Feynman path-integral representation

Our simulations are based on well-established mixed quantum-classical methods in which the electron is described by a fully quantum-statistical mechanical approach whereas the solvent degrees of freedom are treated classically. Details of the method are described elsewhere [27,28], The extent of the electron localization in different supercritical environments can be conveniently probed by analyzing the behavior of the correlation length R(fih/2) of the electron, represented as polymer of pseudoparticles in the Feynman path integral representation of quantum mechanics. Using the simulation trajectories, R is computed from the mean squared displacement along the polymer path, R2(t - t ) = ( r(f) - r(t )l2), where r(t) represents the electron position at imaginary time t and 1/(3 is Boltzmann constant times the temperature. 

The resultant solution is the Feynman path-integral representation of the one electron Green s function (Feynman (1965) Gelfand (I960)). 

Polarization fluctuations of a certain type were considered in the configuration model presented above. In principle, fluctuations of a more complicated form may be considered in the same way. A more general approach was suggested in Refs. 23 and 24, where Eq. (16) for the transition probability has been written in a mixed representation using the Feynman path integrals for the nuclear subsystem and the functional integrals over the electron wave functions of the initial and final states t) and t) for the electron ... 

Yet, the Feynman-Kleinert partition function is to be unfolded within the actual periodic path integral representation, with the help of Eqs. (2.28) and (2.50)... 

Equilibrium properties can be determined from the partition function Zq and this quantity can, in turn, be computed using Feynman s path integral approach to quantum mechanics in imaginary time [86]. In this representation of quantum mechanics, quantum particles are mapped onto closed paths r(f) in imaginary time f, 0 f )8ft. The path integral expression for the canonical partition function of a quantum particle is given by the P 00 limit of the quantum path discretized into P segments. 

By comparing analogous terms in ( , x) and Q, we see that we can think of the partition function as a path integral over periodic orbits that recur in a complex time interval equal to i s flh/i = — ifih. There is no claim here that the closed paths used to generate Q correspond to actual quantum dynamics, but simply that there is an isomorphism. We therefore can refer to the equation above as the discretized path-integral (DPI) representation of the partition function. Using Feynman s notation, we have in the infinite-P limit... 

---