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arXiv:hep-th/9610209

Transition Amplitude within the Stochastic Quantization Scheme -- Perturbative Treatment

Kazuya Yuasa, Hiromichi Nakazato

Published Oct 28, 1996• 0 citations
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Abstract

The quantum mechanical transition amplitudes are calculated perturbatively on the basis of the stochastic quantization method of Parisi and Wu. It is shown that the stochastic scheme reproduces the ordinary result for the amplitude and systematically incorporates higher-order effects, even at the lowest order.
DOI
10.48550/arxiv.hep-th/9610209

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The stochastic quantization method is developed in a possible “Heisenberg” operator theory of stochastic processes so designed as to keep a formal analogy to quantum mechanics. The general theory is first formulated for stochastic processes of the Wiener type and then its application to quantization of boson fields is presented. It seems that the operator formalism is convenient to examine implications of the stochastic quantization method. Within the theoretical framework we develop the “Feynman-Dyson” approach to field propagators in perturbation theory. This approach enables us to simplify and systematize practical calculations in comparison with the Langevin equation method. In the Appendix we formulate the Green function approach to field propagators using the characteristic functional.
Transition amplitudes and probabilities for the harmonic oscillator with a forcing function proportional to cos(ωt) beginning at time zero are calculated to lowest nonvanishing order using time-dependent perturbation theory. The results are compared with the exact amplitudes and probabilities. When the exact amplitude is expanded in a Taylor series in powers of the coupling constant, the individual terms turn out to be the perturbation amplitudes, showing that the complete series of perturbation amplitudes converges to the exact amplitude.
We incorporate a solver for the fragment problem with accuracy beyond coupled cluster singles and doubles (CCSD) into the previously proposed static embedding framework, MPCC. To this end, we employ a CCSDT solver for the fragment subsystem. For the environment subsystem, we construct a perturbative estimate of the triples amplitudes, explicitly accounting for feedback from all fragment amplitudes. The resulting approach is denoted MPCCSDT(pt). We further introduce a more complete formulation in which feedback from the environment amplitudes to the fragment amplitudes is also included. This scheme involves an iterative treatment of the environment triples amplitudes and is denoted MPCCSDT(it). In addition, we assess the accuracy of the previously proposed low-level method by introducing a modified low-level approach that incorporates a lowest-order treatment of selected long-range effects, including spin fluctuations and charge polarization. All resulting approaches may be viewed as post-CCSD(T) methods. We therefore consider test cases for which CCSD(T) exhibits substantial deviations from CCSDT. Our results demonstrate that inclusion of triples amplitudes at the fragment level alone is insufficient; a perturbative treatment of the environment triples amplitudes is required. For many energy-difference applications, feedback from the environment triples amplitudes to the fragment amplitudes, is not essential, but it does play a role in the very challenging molecules. A very interesting finding from our study is that in some challenging cases, we need an improved (second-order) perturbative method for the SD amplitudes, going beyond the first-order one used in our earlier work.