Correlations between spectra with different symmetry: any chance to be observed?

A standard assumption in quantum chaology is the absence of correlation between spectra pertaining to different symmetries. Doubts were raised about this statement for several reasons, in particular, because in semiclassics spectra of different symmetry are expressed in terms of the same set of periodic orbits. We reexamine this question and find absence of correlation in the universal regime. In the case of continuous symmetry the problem is reduced to parametric correlation, and we expect correlations to be present up to a certain time which is essentially classical but larger than the ballistic time.


Introduction
The topic we are to consider is the cross-correlation between spectra of different symmetry for so-called quantum chaotic systems, i.e. quantum systems, whose classical analogue is chaotic. The instinctive reaction would be to say that states of different symmetry live in different spaces, don't talk to each other and cannot conceivably display any correlation [1]. Doubts appear when one remembers that in the semiclassical approach the level correlations of different symmetries can be expressed in terms of sums over the same set of periodic orbits, and such "brotherhood in the parent orbits" may introduce a correlation. Although independence is habitually assumed, the question deserves a closer look. In nuclear physics some evidence has been assembled that correlations in two-body random ensembles with finite sets of states exist, which is not too surprising considering the small number of independent two-body matrix elements [2], and certain nuclear scattering data support the existence of correlations to some extent [3] but a study of correlations between spectra of opposite parity in the limit of many particles revealed no correlations [4].
These findings make it all the more desirable to analyze the usual assumption of independence in the semi-classical limit in some detail, as we shall do here. We get basically a negative result, both for the case of discrete and the continuous symmetry, i.e., the absence of correlations is confirmed in the universal regime.
These conclusions are to be taken with a grain of salt. Correlations may still exist in certain pathological systems (mostly those with disconnected phase space often due to unusual behavior under time-reversal symmetry [5,6,7]. Our reasoning also does not necessarily apply to arithmetic billiards which have chaotic dynamics, but an exponentially large number of geodesics with identical actions [8]). Nevertheless, it has been shown by explicit evaluation of the Selberg trace sum formula for the correlation function, that eigenvalues for even and odd parity in the modular domain on the surface with constant negative curvature are indeed uncorrelated [9] More importantly, for continuous symmetries correlations may exist up to a certain time which is essentially classical in nature, if the difference of quantum numbers characterizing the irreducible representations is of order one as the quantum numbers themselves get large. Note that this effect can be easily overlooked, as times are often given in terms of the Heisenberg time which, in the semi-classical limit, goes to infinity.
The paper is structured as follows: After laying foundations we discuss the discrete symmetries in what we consider to be the simplest case. This is the reflection symmetry in a two-dimensional system when we can pass to a half-space with Dirichlet or Neumann boundary conditions. We obtain the relevant results for these two particular cases using periodic orbit expansions and find the expected absence of correlations. Starting with the diagonal approximation we discuss later the off-diagonal contributions and times beyond the Heisenberg time. After that we study general discrete symmetries. Trivial generalization of the described technique is possible only for groups that induce a decomposition of space into fundamental domains. This is not always true and we shall present the general argument using the symmetry decomposition of periodic orbits, as proposed in [10].
In the next section we pass to continuous symmetry and consider the effect of a change of the index of irreducible representation as parametric correlation [11] within the symmetry-reduced Hilbert space. Again, we find absence of correlations in the universal regime, i.e., on the scale of the Heisenberg time, however correlations on a classical i.e.
-independent time scale are expected. Finally we present some conclusions and an outlook about possible observations.

Definitions: spectral cross-correlation function and the form factor
We start with some definitions that will be essential to the later arguments, and at the same time we fix the notations we shall use. Let be two spectral densities describing either two subspectra of different symmetry of the same Hamiltonian or the spectra of two different Hamiltonians. We shall assume that both spectra are unfolded to the same constant mean level density ∆ (see, e.g., [1]). The precise manner of the unfolding procedure is a somewhat thorny issue and will be discussed later.
The two-level cross-correlation function is defined as, where . . . denotes averaging over an interval of E and smoothing over a window of the dimensionless energy offset ε; since the spectra are unfolded we have ρ (1) It is often more convenient to work with the Fourier transform of R (1,2) (E, ε) with respect to ε. The result is the cross form factor which is a double sum, the dimensionless τ = T /T H is time in units of the Heisenberg time T H = 2π /∆. A semiclassical representation for the spectral correlator and the form factor follows from the Gutzwiller expansion of the fluctuating part of the spectral density in terms of the classical periodic orbits γ, Here S γ , A γ , µ γ are the action, stability coefficient and the Maslov index of the orbit γ.
Strictly speaking this expression should also include the periodic orbit repetitions; these however become irrelevant in the semiclassical limit. Replacing in (2) both spectral densities by the respective expansions (3) we get, the brackets . . . denote averaging over E and smoothing over a small window of τ . The sum is taken over periodic orbits γ 1 of the system 1 and γ 2 of the system 2. Their periods T γ 1 , T γ 2 must lie in the interval [T, T + δ] whose width δ is small compared with T = τ T H .

