Classical limit of non-integrable systems

Classical limit of non-integrable systems

Mario Castagnino

Institutos de Física de Rosario y de Astronomía y Física del Espacio, Casilla de Correos 67, Sucursal 28, 1628 Buenos Aires, Argentina

ABSTRACT

Self-induced decoherence formalism and the corresponding classical limit are extended from quantum integrable systems to non-integrable ones.

1 Introduction

Decoherence was initially considered to be produced by destructive interference [1]. Later the strategy changed and decoherence was explained as caused by the interaction with an environment [2], but this approach is not conclusive because:

i.- The environment cannot always be defined, e. g. in closed system like the universe.

ii.-There is not a clear definition of the ''cut'' between the 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-induced approach , based in a new version of destructive interference, which will be explained in this talk in its version for non-integrable systems. The essential idea is that this interference is embodied in Riemann-Lebesgue theorem where it is proved that if f(n)Î₁ then

If we use this formula in the case when n = w - w', where w,w' are the indices of the density operator , in such a way that n = 0 corresponds to the diagonal, we obtain a catastrophe, since all diagonal and not diagonal terms would disappear. But, if f(n) = Ad(n)+f₁(n), where now f₁(n)Î1, we have

and the diagonal terms n = 0 remain while the off-diagonal ones vanish. This is the trick we will use below.

2 Weyl-Wigner-Moyal mapping

Let be the phase space. The functions over will be called f(f), where f symbolizes the coordinates of

Then the Wigner transform reads

where Î and (f)Î where is the quantum algebra and the classical one is . We can also introduce the star product

and the Moyal bracket, which is the symbol corresponding to the commutator

so we have

To obtain the inverse symb^-1 we will use the symmetrical or Weyl ordering prescription, namely

Then we have an isomorphism between the quantum algebra and the classical one

The mapping so defined is the Weyl-Wigner-Moyal symbol.

For the state we have

and it turns out that

Namely the definition Î, as afunctional on , is equal to the definition symbrÎ, as afunctional on .

3 Decoherence in non integrable systems

3.1 Local CSCO.

a.- When our quantum system is endowed with a CSCO of N+1 observables, containing the underlying classical system is integrable. In fact, let N+1-CSCO be {1,...,N} the Moyal brackets of these quantities are

where I, J,... = 0,1,...,N and = 0. Then when ® 0 from Eq.(1) we know that

then as H(f) = O₀(f) the set {O_I(f)} is a complete set of N+1 constants of the motion in involution, globally defined over all , and therefore the system is integrable. q. e. d.

b.- If this is not the case N+1 constants of the motion in involution {H,O₁,...,O _N} always exist locally, as can be shown integrating the system of equations (3). Then, if f_iÎ there is maximal domain of integration _fñaround f_iÎ where these constants are defined. In this case the system in non-integrable. Moreover we can repeat the procedure with the system

Then we can extend the definition of the constant {H,O₁,...,O _N}, defined in each _fñ, outside _fñ as null functions. Their Weyl transforms {

₁,...,_N} can be considered as a local N+1-CSCOs related each one with a domain _fñ that we will call {, _1fi,...,_Nfi} (we consider that is always globally defined).

c.-We also can define an ad hoc positive partition of the identity

where I_fi(f) is the characteristic function or index function, i.e.:

where the domains D_fiÌ_fñD_fiÇ D_fj = Æ. Then å_iI_fi(f) = 1. Then we can define A_fi(f) = A(f)I_fi(f) and

and using symb^-1

We can further decompose

where the |jñ_fi are the corresponding eigenvectors of the local N+1-CSCO of D_fiÌ_fñ where a local N+1-CSCO is defined.. So

all over . It can be proved that for i ¹ k it is

so the last decomposition is orthonormal, thus decomposition (5) generalizes the usual eigen-decomposition of integrable system to the non-integrable case. We will use this decomposition below.

3.2 Decoherence in the energy.

a.- Let us define in each D_fi a local N+1-CSCO {, } (as we have said we consider that is always globally defined) as

where we have used decomposition (5). The energy spectrum is 0 < w < ¥ and m_Ifi = {m_1fi,...,m_Nfi},m_IfiÎ. Therefore

where, from the orthonormality of the eigenvector and Eq.(5), we have

áw,m| _fi |w',m 'ñ_fj =d(w - w') d_mm'd_ij

b.- A generic observable, in the orthonormal basis just defined, reads:

where Õ(w,w')_fimm' is a generic kernel or distribution in w, w'. As explained in the introduction, the simplest choice to solve our problem is the van Hove choice [4].

where we have a singular and a regular term, so called because the first one contains a Dirac delta and in the second one the O(w,w')_fimm' are ordinary functions of the real variables w and w'. As we will see these two parts appear in every formulae below. So our operators belong to an algebra and they read

The observables are the self adjoint O^†= O operators. These observables belong to a space Ì . This space has the basis {|w,m,m')_fi, |w,w',m,m')_fi} defined as:

|w, m, m')_fi |w, mñ _fiáw, m'|_fi ,

|w, w', m, m')_fi |w, mñ _fiáw', m'|_fi

c.- Let us define the quantum states Î Ì , where img is a convex set. The basis of is {(w,mm'|_fi, (ww',mm'|_fi} and its vectors are defined as functionals by the equations:

(w, m, m'| _fi |h, n, n')_fi = d(w - h)d_mn d_m'n'd_ij,

(w, w', m, m'| _fi | h, h', n, n')_fj =

d(w - h) d(w' - h') d_mn d_m'n'd_ij,

and all others (.|.) are zero. Then, a generic quantum state reads:

We require that:

where

dwå_im|w,mñ _fiáw,m|_fi is the identity operator. Then, in fact,Î , where is a convex set, and we have

If we now take the limit t ® ¥ and use the Riemann-Lebesgue theorem, being O(w,w') and _fimm' regular (namely '_fimm'O(w,w')Î in the variable n = w - w'), we arrive to

or to the weak limit

where only the diagonal-singular terms remain showing that the system has decohered in the energy.

