## Abstract

The concept of integrability of a quantum system is developed and studied. By formulating the concepts of quantum degree of freedom and quantum phase space, a realization of the dynamics is achieved. For a quantum system with a dynamical group G in one of its unitary irreducible representative carrier spaces, the quantum phase space is a finite topological space. It is isomorphic to a coset space G/R by means of the unitary exponential mapping, where R is the maximal stability subgroup of a fixed state in the carrier space. This approach has the distinct advantage of exhibiting consistency between classical and quantum integrability. The formalism will be illustrated by studying several quantum systems in detail after this development.

### Keywords

- classical
- quantum
- chaos
- integrability
- conservation law
- algebra

## 1. Introduction

In classical mechanics, a Hamiltonian system with

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

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, 9, 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, 12, 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, 15, 16]. The close relationship of the angular momentum and the

### 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

The corresponding quantum operator is found by replacement of

So

The Coulomb interaction has another constant of the motion associated with the Runge-Lenz vector

If the

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

There are then

The minus one on the right in (5) is not present in classical mechanics. The mutual commutation relations are given in terms of

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

The generators of the rotation group in an

In (7),

The first bracket in (8) holds if all four indices are different.

Define the reduced Runge-Lenz vector to be

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

and Casimir operator

The classical trajectory is thus uniquely defined with the 6 components of

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

The respective eigenvalues of these operators are

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

The respective eigenvalues of these operators are

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

This set has the following associated subgroup chain,

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.

## 2. Quantum degrees of freedom

The time evolution of a system in classical mechanics in time is usually represented by a trajectory in phase space and the dynamical variables are functions defined on this space. The dimension of phase space is twice the number of degrees of freedom, and a point represents a physical state. The space is even-dimensional and it is endowed with a symplectic Poisson bracket structure. Dynamical properties of the system are described completely by Hamilton’s equations within this space.

For a quantum system, on the other hand, the dynamical properties are discussed in the setting of a Hilbert space. Dynamical observables are self-adjoint operators acting on elements of this space. A physical state is represented by a ray of the space, so the Hilbert space plays a role similar to phase space for a classical system. The Hilbert space cannot play the role of a quantum phase space since its dimension does not in general relate directly to degrees of freedom. Nor can it be directly reduced to classical phase space in the classical limit. Let us define first the quantum degrees of freedom as well as giving a suitable meaning to quantum phase space.

Suppose

*Definition 1*: (Quantum Dynamical Degrees of Freedom) Let

Since the members of

The physical and mathematical considerations for defining the dimension of the nonfully degenerate operator subset

A given quantum system generally has associated with it a well-defined dynamical group structure due to the fact that the mathematical image of a quantum system is an operator algebra

The Hamiltonian

The

From group representation theory, it will be given that a total of

For each subgroup chain

*Definition 2*: For a quantum system with

## 3. Quantum integrability and dynamical symmetry

Quantum phase space defined here can be compact or noncompact depending on the finite or infinite nature of the Hilbert space. A consequence of this development is that the classical definition of integrability can in general be directly transferred to the quantum case.

*Definition 3*: (Quantum Integrability) A quantum system with

Any set of variables that commute may be put in the form of a complete set of commuting observables

The link with the dynamical group structure can be developed. This specifies exactly the integrability of a quantum system. To this end the definition of dynamical symmetry is needed.

*Definition 4*: (Dynamical Symmetry) A quantum system with dynamical group

The index of a particular subgroup chain

*Proposition 1*: (Quantum Integrability) A quantum system with dynamical group

To prove this, note that it can be broken down into two cases or subgroup classes for a given dynamical group

First consider the case in which

When

There are always

For a non-canonical subgroup chain

When the system is characterized by the dynamical symmetry of

Based on this proposition, it can be stated that nonintegrability of a quantum system involves the breaking of the dynamical symmetry of the system. It may be concluded that dynamical symmetry breaking can be said to be a property which characterizes quantum nonintegrability.

