**1. Introduction**

In classical mechanics, a Hamiltonian system with *N* degrees of freedom is defined to be integrable if a set of *N* constants of the motion *Ui* which are in involution exist, so their Poisson bracket satisfies *Ui*, *U <sup>j</sup>* <sup>¼</sup> 0, *<sup>i</sup>*, *<sup>j</sup>* <sup>¼</sup> 1, … , *<sup>N</sup>*. For an integrable system, the motion is confined to an invariant two-dimensional torus in 2*N*-dimensional phase space. If the system is perturbed by a small nonintegrable term, the KAM theorem states that its motion may still be confined to the *N*-torus but deformed in some way [1–3]. The first computer simulation of nonequilibrium dynamics for a finite classical system was carried out by Fermi and his group. They considered a one-dimensional classical chain of anharmonic oscillators and found it did not equilibrate.

Classically, chaotic motion is longtime local exponential divergence with global confinement, a form of instability. Confinement with any kind of divergence is produced by repeatedly folding, a type of mixing that can only be analyzed by using probability theory. The motion of a Hamiltonian system is usually neither completely regular nor properly described by statistical mechanics, but shows both regular and chaotic motion for different sets of initial conditions. There exists generally a transition between the two types of motion as initial conditions are changed which may exhibit complicated behavior. As entropy or the phase space area quantifies the amount of decoherence, the rate of change of the phase space area quantifies the decoherence rate. In other words, the decoherence rate is the rate at which the phase space area changes.

It is important to extend the study of chaos into the quantum domain to better understand concepts such as equilibration and decoherence. Both integrable as well as nonintegrable finite quantum systems can equilibrate [4, 5]. Integrability does not seem to play a crucial role in the structure of the quasi-stationary state. This is in spite of the fact that integrable and nonintegrable quantum systems display different level-spacing statistics and react differently to external perturbation. Although integrable systems can equilibrate, the main difference from nonintegrable systems may be longer equilibration times. This kind of behavior is contrary to integrable classical finite systems that do not equilibrate at all. Nonintegrable classical systems can equilibrate provided they are chaotic.

The properties of a quantum system are governed by its Hamiltonian spectrum. Its form should be important for equilibration of a quantum system. The equilibration of a classical system depends on whether the system is integrable or not. Integrable classical systems do not tend to equilibrate, they have to be nonintegrable. Quantum integrability in *n* dimensions may be defined in an analogous way requiring the existence of *n* mutually commuting operators, but there is no corresponding theorem like the Liouville theorem. An integrable system in quantum mechanics is one in which the spectral problem can be solved exactly, and such systems are few in number [6, 7].

In closed classical systems, equilibration is usually accompanied by the appearance of chaos. Defining quantum chaos is somewhat of an active area of study now. The correspondence principle might suggest we conjecture quantum chaos exists provided the corresponding classical system is chaotic and the latter requires the system to be nonintegrable. Classical chaos does not necessarily imply quantum chaos, which seems to be more related to the properties of the energy spectrum.

It was proposed that the spectrum of integrable and nonintegrable quantum systems ought to be qualitatively different. This would be seen in the qualitative difference of the density of states. At a deeper level, one may suspect that changes in the energy spectrum as a whole may be connected to the breaking of some symmetry or dynamical symmetry. This is the direction taken here [8–10].

It is the objective to see how algebraic and geometric approaches to quantization can be used to give a precise definition of quantum degrees of freedom and quantum phase space. Thus a criterion can be formulated that permits the integrability of a given system to be defined in a mathematical way. It will appear that if the quantum system possesses dynamical symmetry, it is integrable. This suggests that dynamical symmetry breaking should be linked to nonintegrability and chaotic dynamics at the quantum level [11–13].

Algebraic methods first appeared in the context of the new matrix mechanics in 1925. The importance of the concept of angular momentum in quantum mechanics was soon appreciated and worked out by Wigner, Weyl and Racah [14–16]. The close relationship of the angular momentum and the *SO*ð Þ3 algebra goes back to the prequantum era. The realization that *SO*ð Þ 4 is the symmetry group of the Kepler problem was first demonstrated by Fock. A summary of the investigation is as follows. To familiarize those who are not familiar with algebraic methods in solving quantum problems, an introduction to the algebraic solution of the hydrogen atom is presented as opposed to the Schrödinger picture. This approach provides a platform for which a definition of quantum integrability of quantum systems can be established. Thus, at least one approach is possible in which a definition of concepts such as quantum phase space, degrees of freedom as well as how an idea of quantum integrability and so forth can be formulated [17–21]. After these issues are addressed, a number of quantum models will be discussed in detail to show how the formalism is to be used [22–24].

