M. It can be proved that for i 6= k it is〈j|φi |j〉φk = 0so the last decomposition is orthonormal, thus decomposi-tion (5) generalizes the usual eigen-decomposition of inte-grable system to the non-integrable case. We will use thisdecomposition below.3.2 Decoherence in the energy.a.- Let us define in each Dφi a local N + 1−CSCO {Ĥ,Ôφi} (as we have said we consider that Ĥ is always glob-ally defined) asĤ =∫ ∞0ω∑im|ω,m〉φi〈ω,m|φidω,ÔφiI =∫ ∞0∑mOmIφi |ω, m〉φi〈ω,m|φidωwhere we have used decomposition (5). The en-ergy spectrum is 0 ≤ ω < ∞ and mIφi ={m1φi , ..., mNφi},mIφiεN. ThereforeĤ|ω, m〉φi = ω|ω, m〉φi , ÔφiI |ω, m〉φi = OmIφi |ω, m〉φiwhere, from the orthonormality of the eigenvector andEq.(5), we have〈ω, m|φi |ω′, m′〉φj = δ(ω − ω′)δmm′δijb.- A generic observable, in the orthonormal basis justdefined, reads:Ô =∑imm′∫ ∞0∫ ∞0dωdω′Õ(ω, ω′)φimm′ |ω,m〉φi〈ω′,m′|φiwhere Õ(ω, ω′)φimm′ is a generic kernel or distribution inω, ω′. As explained in the introduction, the simplest choiceto solve our problem is the van Hove choice [4].Õ(ω, ω′)φimm′ = O(ω)φimm′δ(ω − ω′) + O(ω, ω′)φimm′(6)where we have a singular and a regular term, so called be-cause the first one contains a Dirac delta and in the secondone the O(ω, ω′)φimm′ are ordinary functions of the realvariables ω and ω′. As we will see these two parts appear inevery formulae below. So our operators belong to an algebraÂ and they readÔ =∑imm′∫ ∞0dωO(ω)φimm′ |ω, m〉φi〈ω,m′|φi+Brazilian Journal of Physics, vol. 35, no. 2B, June, 2005 377∑imm′∫ ∞0∫ ∞0dωdω′O(ω, ω′)φimm′ |ω,m〉φi〈ω′,m′|φiThe observables are the self adjoint O† = O operators.These observables belong to a space Ô ⊂ Â . This spacehas the basis {|ω, m,m′)φi , |ω, ω′,m, m′)φi} defined as:|ω,m, m′)φi .= |ω, m〉φi〈ω, m′|φi ,|ω, ω′,m, m′)φi .= |ω, m〉φi〈ω′,m′|φic.- Let us define the quantum states ρ̂ ∈ Ŝ⊂Ô′ , where Ŝ is aconvex set. The basis of Ô′ is {(ω, mm′|φi , (ωω′,mm′|φi}and its vectors are defined as functionals by the equations:(ω, m, m′|φi |η, n, n′)φj = δ(ω − η)δmnδm′n′δij ,(ω, ω′, m,m′|φi |η, η′, n, n′)φj =δ(ω − η)δ(ω′ − η′)δmnδm′n′δij ,and all others (.|.) are zero. Then, a generic quantum statereads:ρ̂ =∑imm′∫ ∞0dωρ(ω)φimm′(ω,mm′|φi+∑imm′∫ ∞0dω∫ ∞0dω′ρ(ω, ω′)φimm′(ωω′,mm′|φiWe require that:ρ(ω, ω′)φimm′ = ρ(ω′, ω)φim′m,ρ(ω, ω)φimm ≥ 0,(ρ̂|Î) =∑im∫ ∞0dωρ(ω)φi = 1, (7)where Î =∫∞0dω∑im |ω,m〉φi〈ω, m|φi is the identity op-erator. Then, in fact, ρ̂ ∈ Ŝ , where Ŝ is a convex set, and wehave〈Ô〉bρ(t) = (ρ̂(t)|Ô) =∑imm′∫ ∞0dωρ(ω)φimm′O(ω)φimm′+∑imm′∫ ∞0dω∫ ∞0dω′ ρ(ω, ω′)φimm′× ei(ω−ω′)t/~O(ω, ω′)φimm′ (8)If we now take the limit t → ∞ and use the Riemann-Lebesgue theorem, being O(ω, ω′) and ρ(ω, ω′)φimm′ regu-lar (namely ′ρ(ω, ω′)φimm′O(ω, ω′)εL1 in the variable ν =ω − ω′), we arrive tolimt→∞〈Ô〉bρ(t) = (ρ̂∗|Ô) =∑imm′∫ ∞0dωρ(ω)φimm′O(ω)φimm′or to the weak limitW limt→∞ρ̂(t) = ρ̂∗ =∑imm′∫ ∞0dωρ(ω)φimm′(ω,m, m′|φiwhere only the diagonal-singular terms remain showing thatthe system has decohered in the energy.Remarksi.- It looks like that decoherence takes place without acoarse-graining, or an environment. It is not so, the vanHove choice (6) and the mean value (8) are a restriction ofthe information as effective as the coarse-graining is to pro-duce decoherence.ii.