Discrete symmetry
To illustrate the central point of our argument we shall start with the simplest possible example of a system with a single reflection symmetry.

A simple example: reflection symmetry
Consider the spectral correlation between the even and odd spectra in a chaotic system with two degrees of freedom such as the cardioid billiard [12,13,14], see Fig. 1. On the line of symmetry the even and odd wave functions either have zero normal derivative or themselves turn into zero. The subspectra of definite parity can be found separately by solving the Schrödinger equation in the desymmetrized billiard (the fundamental domain) obtained by cutting the billiard along the symmetry line and dropping one of the halves. Imposing the Neumann (resp. Dirichlet) boundary condition on the cut we shall obtain the even (resp. odd) part of the spectrum. The semiclassical level density consists of the smooth part (the Weyl term) and the fluctuating part which is the Gutzwiller sum over periodic orbits of the desymmetrized billiard with the mirror-reflecting wall at the symmetry line. The difference between the even and odd cases in the Weyl term is responsible for the fact that the N-th eigenvalue in the symmetric sector E S N lies lower than the N-th eigenvalue in the antisymmetric sector E A N by an amount of the order √ E A,S N , see [15] and reference therein. This issue has some consequences but can be taken care by an appropriate unfolding procedure. In the Gutzwiller sum the even and odd cases differ by the effective Maslov indices of the periodic orbits since each reflection in the Dirichlet and Neumann boundary contributes π or zero, respectively, to the phase of the orbit contribution.
We shall denote by n γ the number of strikes of the periodic orbit γ of the desymmetrized billiard against the line of symmetry. It is easy to see that orbits with even n γ correspond to the periodic orbits of the full billiard with the same period; the latter are non-symmetric unless γ is self-retracing. On the other hand orbits with odd n γ are associated with symmetric orbits of the full billiard whose period is doubled compared with γ. It is physically reasonable to assume (although we don't know whether the result is proven) that the number of orbits γ with periods in a certain interval [T, T + δ] with T large may not depend on parity of n γ , as well as on parity of the number of visits to any side of the desymmetrized billiard.
The semiclassical cross form factor now becomes a double sum over the same set of periodic orbits of the desymmetrized system. In the following expression summation over γ 1 and γ 2 is connected with the Gutzwiller expansion of the even and odd spectral densities, respectively, the factor (−1) nγ 2 emerges because each visit to the symmetry line with the Dirichlet boundary condition yields the Maslov phase π.

Diagonal approximation
Dropping in (5) all summands with γ 1 not coinciding with γ 2 (up to time reversal since we have time reversal invariance) we come to the diagonal approximation, It differs from Berry's diagonal approximation for autocorrelation form factor [16] by the presence of the sign factor (−1) γ leading to the destructive interference among the contributions. Assuming that for the long orbits with very large n γ the number of orbits with even and odd n γ is the same and not correlated with the stability coefficients we obtain K gu,diag (τ ) = 0. A warning has to be made here. Our basic argument (even and odd spectral densities are created by the same set of orbits which contribute with the same action but a different Maslov index) implicitly assumes ergodicity of the classical motion and loses its validity if ergodicity is violated. Consider, e.g., a particle moving in a symmetric double-well potential in two dimensions. For energies below the dividing barrier the classical motion is clearly non-ergodic since the available phase space disconnects into two symmetric domains; even and odd levels in the quantum problem are then almost degenerate, up to the tunnelling splitting. Consequently the cross g ←→ u form factor will practically coincide with the autocorrelation form factor of the spectrum in a single well.
Another trivial issue should also be pointed out here: for very short times (ballistic times) corresponding to the period of the shortest classical orbits, there are no more cancellations between different orbits and correlations certainly arise.

