Imaginary time

In the diflfiision QMC (DMC) method [114. 119], the evolution of a trial wavefiinction (typically wavefiinctions of the Slater-Jastrow type, for example, obtained by VMC) proceeds in imaginary time, i = it, according to the time-dependent Schrodinger equation, which then becomes a drfifiision equation. All... 

Figure B3.3.11. The classical ring polymer isomorphism, forA = 2 atoms, using/ = 5 beads. The wavy lines represent quantum spring bonds between different imaginary-time representations of the same atom. The dashed lines represent real pair-potential interactions, each diminished by a factor P, between the atoms, linking corresponding imaginary times.

Computations done in imaginary time can yield an excited-state energy by a transformation of the energy decay curve. If an accurate description of the ground state is already available, an excited-state description can be obtained by forcing the wave function to be orthogonal to the ground-state wave function. 

For purely imaginary time r = it the correlator in (3.110) is expressible via the path integral... 

In accordance with the one-dimensional periodic orbit theory, any orbit contributing to g E) is supposedly constructed from closed classical orbits in the well and subbarrier imaginary-time trajectories. These two classes of trajectories are bordering on the turning points. For the present model the classical motion in the well is separable, and the harmonic approximation for classical motion is quite reasonable for more realistic potentials, if only relatively low energy levels are involved. 

T = 0, it will not necessarily hit the point Q imaginary-time equations of motion... 

As long as the system can be described by the rate constant - this rules out the localization as well as the coherent tunneling case - it can with a reasonable accuracy be considered in the imaginary-time framework. For this reason we rely on the Im F approach in the main part of this section. In a separate subsection the TLS real-time dynamics is analyzed, however on a simpler but less rigorous basis of the Heisenberg equations of motion. A systematic and exhaustive discussion of this problem may be found in the review [Leggett et al. 1987]. 

The influence functional theory, as it was formulated by Feyman and Vernon, relies on the additional assumption concerning factorization of the total (system and bath) density matrix in the past. Without this assumption the theory requires a triple path integral, with one thermal integration over the imaginary time axis [Grabert et al. 1988]. 

If we fix a realization of the path Q(x), then, when performing the path integration over q, the particle may be treated as acted on by a time-dependent potential V-miiQix), q). From traditional quantum mechanics it is clear that this integration is equivalent to the solution of the time-dependent Shrodinger equation in imaginary time. 

Consider a potential like that in fig. 19. In the vicinity of the parabolic well (t -> 0) at )S > (Oq the instanton solution (3.34), corresponding to a harmonic oscillator in imaginary time, is given by... 

A successful method to obtain dynamical information from computer simulations of quantum systems has recently been proposed by Gubernatis and coworkers [167-169]. It uses concepts from probability theory and Bayesian logic to solve the analytic continuation problem in order to obtain real-time dynamical information from imaginary-time computer simulation data. The method has become known under the name maximum entropy (MaxEnt), and has a wide range of applications in other fields apart from physics. Here we review some of the main ideas of this method and an application [175] to the model fluid described in the previous section. 

In mean field approximation we obtain for the imaginary-time correlation functions [296]... 

FIG. 8 PIMC results (symbols) of the imaginary-time correlations G r) versus imaginary time for densities p = 0.1,0.2,..., 0.7 from bottom to top the temperature is T = 1. The full line shows the results for Q r) according to the lowest-order virial expansion the dashed lines give the MF values of Q r) for the densities p = 0.7, 0.6, and 0.5 from top to bottom. (Reprinted with permission from Ref. 175, Fig. 1. 1996, American Physical Society.)... 

Besides the deviation mentioned above, the main problem with the dynamical information from the MF approximation is that it contains only one positive frequency and so the resulting real-time correlations cannot be damped or describe localizations on one side of the double well due to interference effects, as one expects for real materials. Thus we expect that the frequency distribution is not singly peaked but has a broad distribution, perhaps with several maxima instead of a single peak at an average mean field frequency. In order to study the shape of the frequency distribution we analyze the imaginary-time correlations in more detail. 

M uj) is the default model, by which additional knowledge about system properties can be incorporated. Minimum additional knowledge is equivalent to M uS) = const. Without data, 5" is maximized by A uj) = M uj). measures the deviation of the time correlation function Q computed from a proposed A via Eq. (32) from the PIMC value G at the point in imaginary time,... 

These are Green s functions diffusing in what is interpreted as an imaginary time [3, according to the Bloch equation, dp/8(3 = JYp (a diffusion-type partial differential equation). These Green s functions satisfy the equation... 

The role of finite temperature in quantum chaos is studied within the imaginary time formalism via quantum action approach (Caron et al 2001). 

Just fifty years ago, with an acclaimed paper by Matsubara (T. Matsubara, 1955), there was the emergence of a systematic approach for the quantum field theory at finite temperature (T 0), presently well-known as the imaginary time approach. Since then the development... 

Let us analyse the component G k ft)11 closely to address the connection of TFD with the imaginary time formalism. Using the fact that n(ko (3) in Eq. (23) may be written as... 

The preceding results show that the equilibrium TFD is equivalent to the Matsubara imaginary-time formalism (for a detailed discussion, see the chapter by Santana et. al. in this Proceedings). Matsubara formalism has been used also to consider spatial compactification in field theoretical models (A.P.C. Malbouisson et.al., 2002 A.P.C. Malbouisson et.al., 2002 A.P.C. Malbouisson et.al., 2004). 

Replacing t by -ir yields the imaginary-time Schrodinger equation... 

---