-Theoretically decoherence takes place at t → ∞.Nevertheless, for atomic interactions, the characteristic de-coherence time is tD = 10−15s [5]. For macroscopic sys-tems this time is even smaller (e.g., 10−38s). Models withtwo characteristic times (decoherence and relaxation) canalso be considered [6].3.3 Decoherence in the other variables.By a change of basis we can diagonalize the ρ(ω)φimm′ inm and m′:ρ(ω)φimm′ → ρ(ω)φipp′ = ρφi(ω)p δpp′ .in a new basis orthonormal {|ω, p〉φi}. Thereforeρφi(ω)p δpp′ .is now diagonal in all its coordinates in a fi-nal local pointer basis in each Dφi , which, in the case ofthe observables is { |ω, p, p′)φi , |ω, ω′, p, p′)φi} (i. e. essen-tially {|ω′, p′〉φi}), so in this pointer basis we have obtaineda boolean quantum mechanics with no interference termsand we have the weak limit:W limt→∞ρ̂(t) = ρ̂∗ =∑ip∫ ∞0dωρφi(ω)p(ω, p, p|φior in the case of P̂ with continuous spectra:W limt→∞ρ̂(t) = ρ̂∗ =∑i∫ ∞0dω∫pεDφidpNρ(ω)φi(ω, p, p|φi (9)the only case that we will consider below.378 Mario Castagnino4 The classical statistical limita.- Let us now take into account the Wigner transforms.There is no problem for regular operators which are con-sidered in the standard theory. Moreover these operators areirrelevant since they disappear after decoherence.b.- So we must only consider the singular ones asÔS =∑i∫pεDφidpN∫ ∞0Oφi(ω, p)|ω, p〉φi〈ω, p|φidωwhere now the P̂ have continuous spectra. SoÔS =∑iOφi(Ĥ, P̂φi) =∑iÔSφiBut Ĥ, P̂φi commute thussymbÔS = OS(φ) =∑iOφi(H(φ), Pφi(φ)) + 0(~2)and if Oφi(ω, p) = δ(ω − ω′)δ(p− p′) we havesymb|ω′, p′〉φi〈ω′, p′|φi = δ(H(φ)− ω′)(Pφi(φ)− p)(really up to 0(~2), but for the sake of simplicity we willeliminate these symbols from now on).Let us now consider the singular dual, the symbρ̂S asthe functional on M that must satisfy Eq.(2) that now reads(symbρ̂S |symbÔS) = (ρ̂S |ÔS)Then we define a density function ρS(φ) = symbρ̂S=∑i ρφiS(φ) such that∑i∫dφ2(N+1)ρφiS(φ)OφiS(φ) =∑i∫pεDφi∫ ∞0ρφi(ω, p, )Oφi(ω, p)dωdpN (10)ρ̂S , is constant of the motion, so ρφi(φ) =f(H(φ), Pφi(φ)). Then we locally define at Dφi the lo-cal action-angle variables (θ0, θ1, ..., θN , J0φi , J1φi, ..., JNφi),where J0φi , J1φi, ..., JNφi would just be H, Pφi1, ..., PφiNand we make the canonical transformation φa →θ0φi , θ1φi, ..., θNφi , H, Pφi1, ..., PφiN so thatdφ2(N+1) = dq(N+1)dp(N+1) = dθ(N+1)φi dHdPNφiNow we will integrate of the functions f(H, Pφi) =f(H, Pφi , ..., Pφi) using the new variables.∫Dφidφ2N+2f(H, Pφi) =∫DφidθN+1φi dHdPNφi f(H, Pφi)=∫DφidHdPNφi Cφi(H, Pφi)f(H, Pφi)where we have integrated the angular variablesθ0φi , θ1φi, ..., θNφi , obtaining the configuration volumeCφi(H, Pφi) of the portion of the hypersurface defined by(H = const., Pφi = const.) and contained in Dφi . SoEq.(10) reads∑i∫pεDφi∫ ∞0ρφi(ω, p, )Oφi(ω, p)dωdpN =∑i∫dHdPNφi Cφi(H,Pφi)ρφiS(H, Pφi)OφiS(H, Pφi)for any Oφi(ω, p) so ρSφi(H, P ) =1Cφiρφi(H, P ) forφεDφ〉 andρS(φ) = ρ∗(φ) =∑iρφi (H(φ), Pφi(φ))Cφi(H,Pφi)Putting ρφi(ω, p) = δ(ω−ω′)δN (p− p′) for some i and allother ρφj (ω, p) = 0 for j 6= i, we havesymb(ω′, p′, (φ)|φi =δ (H(φ)− ω′) δ(N)(P (φ)− p′φi)Cφi(H, Pφi)c.