Off-diagonal contributions
The off-diagonal contribution for the auto form factor differing from (5) by the absence of the sign factor (−1) nγ is known to stem from pairs of the so called orbit-partners. A partner γ ′ of an orbit γ is the orbit which practically coincides with γ everywhere but in the "encounters", i.e., a set of two or more orbit stretches almost parallel and abnormally close to each other. The lengths of γ and γ ′ can be so close that their action difference can be comparable to or smaller than . Consequently the corresponding contributions to the auto form factor will avoid the destructive interference which, in the case of randomly composed pairs, would lead to annihilation of the contribution. The leading off-diagonal correction is due to the so called Sieber-Richter pairs [17]; higher order terms giving the auto form factor as a series in powers of τ are calculated in [18]. Analytically continuing the result one gets the auto form factor for times smaller than the Heisenberg time, i.e., for τ < 1; for larger times a different approach is needed (see the following subsection).
Let us now address the cross form factor. Again, contributions of randomly composed pairs are expected to average to zero. Limiting summation only to pairs of the orbit-partners we may reduce (5) to The inner sum is taken over all partners γ ′ of the orbit γ including the diagonal ones.
Since the partner orbit coincides with the original orbit everywhere except the relatively short encounter stretches, the Maslov indices of γ and γ ′ , coincide, apart from the contribution of reflections from the Dirichlet boundary. Again, we assume that there is no correlation between parity of the number of strikes of the orbit γ against the cut and the value of the sum over partners of γ multiplied by its stability factor. Hence the expected value of K gu (τ ), with the off-diagonal contributions taken into account, is zero, at least for τ < 1.

Times larger than the Heisenberg time
The fact that maxima (and in the case of GSE, even singularities) of form factors can occur at the Heisenberg time, makes it necessary to investigate the long-time behaviour in the discrete symmetry case. Semiclassical investigation of spectral auto and cross correlation beyond the Heisenberg time was started by Bogomolny and Keating [19]. Complete expansion of the autocorrelation functions using the formalism of the 4-determinantal generating functions is given in [20,21]; the results of the latter can be generalized to cross-correlations. The appropriate generating function of the two-point cross-correlation functions for the spectra of Hamiltonians H (1) , H (2) is a ratio of four spectral determinants, where . . . E as usual indicates averaging over the center energy E and smoothing over small windows of the energy offsets from E; unlike the generating function in the problem of autocorrelation it is not symmetric with respect to the exchange of ε C and ε D . Using the identity and proportionality between ImTr (E − H) −1 and the level density ρ (E) we obtain the cross-correlation function as the second derivative, Here η is a small positive number, and the symbol " " denotes the substitution of the arguments, The semiclassical approximation for the generating function follows from that for the spectral determinants. Using the so called "Riemann-Siegel lookalike" asymptotics of det (E − H) introduced by Berry and Keating [22,23,24] one can show that the semiclassical estimate of Z (12) consists of two terms. One of these substituted into (11) leads to a correlation function whose Fourier transformation reproduces the form factor for small times. The second term in Z (12) is responsible for the behavior of the form factor for times larger than T H . Each term in the semiclassical generating function is obtained as an infinite product over the periodic orbits, averaged over the central energy E. The leading order contribution is obtained by the diagonal approximation which, in the formalism of the generating functions, means that the factors associated with different periodic orbits are assumed independent (with the obvious exception of the mutually time-reversed orbits); consequently averages of products over the periodic orbits are replaced by the products of averages.
The diagonal approximation for the generating function describing the gu−correlation is deduced in the same way, step by step, as in the case of autocorrelation; for the lack of space we address the readers to the papers [20,21]. At a certain stage one gets, The factor g in the exponents is equal to 2 for systems with time reversal allowed (like the cardioid billiard) where it reflects the existence of pairs of mutually time reversed orbits; in the absence of time reversal g = 1. The exponents are given by We denote here by τ γ = T γ /T H the period of a periodic orbit γ in terms of the Heisenberg time; F γ is its stability coefficient. As before n γ is the number of visits of the orbit γ to the cut in the desymmetrized billiard; if the factors (−1) nγ were dropped we would return to the case of autocorrelation.
It remains to perform summation over the periodic orbits in the exponents of Z I , Z II . Again, we assume absence of correlation between parity of n γ , on the one hand, and the orbit period and the stability coefficient, on the other hand. Then the summands containing the factor (−1) nγ disappear after summation over γ such that γ ϑ I,γ → 0, consequently the term Z I in the generating function is an irrelevant constant. The sum in the second line of (11) can be evaluated using the sum rule of Hannay and Ozorio di Almeida [25] for the stability coefficients F γ , with γ denoting Euler constant. The lower integration limit is τ 0 = T 0 /T H with T 0 standing for some minimal period above which periodic orbits behave approximately ergodically; in the semiclassical limit τ 0 tends to zero. The second term in the generating function is thus, The respective two-point correlation function is zero in the presence of time reversal when g = 2. Without time reversal (g = 1) the term Z II contributes like const τ 2 0 e i2ε to the cross-correlation function or, equivalently, like a delta-peak to the time dependent cross form factor K gu (τ ) at τ = 1, i.e., at the Heisenberg time. However, since the term is proportional to τ 2 0 this contribution vanishes in the semiclassical limit. Therefore, as could be expected, correlations between the even and odd part of the spectrum do not exist for times either smaller or larger than the Heisenberg time.

