## 1. Introduction

Chaos and many ideas from the study of this area have permeated a very large number of areas of science especially those which rely on mathematics. It is hoped this will illustrate how deeply and powerfully these ideas have influenced such areas as chemistry and physics.

Nature seems to be far too complicated to be linear at all levels all of the time. The exact laws of nature cannot be linear, nor can they be derived from such, to quote Einstein. Quantum mechanics, which is formally linear, is believed to be the underlying system to understand nature [1, 2, 3]. These seemingly conflicting views urge one to ask whether quantum mechanics can encompass nonlinear phenomena as well. This question is related to the study of classical nonlinear phenomena [4, 5]. This leads one to wonder about the behavior of a quantum system if the classical version is chaotic. To understand chaos in quantum mechanics requires a more rigorous formulation of the fundamental structures of quantum theory [6, 7]. To do this, one needs to formulate the quantum-classical correspondence, and at present, such a formulation is lacking.

In classical mechanics a Hamiltonian system with

Attempts to investigate quantum chaos have focused on the quantization of classical nonintegrable systems. Since the former in principle is only a limiting case of the latter and most realistic quantum systems do not have a classical counterpart, the latter approach is more general and natural. The classical limit is most often approached by using Ehrenfest’s theorem, and three popular ways to study the classical limit are given as follows. The Schrödinger approach is to develop a wave packet whose time evolution follows classical trajectories, so the time evolution of the coordinate and momentum expectation values solves not only Hamilton’s equations but also Schrödinger’s equation. Dirac’s approach is to construct a quantum Poisson bracket such that the basic structure of classical and quantum mechanics is placed in one-to-one correspondence. The third approach is the Feynman path integral formalism, which expresses quantum mechanics in terms of classical concepts by integrating overall possible paths for a given initial and final state.

The problem may be reviewed based on the axiomatic structure of quantum mechanics, out of which the quantum dynamical degrees of freedom are defined and permit the construction of quantum phase space. This allows us to propose an idea for what quantum integrability is as well as its relationship with dynamical symmetry.

Quantum chaos is related to the question of the quantum-classical correspondence at two levels, kinematical and dynamical. The kinematical quantum-classical correspondence is a kind of reconciliation of the quantum and classical degrees of freedom and their associated geometrical structures.

Consistency of quantum theory implies there must exist a fundamental structure which can be used to determine the system’s Hilbert space structure before solving the quantum dynamical equations. The axiomatic structure of quantum mechanics implies such a fundamental structure is simply the given algebraic structure of the system. The quantum mechanical Hilbert space is realized as a unitary irreducible representation of an algebra denoted as

A quantum system possesses a well-defined dynamical group

To see what can be extracted from this statement, consider now some nontrivial examples. In particular, let us clarify the idea of quantum dynamical degrees of freedom. The non-fully degenerate quantum numbers are defined by the nonconstant eigenvalues of a complete set of commuting operators in the associated basis.

The harmonic oscillator whose dynamical group is the Heisenberg-Weyl group

Next consider the spin system whose dynamical group is

In the central potential problem, the dynamical group is

The hydrogen atom and the relativistic free Dirac particle are perhaps the simplest and most realistic both having the dynamical group

For the harmonic oscillator, to construct the phase space, the fixed state used is the vacuum state. The phase space is then

The phase space is not complicated, just a one-dimensional complex space or two-dimensional real space.

For a spin system, the dynamical group is

where

where

## 2. Quenched quantum mechanics

The dynamical correspondence of quantum-classical mechanics is a fundamental idea which should be addressed. In order to study the resultant singularity structures which result in a transition to chaos, it must be stated more precisely what this limit entails. Quenched quantum mechanics suggests a possible origin for a parameter which maps out this limit. Instead of considering

In Eq. (3),

To study quenched quantum mechanics, the propagator is expressed as

In Eq. (4),

where *v* is the one-form of

The quantum equations depend on

This limit may be divergent, since the phase space derived from the quantum geometry has not been scaled. Scaled canonical coordinates must be introduced to obtain a convergent limit as

Expectation values of observables in coherent states can have correct dimensions in terms of

The difference between semi-quantal dynamics and classical mechanics is called the quantum fluctuation or correlation

## 3. Dynamical symmetry

Let us discuss now some basic concepts related to chaos. One way to proceed is to study the behavior of quantum systems at the semi-quantal level by explicitly exploring the dynamical effects of quantum fluctuations on classical chaos. It would be good to find some general set of conditions which determine without great effort whether systems become chaotic and when.

The central idea of quantum integrability is dynamical symmetry. Integrability is a fundamental concept in the study of dynamical systems. Usually, the function of symmetry restricts the possible forms of Lagrangian, but not the associated dynamics.

*A quantum system* with dynamical group

Dynamical symmetry is less restrictive on the system than pure, since the Hamiltonian and ground state are not necessarily invariant under a transformation of

*A quantum system* with *a quantum system* with a dynamical group

Consider the example of an

The generators of the dynamical group

It follows that

where

where

The

There is an important consequence of the results mentioned above. Nonintegrability of a quantum system implies breaking of dynamical symmetry. This means that if chaos is present, dynamical symmetry of the system must be broken.

To develop this idea, if a system with

Let us say that *chaos will appear* in a nonintegrable system when the breaking of the dynamical symmetry is accompanied by a structural phase transition. So if a structural phase transition takes place in a quantum system such that certain control parameters are altered, then it passes from one dynamical symmetry limit to another. Different dynamical symmetries connote different toroidal structures in

Let us present a simple model which consists of two-spin coupled system governed by the Hamiltonian:

In (14), *α* is a coupling constant. This system has the possible dynamical symmetries:

The Hilbert space basis which carries the irreducible representations *i*) and (*ii*).

The dynamical symmetries of *α* = 0, *H* has symmetries (*i*) and (*ii*). For *α* = 1, the system is just in (*i*). However, when

where the canonical coordinates

and

Finally, it may be stated that to understand quantum chaos, one has to understand the dynamical behavior of a nonintegrable system when it deviates from the classical dynamics by taking into account nonvanishing quantum fluctuations. It may be asked whether the global phase space structure of classical dynamics can survive when quantum fluctuations are included. There is also the question of what governs the evolution of quantum fluctuations. It is required to have on hand a procedure which allows one to obtain the classical limit from a quantum system when one can only compute the deviations of the dynamics both close to and far from the classical limit. These deviations provide knowledge as to whether quantum fluctuations may alter classical dynamics and in what way. This is also deepening the understanding of the quantum-classical correspondence. Based on this, it may be asked whether the global phase space structure of classical mechanics can survive when quantum fluctuations are included and what actually governs the evolution of quantum fluctuations.