The following article entitled: 'W H A T  I S Quantum Chaos?' by Ze’ev Rudnick bublished by the American Mathematical Society gives a very interesting discussion of the quantum chaos. One way to see the effect of the classical dynamics

is to study local statistics of the energy

spectrum, such as the level spacing distribution

P (s), which is the distribution function of nearestneighbor

spacings λn+1 − λn as we run over all

levels. In other words, the asymptotic proportion

of such spacings below a given bound x is

R x

−∞ P (s)ds. A dramatic insight of quantum chaos

is given by the universality conjectures for P (s):

• If the classical dynamics is integrable, then

P (s) coincides with the corresponding quantity for

a sequence of uncorrelated levels (the Poisson ensemble)

with the same mean spacing: P (s) = ce−cs

,

c = ar ea/4π (Berry and Tabor, 1977).

• If the classical dynamics is chaotic, then P (s)

coincides with the corresponding quantity for the

eigenvalues of a suitable ensemble of random

matrices (Bohigas, Giannoni, and Schmit, 1984).

Remarkably, a related distribution is observed for

the zeros of Riemann’s zeta function.

Not a single instance of these conjectures is

known, in fact there are counterexamples, but

the conjectures are expected to hold “generically”,

that is unless we have a good reason to think otherwise.

A counterexample in the integrable case

is the square billiard, where due to multiplicities in the spectrum, P (s) collapses to a point

mass at the origin. Deviations are also seen in the

chaotic case in arithmetic examples. Nonetheless,

empirical studies offer tantalizing evidence for

the “generic” truth of the conjectures, as Figures

3 and 4 show.

Some progress on the Berry-Tabor conjecture in

the case of the rectangle billiard has been achieved

by Sarnak, by Eskin, Margulis, and Mozes, and by

Marklof. However, we are still far from the goal

even there. For instance, an implication of the

conjecture is that there should be arbitrarily large

gaps in the spectrum. Can you prove this for

rectangles with aspect ratio √4

5?

The behavior of P (s) is governed by the statistics

of the number N(λ, L) of levels in windows

whose location λ is chosen at random, and whose

length L is of the order of the mean spacing

between levels. Statistics for larger windows also

offer information about the classical dynamics and

are often easier to study. An important example

is the variance of N(λ, L), whose growth rate is

believed to distinguish integrability from chaos [1]

(in “generic” cases; there are arithmetic counterexamples).

Another example is the value distribution

of N(λ, L), normalized to have mean zero and variance

unity. It is believed that in the chaotic case

the distribution is Gaussian. In the integrable case

it has radically different behavior: For large L, it

is a system-dependent, non-Gaussian distribution

[2]. For smaller L, less is understood: In the case

of the rectangle billiard, the distribution becomes

Gaussian, as was proved recently by Hughes and

Rudnick, and by Wigman. See the ful article at the webpage:

http://www.ams.org/notices/200801/tx080100032p.pdf

The following article entitled: 'W H A T  I S Quantum Chaos?' by Ze’ev Rudnick bublished by the American Mathematical Society gives a very interesting discussion of the quantum chaos. One way to see the effect of the classical dynamics

is to study local statistics of the energy

spectrum, such as the level spacing distribution

P (s), which is the distribution function of nearestneighbor

spacings λn+1 − λn as we run over all

levels. In other words, the asymptotic proportion

of such spacings below a given bound x is

R x

−∞ P (s)ds. A dramatic insight of quantum chaos

is given by the universality conjectures for P (s):

• If the classical dynamics is integrable, then

P (s) coincides with the corresponding quantity for

a sequence of uncorrelated levels (the Poisson ensemble)

with the same mean spacing: P (s) = ce−cs

,

c = ar ea/4π (Berry and Tabor, 1977).

• If the classical dynamics is chaotic, then P (s)

coincides with the corresponding quantity for the

eigenvalues of a suitable ensemble of random

matrices (Bohigas, Giannoni, and Schmit, 1984).

Remarkably, a related distribution is observed for

the zeros of Riemann’s zeta function.

Not a single instance of these conjectures is

known, in fact there are counterexamples, but

the conjectures are expected to hold “generically”,

that is unless we have a good reason to think otherwise.

A counterexample in the integrable case

is the square billiard, where due to multiplicities in the spectrum, P (s) collapses to a point

mass at the origin. Deviations are also seen in the

chaotic case in arithmetic examples. Nonetheless,

empirical studies offer tantalizing evidence for

the “generic” truth of the conjectures, as Figures

3 and 4 show.