General discrete symmetries
In this section we give a formal treatment of the problem in the presence of more complicated discrete symmetry groups G. The level density ρ (E) falls into a sum of subdensities ρ m (E) connected with different irreducible representations (IR) m of G. The semiclassical approximation for the fluctuating part of ρ m (E) is provided by the symmetry-adapted Gutzwiller trace formula [10]. In the following, we shall follow the ideas expounded in [10], so that our treatment includes the case in which the reduction of the configuration space M of the system to a fundamental domainM , which tesselates the full M when the symmetry operations are applied to it, is difficult or impossible.
Let us look at the cross form factor defined as in (2) K (m 1 ,m 2 ) (τ ) = k,l e i(E k,m 1 −E l,m 2 )t/ ; (12) throughout this section the argument of the form factor τ = t/T H , i. e., time measured in units of the Heisenberg time. What we want to show is that this quantity vanishes. At the abstract level this can be rewritten as where P m is the projector of the Hilbert space on the IR m. It is given by the expression where χ m (g) is the character corresponding to the IR m and g denotes the group element, which is assumed to act directly on the elements of the Hilbert space. As is well known, traces can be estimated semiclassically as integrals over coherent states |p, q . Assuming that the symmetry operations g have a well defined semiclassical limit, we know how they act on the phase space as well as on the Hilbert space, and thus are able to define such quantities as g(p, q). One therefore obtains where we have used the identity Tr(P AP ) = Tr(AP ) where P is an arbitrary projector.
Since the |p, q are well-localized in phase space, the kind of expressions occurring in (15) only differs from zero when the phase space point (p, q) is (at least approximately) carried over to g(p, q) in time t by the classical evolution. In any case, this reasoning is legitimate when t is less than the Ehrenfest time. Since, in this section, we shall limit ourselves to the diagonal approximation, which is only valid when times are not too large, this is a reasonable approach. Now we must account correctly for all the various contributions to this integral. To this end, let us start by looking at the quantity Tr P m 1 e −iHt/ P m 1 . Let us define as a generalized periodic orbit of period t any segment which connects a phase space point (p, q) with one of its symmetric partners g(p, q) over time t. We call such orbits g-orbits, making explicit reference to the corresponding group element. Let λ be a g-orbit. One then additionally defines g λ to be the minimal element of the symmetry group G such that g λ (p, q) lies on the orbit λ going through (p, q) and such that no other element g ′ of G has the property that g ′ (p, q) lies on the orbit λ between (p, q) and g λ (p, q). Note that in general g λ = g. There is, however, as shown in [10], always an integer n such that g = g n λ . Let (p, q) lie on a g-orbit λ. One then has p, q| e −iHt/ |g(p, q) = A λ e iS λ / (16) where A λ is related to the stability of the orbit in the usual way and the phase e iS λ / contains all the information on the action of the g-orbit as well as its Maslov phases. Integrating the l.h.s. of (16) over p, q leads to where n is the number of iterations of the primitive element leading to g. The factor t arises from the fact that an integration over the whole segment from (p, q) to g(p, q) is possible and yields the same result as starting with (p, q). The sum still runs independently over the various symmetric "copies" of the g-orbit λ, in particular over its iterations which lead eventually to a bona fide periodic orbit. Neglecting repetitions, that is n = 1, yields Let us now multiply this with the complex conjugate of the corresponding expression for m 1 . This gives: One now introduces the diagonal approximation, which involves keeping in this sum only such terms as have a phase equal to one, that is, only to sum over these pairs of λ and µ such that S λ = S µ . Since we consider the different iterations of a g orbit to be different, and since we have already discarded orbits which are non-primitive, and since, in the same vein, we discard such orbits as have a non-trivial symmetry (that is, we assume that the only h ∈ G such that h(p, q) = (p, q) is the identity), it is clear that the total number of such pairs is |G| 2 independently of λ and µ: there are |G| degenerate λ's and the same number of degenerate µ's. The final result is hence where summation is carried out now over one representative g-orbit λ among the |G| degenerate ones. In other terms, the sum corresponds to a desymmetrized problem. Keeping all g-orbits would require a compensating factor of |G| −1 in front of the r.h.s. of (20). One now rewrites this as and we now make a final assumption altogether similar to those that we have been making throughout in this paper: namely, we assume that when the g-orbits are sufficiently long, they are equidistributed on all elements of the group G. This leads to stating that the sum over λ is in fact independent of g. This is, of course, sufficient to guarantee that K (m 1 ,m 2 ) (τ ) = 0 whenever m 1 = m 2 .