- Moreover the symb of Eq.(9) readsρS(φ) = ρ∗(φ) =∑i∫pεDφidp×∫ ∞0dωρφi(ω, p)δ (H(φ)− ω) δ(N) (P (φ)− pφi)Cφi(H, Pφi)(11)So we have obtained a decomposition of ρ∗(φ) =ρS(φ) in classical hypersurfaces (H = ω, Pφi(φ) =pφi), containing chaotic trajectories (since the system isnot integrable), summed with different weight coefficientsρφi (ω, p) /Cφi(H, Pφi).d.- Finally only after decoherence the positive definitediagonal-singular part remains and from Eqs.(7) and (11)we see thatρφi(ω, p) ≥ 0 ⇒ ρ∗(φ) ≥ 0so the classical statistical limit is obtained.5 The classical limitThe classical limit can be decomposed into the followingprocessesQuantum Mechanics− (decohence) −→Boolean Quantum Mechanics−(symb and ~→ 0 ) −→Classical Statistical Mechanics−(choice of a trajectory)−→ Classical MechanicsBrazilian Journal of Physics, vol. 35, no. 2B, June, 2005 379where the first two have been explained. It only remains thelast one: For τ(φ) = θ0φi(φ) and at any fixed t we have∑i∫Dφiδ(τ(φ)−τ0−ωt)δ(θφi(φ)−θφi0−pφit)dτ0dθφi0 = 1then we can include this 1 in decomposition (11) and weobtainρ∗(φ) =∑i∫ρφi(ω, pφi)C(ω, pφi)δ(H(φ)− ω)δ(Pφi − pφi)×δ(τ(φ)−τ0−ωt)δ(θφi(φ)−θφi0−pφit)dωdNpφidτ0dθφi0namely a sum of classical chaotic trajectories satisfying:H(φ) = ω , τ(φ) = τ0 + ωt) ,Pφi = pφi , θφi(φ) = θφi0 + pφitweighted by ρφi (ω,pφi )C(ω,pφi ),where we can choose any one ofthem. In this way the classical limit is completed, in factwe have found the classical limit of a quantum system sincewe have obtained the classical trajectories, so the correspon-dence principle is also obtained as a theorem.6 Conclusioni.- We have defined the classical limit in the non-integrablecase.ii.- Essentially, we have presented a minimal formalismfor quantum chaos [7].iii.- We have deduced the correspondence principle.References[1] N.G. van Kampen, Physica XX, 603 (1954). A. Daneri, A.Loinger, and G.M. Prosperi, Nucl. Phys. 33, 297 (1962).[2] J.P. Paz and W. Zurek, “Environment-induced decoherenceand the transition from classical to quantum”, arXiv: quant-ph/0010011 (2000).[3] M. Castagnino and R. Laura, Phys. Rev., A62, 022107 (2000).M. Castagnino, Physica A 335, 511 (2004). M. Castagnino andO. Lombardi, Stud. Phil. and Hist. Mod. Phys. 35, 73 (2004).M. Castagnino and O. Lombardi, “Self-Induced decoherenceand the classical limit of quantum mechanics”, Phil. of Sci., inpress (2004).[4] L. van Hove, Physica 21, 517 (1955); 23, 441 (1957); 25, 268(1959).[5] M. Castagnino, M. Gadella, R. Liotta, and R. Id Betan, Journ.Phys. A34, 10067 (2001). M. Castagnino, R. Laura, R. Liotta,and R. Id Betan, Journ. Phys. A35, 6055 (2002).[6] M. Castagnino, “The master and Fokker-Plank equations ofthe self-induced approach”, in preparation (2004).[7] M. Castagnino, “Quantum chaos according to the self-inducedapproach”, in preparation (2004).

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Classical limit of non-integrable systems

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