Some progress on the Berry-Tabor conjecture in

the case of the rectangle billiard has been achieved

by Sarnak, by Eskin, Margulis, and Mozes, and by

Marklof. However, we are still far from the goal

even there. For instance, an implication of the

conjecture is that there should be arbitrarily large

gaps in the spectrum. Can you prove this for

rectangles with aspect ratio √4

5?

The behavior of P (s) is governed by the statistics

of the number N(λ, L) of levels in windows

whose location λ is chosen at random, and whose

length L is of the order of the mean spacing

between levels. Statistics for larger windows also

offer information about the classical dynamics and

are often easier to study. An important example

is the variance of N(λ, L), whose growth rate is

believed to distinguish integrability from chaos [1]

(in “generic” cases; there are arithmetic counterexamples).

Another example is the value distribution

of N(λ, L), normalized to have mean zero and variance

unity. It is believed that in the chaotic case

the distribution is Gaussian. In the integrable case

it has radically different behavior: For large L, it

is a system-dependent, non-Gaussian distribution

[2]. For smaller L, less is understood: In the case

of the rectangle billiard, the distribution becomes

Gaussian, as was proved recently by Hughes and

Rudnick, and by Wigman. See the ful article at the webpage:

http://www.ams.org/notices/200801/tx080100032p.pdf

Nonlinear dynamics (``chaos theory'') and quantum mechanics are two of the scientific triumphs of the 20th century. The former lies at the heart of the modern interdisciplinary approach to science, whereas the latter has revolutionized physics. Both chaos theory and quantum mechanics have achieved a fairly large level of glamour in the eyes of the general public. The study of quantum chaos encompasses the application of dynamical systems theory in the quantum regime. In the present article, we give a brief review of the origin and fundamentals of both quantum mechanics and nonlinear dynamics. We recount the birth of dynamical systems theory and contrast chaotic motion with integrable motion. We similarly recall the transition from classical to quantum mechanics and discuss the origin of the latter. We then consider the interplay between nonlinear dynamics and quantum mechanics via a classification and explanation of the three types of quantum chaos.

http://arxiv.org/abs/nlin/0107039

Dear Prof. Mahfuz:

Thank you for providing a great article, which precisely was what I was looking for. This 57-pages paper by Mason Porter is dated 2001, and so it is not the latest research, but still it gives a good introduction about the research area - Quantum Chaos.

Quantum chaos is something that simply does not exist. When I was young I spent one year of my life trying to find quantum chaos in field theories. My conclusion was (then) that we should define from scratch many entities in order ... blah blah... But, now I realize that I was just trapped in a linear trap while I was searching for a pure nonlinear phenomenon. (I think that my failed approach got me away from Physics for many years and now I believe that my professor should know that we cannot find a nonlinear object at the kingdom of linearity). So, dear Sundarapandian be careful.

Here is a summarized review of the whole spectrum of quantum studies including quantum chaos by Martin Gutzwiller in scholarpedia. As he outlined, the name is the concatenation of two important words: quantum - the smallest amount of matter that moves and chaos: the erratic and unpredictable behavior.

But my own take: quantum studies(including quantum chaos) and string theory are  more of mathematical and philosophical in existence and test. Just to recall from historical anecdotes: Einstein though awarded the Nobel prize for his work on quantum theory, he remained hesitant to believe the randomness behaviors of quanta modeled by the theory of quantum and said "God does not play dice" on this issue.  

http://www.scholarpedia.org/article/Quantum_chaos

A recent research paper on this topic:

"On the classical limit of quantum mechanics, fundamental graininess and chaos: Compatibility of chaos with the correspondence principle" by Ignacio Gomeza, and Mario Castagninob (Chaos, Solitons & Fractals, Volume 68, November 2014, Pages 98–113)

Abstract:

"The aim of this paper is to review the classical limit of Quantum Mechanics and to precise the well known threat of chaos (and fundamental graininess) to the correspondence principle. We will introduce a formalism for this classical limit that allows us to find the surfaces defined by the constants of the motion in phase space. Then in the integrable case we will find the classical trajectories, and in the non-integrable one the fact that regular initial cells become “amoeboid-like”. This deformations and their consequences can be considered as a threat to the correspondence principle unless we take into account the characteristic timescales of quantum chaos. Essentially we present an analysis of the problem similar to the one of Omnès (1994,1999), but with a simpler mathematical structure."

http://www.sciencedirect.com/science/article/pii/S0960077914001246