Continuous symmetry
In the presence of continuous symmetry, the problem has one or several integrals of motion due to Noether's theorem. We focus on one of them which can be chosen as an action and whose eigenvalues in the semiclassical limit are given by m, where m is integer. Our question is whether correlation may exist between the energy spectra of two operators, H m and H m+1 . Here we denote by H m the Hamiltonian H reduced to the appropriate symmetry subspace. In the following, we limit ourselves to the case in which the dynamics of these Hamiltonians are all overwhelmingly chaotic in the energy range of interest.
A simple example which we shall use to illustrate our argument is a system with axial symmetry in a 3d space. Introducing cylindric coordinates we can separate the angular part of the eigenfunctions writing Ψ = e imφ ψ (ρ, z) where ψ (ρ, z) is an eigenfunction of the Hamiltonian H m (ρ, z) in the subspace of states with the given zcomponent of the angular momentum L z = m. As stated above, we assume that the 2d classical motion described by this Hamiltonian is chaotic. A moment's reflection will show that this example contains all the relevant features of the more general situation described at the beginning of this section.
In this case, the Hamiltonians H m and H m+1 correspond to the radial Hamiltonians with magnetic quantum number m and m + 1 respectively. The magnetic quantum number enters these Hamiltonians as a parameter, and the well-known theory of parametric correlations can be applied. Special treatment is necessary when m = 0 or m ∼ 1. The difference between the two Hamiltonians due to the centrifugal term W = L 2 z /2Mρ 2 (M is the mass of the particle) is then of the formal order 2 . However the energy shifts associated with the change of m by 1 are of the order ; this discrepancy is due to singularity of the centrifugal term at ρ = 0. The appropriate Gutzwiller expansions in this case use the set of classical periodic orbits calculated for L z = 0, i.e., m-independent. The m-dependence of the spectra is taken into account via an appropriate phase added to the Maslov phase; the phase increment occurs each time the orbit hits the axis ρ = 0, see the reviews [26,27]. Since this additional phase is pseudorandom, the cross-correlation functions between the spectra with different m ∼ 1 vanish due to the same mechanism as between the spectra of different parity in the discrete case.
Let us now turn to the case of |m| ≫ 1; more precisely consider the semiclassical limit → 0, m → ∞ such that the angular momentum projection L z = m remains constant. In this case the centrifugal barrier W would not let the particle approach the z -axis; the singularity at ρ = 0 will thus be deep in the classical shadow. The change of m by 1 brings about change of the Hamiltonian of the order , namely by ∆H m = ∆W = ∂W/∂L z . Before comparing the eigenvalues at different m one must subtract the overall drift of the spectrum upwards caused by the increase of the centrifugal energy W . Note that this issue is similar to the one caused by the short periodic orbits in the discrete case: it arises from a difference in the one-particle densities of the two systems, and is an issue which can also be settled by means of an appropriate unfolding procedure. This quantity can be evaluated semiclassically as the derivative of its average over the energy shell, where dΓ is the element of the phase space. Denoting by S 0 a typical classical action within the given (highly excited) energy range, we can introduce a dimensionless small parameter ξ = /S 0 and write To appreciate the impact of the perturbation associated with H m → H m+1 consider the general problem of parametric correlation in the family of Hamiltonians H ξ = H 0 +ξH ′ where ξ is small and dimensionless; the Hamiltonians H 0 , H ′ are assumed to have finite classical counterparts. Let us consider the cross form factor between the spectra of H m and H m+1 in the semiclassical domain. We shall assume that ξ is so small that the periodic orbits of H m , H m+1 practically coincide; only their action difference has to be taken into account since it is referred to .