Remarks

i.- It looks like that decoherence takes place without a coarse-graining, or an environment. It is not so, the van Hove choice (6) and the mean value (8) are a restriction of the information as effective as the coarse-graining is to produce decoherence.

ii.-Theoretically decoherence takes place at t ® ¥. Nevertheless, for atomic interactions, the characteristic decoherence time is t_D = 10^-15s [5]. For macroscopic systems this time is even smaller (e.g., 10^-38s). Models with two characteristic times (decoherence and relaxation) can also be considered [6].

3.3 Decoherence in the other variables.

By a change of basis we can diagonalize the in m and m':

r(w)_fimm'® r(w)_fipp' = r_fi (w)_p d_{pp' .}

in a new basis orthonormal {|w,pñ_fi}. Therefore r_fi(w)_pd_pp'.is now diagonal in all its coordinates in a final local pointer basis in each D_fi, which, in the case of the observables is { |w,p,p')_fi, |w,w',p,p')_fi} (i. e. essentially {|w',p'ñ_fi}), so in this pointer basis we have obtained a boolean quantum mechanics with no interference terms and we have the weak limit:

or in the case of with continuous spectra:

the only case that we will consider below.

4 The classical statistical limit

a.- Let us now take into account the Wigner transforms. There is no problem for regular operators which are considered in the standard theory. Moreover these operators are irrelevant since they disappear after decoherence.

b.- So we must only consider the singular ones as

where now the have continuous spectra. So

But , commute thus

and if O_fi(w,p) = d(w-w')d(p-p') we have

symb|w', p'ñ_fi áw', p'|_fi = d(H(f) - w') ( P_fi (f) - p)

(really up to 0(2), but for the sake of simplicity we will eliminate these symbols from now on).

Let us now consider the singular dual, the symb

_S as the functional on that must satisfy Eq.(2) that now reads

Then we define a density function r_S(f) = symb

_S = å_ir_fiS(f) such that

, is constant of the motion, so r_fi(f) = f(H(f),P_fi(f)). Then we locally define at D_fi the local action-angle variables (q⁰,q¹,...,q^N, ), where would just be H,P_fi1,...,P_fiN and we make the canonical transformation f^a® , H,P_fi1,...,P_fiN so that

Now we will integrate of the functions f(H,P_fi) = f(H,P_fi,...,P_fi) using the new variables.

where we have integrated the angular variables , obtaining the configuration volume C_fi(H,P_fi) of the portion of the hypersurface defined by (H = const.,P_fi = const.) and contained in D_fi. So Eq.(10) reads

for any O_fi(w,p) so r_Sfi(H,P) = r_fi(H,P) for fÎ_fñ and

Putting r_fi(w,p) = d(w-w')d^N(p-p') for some i and all other r_fj(w,p) = 0 for j ¹ i, we have

c.- Moreover the symb of Eq.(9) reads

So we have obtained a decomposition of r_*(f) = r_S(f) in classical hypersurfaces (H = w, P_fi(f) = p_fi), containing chaotic trajectories (since the system is not integrable), summed with different weight coefficients r_fi( w,p) /C_fi(H,P_fi).

d.- Finally only after decoherence the positive definite diagonal-singular part remains and from Eqs.(7) and (11) we see that

r_fi (w, p) > 0 Þ r_*(f) > 0

so the classical statistical limit is obtained.

5 The classical limit

The classical limit can be decomposed into the following processes

Quantum Mechanics - (decohence) ®

Boolean Quantum Mechanics - (symb and ® 0 ) ®

Classical Statistical Mechanics - (choice of a trajectory)

® Classical Mechanics

where the first two have been explained. It only remains the last one: For t(f) = (f) and at any fixed t we have

then we can include this 1 in decomposition (11) and we obtain

namely a sum of classical chaotic trajectories satisfying:

H(f) = w, t (f) = t0 + wt) ,

P_fi = p_fi, q_fi (f) = q_fi0 + p_fit

weighted by ,where we can choose any one of them. In this way the classical limit is completed, in fact we have found the classical limit of a quantum system since we have obtained the classical trajectories, so the correspondence principle is also obtained as a theorem.

6 Conclusion

i.- We have defined the classical limit in the non-integrable case.

ii.- Essentially, we have presented a minimal formalism for 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 decoherence and 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 and O. Lombardi, Stud. Phil. and Hist. Mod. Phys. 35, 73 (2004). M. Castagnino and O. Lombardi, "Self-Induced decoherence and the classical limit of quantum mechanics'', Phil. of Sci., in press (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 of the self-induced approach'', in preparation (2004).

[7] M. Castagnino, "Quantum chaos according to the self-induced approach'', in preparation (2004).

Received on 6 January, 2005

Publication Dates

• Publication in this collection
  06 Sept 2005
• Date of issue
  June 2005

History

• Received
  06 Jan 2005