{'fraction_non_alphanumeric': 0.032423700455778, 'fraction_numerical': 0.010730975595684533, 'mean_word_length': 4.506353240152478, 'pattern_counts': {'":': 0, '<': 4, '<?xml version=': 0, '>': 5, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 2, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Different Facets of Chaos in Quantum Mechanics

Nowadays there is no universally accepted definition of quantum chaos. In this paper we review and critically discuss different approaches to the subject, such as Quantum Chaology and the Random Matrix Theory. Then we analyze the problem of dynamical chaos and the time scales associated with chaos suppression in quantum mechanics. Summary: 1. Introduction 2. Quantum Chaology and Spectral Statistics 3. From Poisson to GOE Transition: Comparison with Experimental Data 3.1 Atomic Nuclei 3.2 The Hydrogen Atom in the Strong Magnetic Field 4. Quantum Chaos and Field Theory 5. Alternative Approaches to Quantum Chaos 6. Dynamical Quantum Chaos and Time Scales 6.1 Mean-Field Approximation and Dynamical Chaos 7. Conclusions


Introduction
The aim of this paper is to review and discuss various definitions and approaches to quantum chaos. 1−5 As yet, there is no universally accepted definition of quantum chaos. On the contrary, the meaning of classical chaos is beyond question. In classical mechanics a trajectory z(t) in the phase space Ω is chaotic if its maximal Lyapunov exponent λ is positive. The Lyapunov exponent is defined as where ω(t) is a tangent vector to z(t) with the condition that |ω(0)| = 1. The exponential instability of chaotic trajectories implies a continuous frequency spectrum of motion. The continuous spectrum, in turn, implies correlation decay; this property, which is called mixing in ergodic theory, is the most important property of dynamical motion for the validity of the statistical description. 6−8 The problem of quantum chaos arose because the above mentioned condition of continuous spectrum for classical chaos is violated in quantum mechanics. Indeed the energy and the frequency spectrum of any quantum motion, bounded in phase space, are always discrete due to the non-commutative geometry (discreteness) of the phase space. According to the theory of dynamical systems, such motion corresponds to the limiting case of regular motion. It means that there is no classical-like chaos at all in quantum mechanics. 1−5 Nevertheless, we shall show that it is reasonable and useful to apply the word chaos also in quantum mechanics. In the first part of this article some definitions of quantum chaos for stationary systems are discussed, while in the second part, the time evolution of classical and quantum systems is compared.