The additional classical action due to the perturbation H ′ over a stretch of the orbit with duration from t 1 to t 2 would be ∆S ∼ ξ Now the well-established approach is to model the accumulation of ∆S on a trajectory of a chaotic system by a Gaussian stochastic process [28]; below . . . denotes averaging over such a process . This can be understood if we mentally divide the periodic orbit into pieces of duration larger than the Lyapunov time T L = λ −1 where λ is the Lyapunov constant; the action differences accumulated at two such stretches would be uncorrelated. Let t = t 2 − t 1 ≫ T L and assume that there is no systematic action increment, ∆S = 0. The averaged second moment of ∆S will grow proportional to t , where a is a purely classical, system-specific, parameter; its dimensionality is squared action over time. Remembering that in a Gaussian process exp (iΦ) = exp (− Φ 2 /2) we obtain the average of the exponential of the phase difference accumulated on a periodic orbit with the period T γ , e i ∆Sγ = e −a ξ 2 2 2 Tγ . Now let us use the semiclassical expression (4) for the cross-correlation form factor which we shall now denote K ξ (τ ). Limiting ourselves to the diagonal approximation we would get, for the time T = τ T H , = e −a ξ 2 The quantity K diag (τ ) is the autocorrelation form factor in the diagonal approximation, i.e., 2τ or τ depending on the presence or absence of the time reversal. The perturbation ξH ′ leads thus to the decay of the form factor with the characteristic time T decay = 2 2 /aξ 2 .
Applying this result to the correlation between the spectra belonging to two neighboring values of the quantum number m when ξ = /S 0 we obtain the decay time This time is classical, i.e., finite in the semiclassical limit, however it could be much larger than the ballistic time scale, i.e., the period of the shortest periodic orbits (time on which correlations are expected anyway).
The corresponding dimensionless τ decay = T decay /T H turns into zero in in the limit → 0. We therefore find that the cross-correlation between spectra corresponding to different symmetries vanishes in the universal regime. Considering that the periodic orbit expansions of the type (4)-and above all the averaging procedure over large numbers of periodic orbits having nearly the same action-are justified only for times exceeding the Ehrenfest time T E ∼ λ −1 ln S 0 / , the vanishing of the cross-correlation only holds for times larger than T E . For times shorter than T E , and hence in particular for times of the order of classical times, correlations are expected to arise, based on the fact that both symmetry sectors have the same periodic orbits. Such correlations will depend on the specific features of the periodic orbits and will thus be highly systemdependent. A possible connection with a recent discussion of the transition between short time effects and the universal regime [29] is left for future investigation.

Conclusions
In this paper we have given a justification to the common assumption that spectra corresponding to different irreducible representations of a symmetry group of a given quantum chaotic problem are independent in the semi-classical limit. More precisely we have shown in the semi-classical limit that the two-point cross correlation vanishes, a property also sometimes called weak independence. For the specific case of a mirror symmetry we showed that the cross correlation function is zero not only in the diagonal approximation and for short times. The non diagonal terms of the trace formula were also evaluated and the semi-classical form of the spectral determinant was used to show that our result holds also beyond the Heisenberg time. For arbitrary discrete symmetries and for continuous symmetries we limited our argument to the diagonal approximation, but the arguments given for mirror symmetries carry over.
An potentially interesting line for future work appears in continuous symmetries, where the trace formulae reveal an unusual correlation on a classical time scale. The approximation used is clearly not valid on this time scale, but the result might still be a hint, that such correlations exist and are different from the obvious ones resulting from short orbits.