*Classical and Quantum Integrability: A Formulation That Admits Quantum Chaos DOI: http://dx.doi.org/10.5772/intechopen.94491*

## **1.1 The hydrogen atom**

The hydrogen atom is a unique system. In this system, almost every quantity of physical interest can be computed analytically as it is a completely degenerate system. The classical trajectories are closed and the quantum energy levels only depend on the principle quantum number. This is a direct consequence of the symmetry properties of the Coulomb interaction. Moreover, the properties of the hydrogen atom in an external field can be understood using these symmetry properties. They allow a parallel treatment in the classical and quantum formalisms.

The Hamiltonian of the hydrogen atom in atomic units is

$$H\_0 = \frac{\mathbf{p}^2}{2} - \frac{1}{r}.\tag{1}$$

The corresponding quantum operator is found by replacement of **p** by �*i*∇. Due to the spherical symmetry of the system, the angular momentum components are constants of the motion,

$$\mathbf{L} = \mathbf{r} \times \mathbf{p}, \qquad [H\_0, \mathbf{L}] = \mathbf{0}. \tag{2}$$

So *H*0, **L**<sup>2</sup> , *Lz* is a complete set of commuting operators classically, so three quantities in mutual involution, which implies integrability of the system.

The Coulomb interaction has another constant of the motion associated with the Runge-Lenz vector **R**. This has the symmetrized quantum definition

$$\mathbf{R} = \frac{1}{2} (\mathbf{p} \times \mathbf{L} - \mathbf{L} \times \mathbf{p}) - \frac{\mathbf{r}}{r}, \qquad [H\_0, \mathbf{R}] = \mathbf{0}. \tag{3}$$

If the **R** direction is chosen as the reference axis of a polar coordinate system in the plane perpendicular to **L**, one deduces the equation of the trajectory as

$$r = \frac{\mathbf{L}^2}{\mathbf{1} + ||\mathbf{R}|| \cos \theta}. \tag{4}$$

The modulus determines whether the trajectory is an ellipse, a parabola or a hyperbola.

There are then 7 constants of the motion ð Þ **L**, **R**, *H*<sup>0</sup> are not independent and satisfy

$$\mathbf{R} \cdot \mathbf{L} = 0, \qquad \mathbf{L}^2 - \frac{\mathbf{R}^2}{2H\_0} = -\frac{1}{2H\_0} - \mathbf{1}.\tag{5}$$

The minus one on the right in (5) is not present in classical mechanics. The mutual commutation relations are given in terms of *εijk*, the fully antisymmetric tensor as follows,

$$\left[L\_i, L\_j\right] = i\varepsilon\_{ijk}L\_k, \qquad \left[L\_i, R\_j\right] = i\varepsilon\_{ijk}R\_k, \qquad \left[R\_i, R\_j\right] = i\varepsilon\_{ijk}(-2H\_0)L\_k,\tag{6}$$

Let us look at the symmetry group of the hydrogen atom. The symmetry group is the set of phase space transformations which preserve the Hamiltonian and the equations of motion. It can be identified from the commutation relations between constants of motion. For hydrogen, for negative energies, the group of rotations in 4-dimensional space is called *SO*ð Þ 4 .

The generators of the rotation group in an *n*-dimensional space are the ð Þ *n* � 1 *n=*2 components of the *n*-dimensional angular momentum

$$\mathcal{L}\_{ij} = \mathbf{x}\_i \boldsymbol{p}\_j - \mathbf{x}\_j \boldsymbol{p}\_i, \quad 1 \le i, j \le n. \tag{7}$$

In (7), L*ij* is the generator of the rotations in the ð Þ *i*, *j* -plane and has the following commutation relations

$$\left[\mathcal{L}\_{\vec{\eta}}, \mathcal{L}\_{kl}\right] = \mathbf{0}, \qquad \left[\mathcal{L}\_{\vec{\eta}}, \mathcal{L}\_{ik}\right] = i\mathcal{L}\_{jk}.\tag{8}$$