Quantum Chaology and Spectral Statistics
As is well known, in the study of the transition from order to chaos in classical systems a useful tool is the examination of the phase space properties, such as the Poincarè sections. 6 Such plots are not directly available in the case of quantum systems. In many papers, the Berry definition of quantum chaos is adopted: "Quantum Chaology is the study of semiclassical, but not classical, behaviour characteristic of systems whose classical motion exhibits chaos". 8 The idea, also suggested by authors like Percival and Gutzwiller, 2 is to connect the behaviour of the eigenvalues and eigenfunctions of a quantum system to the different structure of the phase space of the corresponding classical system in the regular and chaotic region.
In the context of quantum chaology, the spectral statistics of the energy levels are of great importance. Mehta defined: "A spectral statistic is a quantity which can be calculated from an observed sequence of levels alone, without other information and whose average value and variance are known from the theoretical model. A suitable statistic is one which is sensitive for the properties to be compared or distinguished and is insensitive for other details". 9 In particular, it has been found that the spectral statistics of sys- The most used spectral statistics of the energy levels are P (s) and ∆ 3 (L).
P (s) is the distribution of nearest-neighbour spacings s i = (Ẽ i+1 −Ẽ i ) of the unfolded levelsẼ i . It is obtained by accumulating the number of spacings that lie within the bin (s, s + ∆s) and then normalizing P (s) to unit. As shown by Berry, 10,11 for quantum systems whose classical analogs are integrable, P (s) is expected to follow the Poisson distribution On the other hand, quantal analogs of chaotic systems exhibit the spectral properties of GOE with which is the so-called Wigner distribution. Note that for systems without time-reversal symmetry the GUE predicts P (s) = (32/π 2 )s 2 exp (−4s 2 /π).
The statistic ∆ 3 (L) is defined, for a fixed interval (−L/2, L/2), as the least-square deviation of the staircase function N(E) from the best straight line fitting it: where N(E) is the number of levels between E and zero for positive energy, between −E and zero for negative energy. The ∆ 3 (L) statistic provides a measure of the degree of rigidity of the spectrum: for a given interval L, the smaller ∆ 3 (L) is, the stronger is the rigidity, signifying the long-range correlations between levels. For this statistic the Poissonian prediction is The GOE predicts the same behaviour for L << 1; instead for L >> 1 it gives ∆ 3 (L) = 1 π 2 log L .
In the GUE case one has ∆ 3 (L) = (1/2π 2 ) log L. It is useful to remember that Berry 12 has shown that ∆ 3 (L) deviates from the universal predictions of RMT for very large L.
Another probe, which is generally regarded (see for instance Ref. 14) as very sensitive to the structure of chaotic states, is the transition probability.
For reasons of space, we mention only the pioneering work of French and his coworkers 13 and two more recent works concerning the interacting-boson model 14 (IBM) and the three-level Lipkin, Meshkov, Glick (LMG) model. 15 In both works the results based on the transition probabilities between eigenstates of the system completely agree with the spectral statistics P (s) and It is important to stress that even though the classical system is not known, to distinguish between ordered and chaotic states, the spectral statistics and the transition probabilities can be used. 16 In this section we shall compare the transition Poisson-GOE with the experimental data of two different systems: the atomic nuclei and the Hydrogen atom in a static magnetic field.

Atomic Nuclei
In atomic nuclei, as in other many-body systems, ordered and chaotic states generally coexist. 17

a) The Low Energy Region
The behaviour of spectral statistics near the ground state has been studied by Garret, German, Courtney and Espino 17 and Shriner, Mitchell and Von Egidy 17 . The main results of these authors have been shown in Figure 1 and

b) The High Energy Region
Neutron resonance spectroscopy on a heavy even-even nucleus typically leads to the identification of about 150 to 170 s-wave resonances with J π = 1 2 + located 8-10 MeV above the ground state of the compound system, with average spacings around 10 eV and average total widths around 1 eV. Proton resonance spectroscopy yields somewhat shorter sequences of levels with fixed spin and parity, with typically 60 to 80 members.
For the statistical analysis, it is essential that the sequences be pure (no admixture of levels with different spin or parity) and complete (no missing levels). Only such sequences were considered by Haq, Pandey and Bohigas. 30 Scaling each sequence to the same average level spacing and lumping together all sequences one leads to the "Nuclear Data Ensemble" (NDE), which contains 1726 level spacings.
As shown in Figure 3, the agreement between the experimental data and the GOE predictions is surprisingly good (in the GOE model there are no free parameters).

The Hydrogen Atom in the Strong Magnetic Field
We now discuss the local statistical properties of energy levels of a Hydrogen atom in a uniform strong magnetic field. The Hamiltonian of the system is given by where the magnetic field B breaks the time-reversal symmetry. Although Eq.
where γ = B/B c is the magnetic field in atomic units and B c = 2.35 · 10 5 Tesla. This equation can be numerically solved for different values of the scaled energy ǫ = E/(2γ) 2/3 . Once the eigenvalues have been obtained, the spectral statistics P (s) and ∆ 3 (L) can be calculated. Figure 4 shows the function P (s) for different values of the scaled energy ǫ. Increasing ǫ, a smooth Poisson-GOE transition can be observed. Figure 5 shows the spectral rigidity ∆ 3 (L) in three different energy intervals. For this statistic the transition Poisson-GOE is also very clear.
In addition, a comparison has been made between the theoretical energy levels and the experimental ones; the agreement is excellent.

