Integration paths, classical trajectories

Time reversibility. Newton s equation is reversible in time. Eor a numerical simulation to retain this property it should be able to retrace its path back to the initial configuration (when the sign of the time step At is changed to —At). However, because of chaos (which is part of most complex systems), even modest numerical errors make this backtracking possible only for short periods of time. Any two classical trajectories that are initially very close will eventually exponentially diverge from one another. In the same way, any small perturbation, even the tiny error associated with finite precision on the computer, will cause the computer trajectories to diverge from each other and from the exact classical trajectory (for examples, see pp. 76-77 in Ref. 6). Nonetheless, for short periods of time a stable integration should exliibit temporal reversibility. 

We seek the poles of the spectral function g(E) given by (3.7). In the WKB approximation the path integral in (3.7) is dominated by the classical trajectories which give an extremum to the action functional... 

Since the birth of quantum theory, there has been considerable interest in the transition from quantum to classical mechanics. Because the two formulations are given in a different theoretical framework (nonlinear classical trajectories versus expectation values of linear operators), this transition is far more involved than the naive limit —> 0 suggests. By exploring the classical limit of quantum mechanics, new theoretical concepts have been developed, including path integrals [1], various phase-space representations of quantum mechanics [2], the semiclassical propagator and the trace formula [3], and the notion of quantum... 

Figure 2. Relation between the integration paths and the complexified trajectories of the unperturbed barrier system, (a) Singularities and representative integration paths on the i -plane. b) Complex trajectories obtained along various integration paths depicted in (a) for the case of El < 1. The instanton trajectory in the classically forbidden region in b), as well as the corresponding integration path in (a), is indicated by a hatched halo around it.

The functional above was used already by Gauss [12] to study classical trajectories (which explains our choice of the action symbol). Onsager and Machlup used path integral formulation to study stochastic trajectories [13]. The origin of their trajectories is different from what we discussed so far, which are mechanical trajectories. However, the functional they derive for the most probable trajectories, O [X (t)] is similar to the equation above ... 

Linearization methods start from a path integral representation of the forward and backward propagators in expressions for time correlation function, and combine them to describe the overall time evolution of the system in terms of a set of classical trajectories whose initial conditions are sampled from a quantity related to the Wigner transform of the quantum density operator. The linearized expression for a correlation function provides a powerful tool for describing systems in the condensed phase. The rapid decay of... 

The expression we derive below leads to an action and to a stationary (minimum) condition on the classical path. The optimal path is a discrete approximation to a classical trajectory. Interestingly, in the integral limit (an infinitesimal time step), the action below was used already by Gauss ( ) to compute classical trajectories [14]. At variance with Gauss we keep a finite At. 

Another approach (Newns 1985 Nourtier 1985) is based on the double path integral representation of the scattering probability. The surface dynamics is described in terms of an influence functional that can be found in explicit form in the linear coupling approximation. However, semiclassical evaluation of the path integrals leads to a complex classical trajectory problem that is non-local in time and too difficult for applications. 

A formalism presented in this part unifies both approaches. Path integrals over projectile variables in momentum representation are evaluated in the quasiclassical limit separately before and after the turning point in the spirit of generalized eikonal method. The Faddeev-Popov method (Popov 1983) is used to fix classical trajectories with respect to the symmetry of the problem. The influence functional is treated as a pre-exponential factor. 

The quasiclassical amplitude in the momentum representation does not suffer from this feature, because boundary conditions Pz h) = Pu> Pzitz) = P2z determine the unique classical trajectory for the typical scattering potentials. Such amplitude cannot be obtained as a Fourier transform of the quasiclassical propagator in coordinate representation. So it is necessary to modify the stationary phase method for the evaluation of the path integral in momentmn representation. 

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