# Quantum nonintegrability in finite systems

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Referee this paper: https://doi.org/10.1016/0370-1573(94)00081-D by Wei-Min Zhang, Da Hsuan Feng

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Quantum nonintegrability in finite systems, as viewed from geometry and dynamical symmetry breaking, is discussed in this article. The concept of quantum nonintegrability can be constructed from the mathematical structures of quantum mechanics. It is shown that there is a natural geometrical description for a quantum system, which provides a suitable stage to investigate the time-honored question of quantum-classical correspondence as well as the underlying problem of nonintegrability in quantum mechanics. The implication of dynamical symmetry breaking to quantum nonintegrability and chaos is explored.

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paper authored Feb 4, 2000 to math-ph
edited Feb 3

## 1 Review

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These remarks are rather an extended summery supplemented by some additional thoughts of an educated reader than the judgement of a senior researcher in the field who has kept track of all the existing literature concerning the topics of this paper..

In Quantum nonintegrability in finite systems, Zhang and Feng (1995)
define and investigate the integrability of finite quantum systems
in analogy to the definition of integrability in classical mechanics.
For that purpose they construct the quantum phase space (based on
generalized coherent states) of the quantum system at hand and
derive classical-like equations of motion on it called semiquantal dynamics.

The quantum dynamical degrees of freedom are identified to be the quantum numbers needed
to describe the system. The quantum dynamical phase space is then defined
as the coset space $G/H$, where $G$ is the covering group of g (a Lie algebra of self-adjoint operators) and $H$ is the
maximal stability group with respect to a fixed state
$|\psi_0\rangle$. The quantum phase space has as symplectic structure
with the metric given by
$d^2s = \sum\limits_{ij}dz^idz^j* = \sum\limits_{ij} \frac{\partial^2K(z,z^*)}{\partial z^i\partial z^{j*}}dz^idz^{j*}$
where $K$ is the coherent product
$K(z,z^*)=\sum\limits_n c_n(z)c_n^*(z)$
If $g$ is a semisimple Lie algebra,
$K(z,z^*)= \det(I \pm Z^*Z)^{\pm\Xi}$
where the integer $\Xi$ is a so-called quenching index.
The symplectic form $\omega$ is given by
$\omega=i\hbar\sum\limits_{ij}g_{ij}dz^i\wedge dz^{j*}$
and the Poisson bracket is
$\left\{ f,g \right\} = \frac{1}{i\hbar}\sum\limits_{ij}g^{ij} \left(\frac{\partial f}{\partial z^i}\frac{\partial g}{\partial z^{j*}}- \frac{\partial g}{\partial z^i}\frac{\partial f}{\partial z^{j*}}\right)$
Wave functions and operators of the quantum system can be expressed
on the quantum phase space by means of the (generalized) coherent states
$|\psi\rangle = \int\limits_{G/H}\|z\rangle f(z^*)du(z)$
and
$A=\int\|z\rangle\langle z\|A\|z\rangle\langle z'\|du(z)du(z')$
Using the stationary phase approximation for the effective quantum action,
Hamilton-like equations can be obtained on the quantum phase space $G/H$.
The classical limit of the quantum system, if it exists, can be studied
by letting $\Xi \to \infty$
It is then shown, that a quantum system described by a dynamical group $G$
is integrable if it possesses a dynamical symmetry of $G$.
For a nonintegrable system, if the dynamical symmetry breaking is accompanied
by a structural phase transition, chaotic motion occurs.
For a nonintegrable non-compact quantum system, quantum fluctuation enhance the chaotic behavior whereas for compact nonintegrable quantum systems chaotic behavior is suppressed by quantum fluctuations.

The explicit construction of a quantum phase space by means of generalized coherent states presented in this paper seems highly interesting and mathematically sound and the further investigations make a lot of sense. Studying the classical limit by letting the quenching index go to infinity seems less problematic than letting $\hbar \to 0$. The criterium of dynamical symmetry breaking plus structural phase transition for the occurance of quantum chaos, illustrated by numerical simulations for some representative examples, is rather cute.

In addition to extending the investigations presented in this paper to the quantum field theory context, it would also be interesting to commpare the methods presented here to the new framework of coherent spaces as presented by Arnold Neumaier for example in his Introduction to coherent spaces and Introduction to coherent quantization.

reviewed Feb 3 by (5,240 points)
edited Feb 4 by Dilaton