Quantum Chaos and Field Theory
In the last few years there has been much interest in chaos in field theories.
It is now well known that the spatially uniform limits of scalar electrodynamics and Yang-Mills theory exhibit classical chaotic motion. 34 In this section we discuss quantum chaos in a field-theory schematic model, namely the spatially homogeneous SU(2) Yang-Mills-Higgs (YMH) system. 27−29 The La-grangian density of the SU(2) YMH system is given by where with (2) algebra, and where the potential of the scalar field (the Higgs field) is In the (2+1)-dimensional Minkowski space (µ = 0, 1, 2) and with spatially homogeneous Yang-Mills and the Higgs fields In the gauge A a 0 = 0 and using the real triplet representation for the Higgs field one obtains . When µ 2 > 0, the potential V has a minimum at | φ| = 0, but for µ 2 < 0 the minimum is at which is the non zero Higgs vacuum. This vacuum is degenerate, and after spontaneous symmetry breaking the physical vacuum can be chosen φ 0 = (0, 0, v). If A 1 1 = q 1 , A 2 2 = q 2 and the other components of the Yang-Mills fields are zero, in the Higgs vacuum the Hamiltonian of the system reads where p 1 =q 1 and p 2 =q 2 . Here w 2 = 2g 2 v 2 is the mass term of the Yang-Mills fields. This YMH Hamiltonian is a toy model for classical non-linear dynamics, with the attractive feature that the model emerges from particle physics. At low energy the motion near the minimum of the potential where the Gaussian curvature is positive, is periodic or quasiperiodic and is separated from the instability region by a line of zero curvature; if the energy is increased, the system will be for some initial conditions in a region of negative curvature, where the motion is chaotic. According to this scenario, the energy E c of chaos-order transition is equal to the minimum value of the line of zero Gaussian curvature K(q 1 , q 2 ) on the potential-energy surface. It is easy to show that the minimal energy on the zero-curvature line is given by: and by inverting this equation one obtains v c = (E/6g 2 ) 1/4 . There is an order-chaos transition by increasing the energy E of the system and a chaosorder transition by increasing the value v of the Higgs field in the vacuum.
Thus, there is only one transition regulated by the sole parameter E/(g 2 v 4 ).
It is important to point out that in general the curvature criterion guarantees only a local instability 16) and should therefore be combined with the Poincarè sections 17) . Chaotic regions on the surface of the section are characterized by a set of randomly distributed points, and regular regions by dotted or solid curves. Figure 6 shows the Poincarè sections, which confirm the analytical predictions of the curvature criterion: the critical value of the onset of chaos is in very good agreement with the Poincarè sections.
This function interpolates between the Poisson distribution (ω = 0) of integrable systems and the Wigner distribution (ω = 1) of chaotic ones, and thus the parameter ω can be used as a simple quantitative measure of the degree of chaoticity. By using the P(s) distribution and the Brody function it is possible to give a quantitative measure of the degree of quantal chaoticity of the system. The numerical calculations of Figure 6 and 7 clearly show the quantum chaos-order transition and its connection to the classical one.

Alternative Approaches to Quantum Chaos
A different approach to quantum chaos has been discussed by Sakata and his coworkers. 35 The example discussed by Sakata is a system of an even 2) Quantum KAM states: If |i > is described perturbatively starting from the |µ, ν >-basis state, then it is a quantum KAM state, see Figure 8 In our opinion these two approaches are quite similar and the Sakata approach also gives a simpler picture of the three cases discussed (regular, KAM, chaotic).

