Review of Feynman’s Path Integral in Quantum Statistics:from the Molecular Schrödinger Equation to Kleinert’s Variational Perturbation Theory

Feynman’s path integral reformulates the quantum Schrödinger differential equation to be an integral equation.It has been being widely used to compute internuclear quantum-statistical effects on many-body molecular systems.In this Review,the molecular Schrödinger equation will first be introduced,together with the BornOppenheimer approximation that decouples electronic and internuclear motions.Some effective semiclassical potentials,e.g.,centroid potential,which are all formulated in terms of Feynman’s path integral,will be discussed and compared.These semiclassical potentials can be used to directly calculate the quantum canonical partition function without individual Schrödinger’s energy eigenvalues.As a result,path integrations are conventionally performed with Monte Carlo and molecular dynamics sampling techniques.To complement these techniques,we will examine how Kleinert’s variational perturbation(KP)theory can provide a complete theoretical foundation for developing non-sampling/non-stochastic methods to systematically calculate centroid potential.To enable the powerful KP theory to be practical for many-body molecular systems,we have proposed a new path-integral method:automated integrationfree path-integral(AIF-PI)method.Due to the integration-free and computationally inexpensive characteristics of our AIF-PI method,we have used it to perform ab initio path-integral calculations of kinetic isotope effects on proton-transfer and RNA-related phosphoryl-transfer chemical reactions.The computational procedure of using our AIF-PI method,along with the features of our new centroid path-integral theory at the minimum of the absolute-zero energy(AMAZE),are also highlighted in this review.

How to Explicitly Calculate Feynman and Wheeler Propagators in the ADS/CFT Correspondence

We discuss, giving all necessary details, the boundary-bulk propagators. We do it for a scalar field, with and without mass, for both the Feynman and the Wheeler cases. Contrary to standard procedure, we do not need here to appeal to any unfounded conjecture (as done by other authors). Emphasize that we do not try to modify standard ADS/CFT procedures, but use them to evaluate the corresponding Feynman and Wheeler propagators. Our present calculations are original in the sense of being the first ones undertaken explicitly using distributions theory (DT). They are carried out in two instances: 1) when the boundary is a Euclidean space and 2) when it is of Minkowskian nature. In this last case we compute also three propagators: Feynman’s, Anti-Feynman’s, and Wheeler’s (half advanced plus half retarded). For an operator corresponding to a scalar field we explicitly obtain, for the first time ever, the two points’ correlations functions in the three instances above mentioned. To repeat, it is not our intention here to improve on ADS/CFT theory but only to employ it for evaluating the corresponding Wheeler’s propagators.