Let us summarize what has been found as to what quantum mechanics tells us. In a given quantum system with dynamical Lie group

## 4. Quantum phase space

It is of interest then to develop a model for phase space for quantum mechanics which may be regarded as an analogue to classical physics. By what has been said so far, the Hilbert space

In (25), the subscript

To construct the quantum phase space from the quantum dynamical degrees of freedom for an arbitrary quantum system, the elementary excitation operators can be obtained from the structures of

Moreover

With respect to

The

The homogeneous space

The function

The functions

and

In (32)

Corresponding to this two form there is a Poisson bracket which is given by

In (34)

### 4.1 Phase space quantum dynamics

Based on what has been stated about

Here

Then

The

The Bargmann kernel was introduced in (30), and for a semisimple Lie group, the parameters

Here

where

A classical-like framework or analogy has been established in the form of a quantum phase space specified by

This equation can be replaced by Hamilton’s equations

In (42) and (43),

Clearly, if suitable conditions hold the phase space representation of the commutator of any two operators is equal to the Poisson bracket of the phase space representation of these two operators so that

Then the phase space representation of the Heisenberg equation

given by (42) is therefore equivalent to (43). In (45),

and

Let us discuss integrability and dynamical symmetry. A quantum system with

It follows that in the classical limit which has been formulated,

Together with the Hamilton equations, (47) also formally defines classical integrability, so quantum integrability is completely consistent with the classical theory. In the classical analogy, the group structure of the system is defined by Poisson brackets. The concept of dynamical symmetry is naturally preserved in the classical analogy, so the theorem on dynamical symmetry and integrability is also meaningful for the classical analogy. If the Hamiltonian has the symmetry

in the phase space representation, it holds that

To put this concisely, we write

where

## 5. Applications to physical systems

### 5.1 Harmonic oscillator

The harmonic oscillator has dynamical group

With generators

It is useful to introduce the standard canonical position and momentum coordinates

The Glauber coherent states can be realized by the states

The normalization constant in (55) is the Bargmann kernel

The phase space representation of the wavefunction

By Wick’s Theorem, it is always possible to write an operator

The phase space representation of

In the case

The corresponding algebraic structure of

Here

For the forced harmonic oscillator, the classical analogy of the Hamiltonian is given by

Hamilton’s equations in (44) can be used to evaluate the

Hence combining these two derivatives, we obtain

Multiplying both sides by the integrating factor

If the initial state is

and

This seems to imply the classical analogy provides an exact quantum solution if the Hamiltonian is a linear function of the generators of

### 5.2 SU 2 spin system

The phase space structure of a spin system will be constructed and as well the phase-space distribution and classical analogy.

Since the dynamical group of the spin system is

Any state

The generalized Bargmann kernel on

Given the canonical coordinates

there obtains the bracket

where

The phase space representation of the state

The phase space representation of an operator

When the operator

These can also be given in terms of

The classical analogy of an observable

The classical limit is found by taking

### 5.3 SU 1 1 quantum systems and a two-level atom

A two-level atom is considered which interacts with two coupled quantum systems that can be represented in terms of a

It is similar to **5.1**, so we sketch the physical situation. The

The corresponding Casimir

Given that this is the Lie algebra, it can be said that the Fock space is spanned by the set of vectors

It is the case that

where the Bargmann index

Using these operators, (80) is written in terms of

The Heisenberg equations of motion obtained from (85) gives

The following two operators

Hamiltonian (80) can now be put in the equivalent form

where

The constant

As

In the space of the two-level eigenstates

The operators

where

The coherent atomic state

where

Since the operators

Then applying

This calculation implies that the normalization constant

The new state is of the form,

If it is assumed that at

The reduced density matrix is constructed from this

## 6. Summary and conclusions

Explicit structures for quantum phase space have been examined. Quantum phase space provides an inherent geometric structure for an arbitrary quantum system. It is naturally endowed with sympletic and quantum structures. The number of quantum dynamical degrees of freedom has a great effect on determining the quantum phase space. Inherent properties of quantum theory, the Pauli principle, quantum internal degrees of freedom and quantum statistical properties are included. A procedure can be stated for constructing this quantum phase space and canonical coordinates should be derivable for all semi-simple dynamical Lie groups with Cartan decomposition. The coset space