Dynamical Quantum Chaos and Time Scales
In this section we analyze in more detail the previously discussed problems of the chaotic time evolution for a quantum system. The value of the break time t B depends on the model studied and its exact behaviour is still controversial, but can be estimated from the Heisenberg indetermination principle as where ∆E is the mean spacing of energy levels. The discrete spectrum of the Liouville operator cannot be resolved if t < t B , i.e. t B is the time at which the quantal evolution (of a wave packet, for example) "realizes" that the spectrum of the evolution operator is discrete. According to the Thomas-Fermi rule, ∆E ∝h N , where N is the number of degrees of freedom, i.e. the dimension of the configuration space. So, ash → 0, the break time diverges as t B ∼h 1−N , and it does so faster, the higher N is.
In the case of classical chaos, another time scale, the random time scale t R , much shorter than t B , can be introduced to estimate the time at which classical exponential spreading reaches the quantal resolution of the phase space. Thus, it is the full time for the exponential spreading of the minimum initial wave packet. As shown by Berman and Zaslavski 41 the random time (called the breaking time by the two authors) follows the logarithmic law where S is a classical action and λ is the Lyapunov exponent of the system. In particular, Berman and Zaslavski have studied two different models, which allow one to calculate the random time scale t R . The first model is a periodically kicked oscillator, which gives t R = ln(cost/h). The second is an ensemble of N atoms interacting with light in the resonant cavity. For this model t R = ln (cost N).

Mean-Field Approximation and Dynamical Chaos
Let us consider a N-body quantum system with HamiltonianĤ. The exact time-dependent Schrödinger equation can be obtained by imposing the quantum last action principle on the Dirac action where ψ is the many-body wavefunction of the system. 42 Looking for stationary points of S with respect to variation of the conjugate wavefunction ψ * gives ih ∂ ∂t ψ =Ĥψ .
As is well known, it is usually impossible to obtain the exact solution of the many-body Schrödinger equation and some approximation must be used.
In the mean-field approximation the total wavefunction is assumed to be composed of independent particles, i.e. it can be written as a product of single-particle wavefunctions φ j . In the case of identical fermions, ψ must be antisymmetrized. By looking for stationary action with respect to variation of a particular single-particle conjugate wavefunction φ * j one finds a timedependent Hartree-Fock equation for each φ j : whereĥ is a one-body operator. 42 The main point is that, in general, the onebody operatorĥ is nonlinear. Thus the Hartree-Fock equations are non-linear (integro-)differential equations. These equations can give rise, in some cases, to chaotic behaviour (dynamical chaos) of the mean-field wavefunction.
In the mean-field approximation the mathematical origin of dynamical chaos resides in the nonlinearity of the Hartree-Fock equations. These equations provide an approximate description, the best independent-particle description, which describes, for a certain time interval, the very complicated evolution of the true many-body system. Two questions then arise: 1) Does this chaotic behavior persist in time?
2) What is the best physical situation to observe this kind of nonlinearity?
To answer the first question, it should be stressed that, as shown pre- Concerning the second question, it is useful to remember that, in the thermodynamic limit, i.e. when the number N of particles tends to infinity at constant density, the energy spectrum of the system is, in general, continuous and true chaotic phenomena are not excluded. 43 When the mean-field theory is a good approximation of the exact manybody problem, one can use the nonlinear mean-field equations to estimate the transient chaotic behaviour of the many-body system. An important case where such an approach could be applied is the Bose-Einstein condensate of weakly-interacting alkali-metal atoms. 44 In particular, we suggest that the collective oscillations of the Bose condensate 45,46 can give rise to dynamical chaos.

Conclusions
In this paper we have reviewed various definitions of quantum chaos. In our opinion the simplest and clearest approach is that of Quantum Chaology, increases with the number of microscopic degrees of freedom explains the chaotic behaviour of macroscopic systems, without invoking a role for the observer or the environment.
The study of quantum chaos for many-body or continuum systems (field theory) is a very promising field of research which can also help to better understand the foundations of quantum theory and statistical mechanics. * * * One of us (L.S.) is greatly indebted to Prof. M. Robnik for many suggestions.