The first bracket in (8) holds if all four indices are different. Define the reduced Runge-Lenz vector to be

$$\mathbf{R}' = \frac{\mathbf{R}}{\sqrt{-2H\_0}}.\tag{9}$$

The commutation relations (6) are those of a four-dimensional angular momentum with the identification

$$\begin{aligned} \mathcal{L}\_{12} &= \mathcal{L}\_{\mathfrak{x}} & \mathcal{L}\_{23} &= \mathcal{L}\_{\mathfrak{x}} & \mathcal{L}\_{31} &= \mathcal{L}\_{\mathfrak{y}}\\ \mathcal{L}\_{14} &= R\_{\mathfrak{x}'} & \mathcal{L}\_{24} &= R\_{\mathfrak{y}'} & \mathcal{L}\_{34} &= R\_{\mathfrak{x}'}, \end{aligned} \tag{10}$$

and Casimir operator

$$\mathcal{L}^2 = \sum\_{i$$

The classical trajectory is thus uniquely defined with the 6 components of L and **L** � **R**<sup>0</sup> ¼ 0. Any trajectory can be transformed into any other one having the same energy by a 4-dimensional rotation. An explicit realization of this four-dimensional invariance is to use a stereographic projection from the momentum space onto the 4-dimensional sphere with radius *<sup>p</sup>*<sup>0</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffi �2*H*<sup>0</sup> <sup>p</sup> . On this sphere, the solutions to Schrödinger's equation as well as the classical equations of motion are those of the free motion. Schrödinger's equation on the four-dimensional sphere can be separated into six different types of coordinates each associated with a set of commuting operators.

Spherical coordinates correspond to the most natural set, and choosing the quantization axis in the 4 direction and inside the (1,2,3) subspace, the *z*-axis or usual 3-axis as reference axis, the three operators can be simultaneously diagonalized,

$$\mathcal{L}^2 = -\frac{1}{2H\_0} - 1,$$

$$\mathbf{L}^2 = \mathcal{L}\_{12}^2 + \mathcal{L}\_{31}^2 + \mathcal{L}\_{23}^2 = L\_x^2 + L\_y^2 + L\_z^2,\tag{12}$$

$$L\_x = \mathcal{L}\_{12},$$

The respective eigenvalues of these operators are *<sup>n</sup>*<sup>2</sup> � 1, *l l*ð Þ <sup>þ</sup> <sup>1</sup> and *<sup>M</sup>* such that <sup>∣</sup>*M*∣ ≤*<sup>l</sup>* <sup>≤</sup>*<sup>n</sup>* � 1, so the total degeneracy is *<sup>n</sup>*2. It corresponds to a particular subgroup chain given by

$$\text{LSO}(4)\_n \supset \text{SO}(\mathfrak{Z})\_l \supset \text{SO}(\mathfrak{Z})\_{\mathfrak{M}}.\tag{13}$$

*Classical and Quantum Integrability: A Formulation That Admits Quantum Chaos DOI: http://dx.doi.org/10.5772/intechopen.94491*

Other choices are possible, such as other spherical coordinates obtained from the previous by interchanging the role of the 3 and 4 axes. This simultaneously diagonalizes the three operators

$$\mathcal{L}^2 = -\frac{1}{2H\_0} - 1,$$

$$\lambda^2 = \mathcal{L}\_{12}^2 + \mathcal{L}\_{14}^2 + \mathcal{L}\_{24}^2 = R\_x^{\prime 2} + R^{\prime 2}\_{\ \ j} + L\_x^{\prime 2}, \tag{14}$$

$$L\_x = \mathcal{L}\_{12}.$$

The respective eigenvalues of these operators are *<sup>n</sup>*<sup>2</sup> � 1, *λ λ*ð Þ <sup>þ</sup> <sup>1</sup> and *<sup>M</sup>* such that ∣*M*∣ ≤ *λ*≤*n* � 1. The subgroup chain for this situation is

$$\mathrm{SO}(4)\_{\mathfrak{n}} \supset \mathrm{SO}(\mathfrak{3})\_{\mathfrak{k}} \supset \mathrm{SO}(\mathfrak{2})\_{\mathfrak{M}}.\tag{15}$$

Another relevant case is the adoption of cylindrical coordinates on the 4-dimensional sphere associated with the following set of commuting operators

$$\begin{aligned} \mathcal{L}^2 &= -\frac{1}{2H\_0} - 1, \\ \mathcal{L}\_{12} &= L\_x, \\ \mathcal{L}\_{34} &= R'\_x. \end{aligned} \tag{16}$$

This set has the following associated subgroup chain,

$$SO(4) \supset SO(2) \otimes SO(2). \tag{17}$$

In configuration space, this is associated with separability in parabolic coordinates. This is a specific system but it exhibits many of the mathematical and physical properties that will appear here.
