In order to have a complete description of the spatio-temporal evolution of a quantum state, the initial state should be properly specified as the coherent state corresponding to a definite dynamical symmetry. Then we have qualitatively different situations where the initial coherent state evolves under the action of a system having or not having the same dynamical symmetry. In the former case, we have regular motion with the general character which tends to that of the corresponding classical case in the limit ℏ → 0. While in the latter case, we may have energy regions of global stochasticity with characters closely related to the initial dynamical symmetry strongly violated.
In order to have a complete description of the spatio-temporal evolution of a quantum state, the initial state should be properly specified as the coherent state corresponding to a definite dynamical symmetry. Then we have qualitatively different situations where the initial coherent state evolves under the action of a system having or not having the same dynamical symmetry. In the former case, we have regular motion with the general character which tends to that of the corresponding classical case in the limit ℏ → 0. While in the latter case, we may have energy regions of global stochasticity with characters closely related to the initial dynamical symmetry strongly violated.
The notion of quantum-classical correspondence is carefully investigated in order to prepare firm grounds for studying the spatiotemporal evolution of quantum states in the same spirit as for corresponding classical cases. Three relevant problems, (1) the integrability of dynamical equations of quantum systems, (2) the initial minimum uncertainty states one-to-one correspondent to classical phase points, and (3) the effective Planck constants for systems having analogous dynamical properties but exhibiting different quantum effects, have been successfully resolved. Then the solution ${\ensuremath{\rho}}_{\ensuremath{\gamma}}(t)$ of the dynamical equation of a quantum integrable system is shown to be expressed as an analytical functional of the initial minimum uncertainty state ${\ensuremath{\rho}}_{\ensuremath{\gamma}}^{0}$ varying smoothly with $\ensuremath{\gamma}$ and t. Such a general character of the quantum regular motion serves as a reference for the study of quantum irregular motion under the action of perturbed Hamiltonian.
Based on the dynamical symmetry group characterizing the integrability of classical as well as quantum mechanics, quantum dynamics with proper initial conditions was genuinely formulated, and analytical solutions in the form of soliton-like state evolving around a certain invariant torus were obtained. It has been shown that, in case the intrinsic size of the evolving quantum state is significantly smaller than the extent of its evolving orbit, the motion can be satisfactorily treated with classical approximation. Having demonstrated the quantum-classical correspondence of regular motion, the possibility to study quantum chaos on the basis dynamical symmetry breaking was briefly discussed.
Coherent quantum physics is quantum physics in terms of a coherent space and an appropriate coherent product. The kinematical structure and the meaning of the quantities are given by the symmetries (invertible coherent maps) of the coherent space. If Z is a coherent manifold, a canonical symplectic form is often canonically determined by the coherent product through the coherent action principle discussed in Section 6.1. In a quantum mechanical context, Z is indeed a classical phase space or an extended phase space-typically a symplectic manifold, a Poisson manifold, or a circle or line bundle over such a manifold that incorporates the classical action variable (encoding the Berry phase under quantization). For example, the Aharonov-Bohm effect [3] needs the bundle formulation. A coherent space and its quantum space provide a classical and a quantum view of the same physical system, discussed in Section 6.2. Section 6.3 shows that symmetries of coherent spaces can be quantized, that is, promoted to linear operators on the quantum space. This leads to quantum dynamics, which in special (completely integrable) situations can be solved in closed form in terms of classical motions on the underlying coherent space, if the latter has a compatible manifold structure. Close relations to concepts from geometric quantization and Kähler manifolds are pointed out in Section 6.4. In order that the quantum space of a coherent space Z can describe physics, one needs not only a distinguished class of coherent states but a more general concept of state. In Sections 6.5-6.6, we give a general abstract setting for states in Lie ∗-algebras, emphasize the essential mathematical features and the close analogy between classical and quantum physics. We illustrate these generalities with a number of examples in Section 6.7. The coherent action principle is put to use in Section 6.8 for numerical quantum physics and an application to quantum chaos. Spectral issues can, in favorable cases, be handled in terms of dynamical Lie algebras, discussed in Section 6.9.
A major difficulty in comparing quantum and classical behavior resides in the struc- tural differences between the corresponding mathematical languages. The Heisenberg equations of motion are operator equations only formally identical to the classical equations of motion. By taking the expectation of these equations, the well-known Ehrenfest theorem provides identities which, however, are not a closed system of equations which allows to evaluate the time evolution of the system. The formalism of the effective action seems to offer a possibility of comparing quantum and classical evolutions in a system- atic and logically consistent way by naturally providing approximation schemes for the expectations of the coordinates which at the zeroth order coincide with the classical evolution [1]. The effective action formalism leads to equations of motion which differ from the classical equations by the addition of terms nonlocal in the time variable. This means that for these equations an initial value problem is not meaningful and they have to be interpreted in an appropriate way. Here we analyze situations in which the nonlocal terms can be reasonably approximated by local ones, so that the quantum corrections do not modify the locality of classical equations. In the simplest approximation, the effective Lagrangian differs from the corresponding classical one by a renormalization of both the potential- and the kinetic-energy terms. We shall not discuss the causal formalism used, for example, in refs. [2-4], as in the approximation considered this would lead to the same local equations. The present contribution describes the beginning of a systematic study of semiclassical evolutions using the effective action formalism. In the first part, after introducing the formalism of the effective action and its expansion in powers of hbar (loop expansion) in the context of quantum mechanics, we concentrate on the structure of the first-order corrections in li. These corrections are evaluated to the second order in the derivative expansion [5], by two different methods. The first is based on a Euclidean approach [6], the second one on an adiabatic approximation in evaluating functional determinants. In the second part of the article we put the formalism at work, choosing as our case study a two-dimensional (2D) anharmonic oscillator of the kind considered in molecular physics. The results of the simulations show that by increasing lithe effective dynamics tends to regularize the classical motion and becomes qualitatively very similar to the quantum evolution provided the energy is sufficiently small. The evaluation of the effective dynamics in more general cases will be presented in a forthcoming paper.
The starting point of the classical description of motion is the Newton equations that yield a phase space trajectory (rN (t), pN (t)) for a given initial condition (rN (0), pN (0)). Alternatively one may describe classical motion in the framework of the Liouville equation (Section (1.2.2)) that describes the time evolution of the phase space probability density f (rN , pN ; t). For a closed system fully described in terms of a well specified initial condition, the two descriptions are completely equivalent. Probabilistic treatment becomes essential in reduced descriptions that focus on parts of an overall system, as was demonstrated in Sections 5.1–5.3 for equilibrium systems, and in Chapters 7 and 8 that focus on the time evolution of classical systems that interact with their thermal environments. This chapter deals with the analogous quantum mechanical problem. Within the limitations imposed by its nature as expressed, for example, by Heisenbergtype uncertainty principles, the Schrödinger equation is deterministic. Obviously it describes a deterministic evolution of the quantum mechanical wavefunction. The analog of the phase space probability density f (rN , pN ; t) is now the quantum mechanical density operator (often referred to as the “density matrix”), whose time evolution is determined by the quantum Liouville equation. Again, when the system is fully described in terms of a well specified initial wavefunction, the two descriptions are equivalent. The density operator formalism can, however, be carried over to situations where the initial state of the system is not well characterized and/or a reduced description of part of the overall system is desired. Such situations are considered later in this chapter.
According to the exactly formulated concepts of quantum integrability and nonintegrability,an initial coherent state evolving under the action of an integrable Hamiltonian having the same dynamical symmetry is expressed as an analytical function of and . The expectation values of various dynamical variables should be expressed as correlated analytical functions of and . Thus, the general characters of initial coherent state would be maintained during the spatio-temporal evolution under the action of such a system even if the wavepacket is slightly distorted. In contrast, when the integrability condition is destroyed, these features will be lost during the evolution. This theoretical prediction is illustrated with numerical results computed for the two-level Lipkin model with a dou ble-well potential and dynamical explanations are given. And the Eherenfest time is verified with this model.